WITH YOU WHEN YOU FLY:

Aeronautics for 

Introductory Physics

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
2 Aeronautics for Introductory Physics: Table of Contents

Table of Contents

Overview 4

Physics Aeronautics: A Contextual Approach for Learning Physics 4

About NASA Aeronautics 5

About the AAPT 5

Standards and Best Practices 6

Standards 6

 Next Generation Science Standards 6

 Common Core State Standards 6

Theoretical Foundations 7

 Building Conceptual Models of Flight Physics 7

 Modeling Method of Instruction in Physics 8

 Levels of Inquiry 9

 Learning Cycle 10

Scientific Thinking 12

Inquiry Lesson/Lab: Scaling and Paper Airplanes 13

Literary/Data Analysis: Wings According to Size 20

 Successful Flight Test of Shape-Changing Wing Surface 29

Constant Velocity 30

Real-World Applications: Flying Speed 31

 A Day in the Life of Air Traffic Control 34

 Problem Set: Comparative Velocities 37

 Problem Set: Vectors in Relative Velocities Problems 39

Real-World Applications: Estimating the Speed and Distance of an Airplane 43

Real-World Applications: Noise Doppler-Shift Measurement of Airplane Speed 52

 New Acoustics Techniques Clear Path for Quieter Aviation 53

Real-World Applications: Smart Skies 54

 Fly by Math 54

 Line Up With Math 55

Sector 33 55

Air Traffic Operations Lab Answering Big Questions About the Future of Air Travel 58

NASA Tool Helps Airliners Minimize Weather Delays 58

Interactive Demonstration: Pitot-Static Tube 59

Inquiry Lesson/Lab: Bernoulli Effect Lab 65

 Problem Set: Bernoulli’s Equation and Pitot-Static Tubes 73

Discovery Lab: Airplane Dynamics: Engine Thrust, Braking, and Lift 76

Uniform Acceleration 78

Literary/Data Analysis: Prepare for Landing 79

 Shhhh! Keep it Down Please! 84

Application Lesson/Lab: Mobile Accelerometers 85

 NASA-Pioneered Automatic Ground-Collision Avoidance System Operational 90

Terminal Velocity 91

Discovery Lab (Raw): Drag Stations Lab 92

Discovery Lab (Contextual): Drag and Aircraft Design 99

 NASA Researchers to Flying Insects: “Bug Off!” 107

Inquiry Lesson/Lab: Viscosity Tubes 108

 NASA Turns World Cup Into Lesson in Aerodynamics 116

Inquiry Lesson/Lab: The Physics of Soaring (and L/D ratios) 117

 Problem Set: Glider Trajectory Problems 125
Aeronautics for Introductory Physics: Table of Contents 3

Literary/Data Analysis: Flying with Finesse 127

Model Building: Developing a Model for Drag 132

Force Particle – Inertia 137

Inquiry Lesson/Lab: Falling and Air Resistance (Qualitative) 138

 Skyray 48 Takes Flight 143

Real-World Applications: Hanging an Airplane: A Case Study in Static Equilibrium 144

NASA Model Flies at Air and Space 150

Problem Set: Trimmed Aircraft Activity 151

Inquiry Lesson/Lab: Investigating Flight with a Toy Helicopter 157

 Future Helicopters Get SMART 162

Interactive Demonstration: Lift Demonstrations 163

Inquiry Lesson/Lab: Measuring Lift with the Wright Airfoils 165

Glow With the Flow 175

Problem Set: Lift-Velocity Relationship Problems 176

Problem Set: Lift Equation Problems 178

Model Building: Developing a Model for Lift 180

Inquiry Lesson/Lab: Rubber-Band-Driven Airplane Contest 190

Recalling a Record: X-43A Scramjet Sets New Hypersonic Record a Decade Ago 194

Real-World Applications: Helicopter Toy and Lift Estimation 195

Constant Net Force 200

Real-World Applications: Parachute Lab 201

Inquiry Lesson/Lab: Falling and Air Resistance (Quantitative) 206

Hitchhiking Sensors Capture Curiosity’s Entry 212

Problem Set: Simplified Aircraft Motion 213

Problem Set: Forces in a Climb 216

Interactive Demonstration: Demonstration of Newton’s Third Law Using a Balloon Helicopter 219

2-D Combination 221

Real-World Applications: Measuring the Flight Speeds of Fire Bombers from Photos: An In-Class 

Exercise in Introductory Kinematics 222

NASA Tests Radio for Unmanned Aircraft Operations 224

 Problem Set: Aircraft Motion Activity 225

Energy 227

Literary/Data Analysis: A Hard Day’s Flight 228

Literary/Data Analysis: In Wind and Weather 232

NASA Helicopter Drop Test a Smashing Success 235

Rotational Motion 236

Real-World Applications: Circular Motion Studies with a Toy Airplane/The Airplane Experiment 237

Real-World Applications: Measuring the Moment of Inertia of an Airplane 241

 Video: Flight Testing Newton’s Laws – Weight and Balance 242

Additional Resources 243

Literary/Data Analysis Activity: The 747 and its Competitors 244

NASA Resources 247

AAPT Resources 248

 American Association of Physics Teachers 248

 The Physics Teacher magazine 248

Additional Resources 249

Literature Resources 249

Leybold Lessons (and Laboratory Equipment) 249

Other Aeronautics Education Resources 249
4 Aeronautics for Introductory Physics: Overview

Overview

Physics Aeronautics: A Contextual Approach for Learning Physics

There are many right ways to teach physics. Teaching by inquiry in context is 

perhaps the most effective way to effectively teach physics so that it becomes both 

relevant and practical. This educators’ guide addresses a growing need for higherlevel inquiry in the physics classroom, provides a solid content base, and actively 

engages with topics necessary to the development of 21st century skills. 

Context is provided from student experiences with flight as well as ongoing 

aeronautics research.

This document is the result of a Space Act Agreement between NASA and the 

American Association of Physics Teachers. This project is founded on teaching 

ideas presented by seasoned high school and college physics teachers and articles 

that have appeared in the American Association of Physics Teachers’ (AAPT) 

publication, The Physics Teacher. As such, all of these ideas are appropriate 

for high school or introductory university level physics courses, and have been 

reviewed by acting high school and college/university AAPT members. Although 

it is not necessary for lesson and lab implementation to read the published articles 

associated with the activities in this document, all of the articles are freely available 

to AAPT members through the AAPT Publications website and directly from each 

link at the top of each activity.

In addition, practical relevance is interwoven through the guide with applications 

from current, cutting-edge aeronautics research being undertaken by the National 

Aeronautics and Space Administration’s Aeronautics Research Mission Directorate 

(NASA ARMD). Each of the activities in this document is aligned with at least one 

of NASA Aeronautics’ research themes, including green aviation, reducing flight 

delays, revisiting supersonic flight, and designing future aircraft. Inquiry lessons 

presented in this document help students develop concepts, derive fundamental equations, practice reading and data analysis 

skills, and relate their laboratory work to real-world applications in NASA Aeronautics research.

While this document presents itself as an educator’s 

activity guide filled with resources for demonstrations, 

lessons, and labs, instructional approach is also very 

important. Many of these activities are structured as 

paradigm labs that could easily serve as a curricular 

supplement to the Modeling Method of Instruction 

approach. Many of these activities can also be easily 

applied or adapted to align with the Next Generation 

Science Standards (NGSS) and the Common Core 

State Standards (CCSS) for Reading and Writing in 

Technical Subjects. Each activity includes objectives 

and/or guiding questions related to the physics of 

flight, followed by a very detailed listing of the NGSS 

and CCSS addressed by students who successfully 

complete each activity.
Aeronautics for Introductory Physics: Overview 5

About NASA Aeronautics 

NASA Aeronautics works to solve the many challenges 

that still exist in our nation’s air transportation system: air 

traffic congestion, safety, efficiency, and environmental 

impacts. Thanks to advancements in aeronautics developed 

by NASA, today’s aviation industry is better equipped than 

ever to safely and efficiently transport passengers and cargo 

to their destinations. In fact, every U.S. aircraft flying 

today and every U.S. air traffic control tower uses NASAdeveloped technology in some way.

Streamlined aircraft bodies, quieter jet engines, techniques 

for preventing icing, drag-reducing winglets, lightweight 

composite structures, software tools to improve the flow 

of tens of thousands of aircraft through the sky, and so 

much more are an everyday part of flying thanks to NASA 

research that traces its origins back to the earliest days 

of aviation.

We are committed to transforming aviation by dramatically 

reducing its environmental impact, improving efficiency 

while maintaining safety in more crowded skies, and paving 

the way to revolutionary aircraft shapes and propulsion.

Inside cockpits, cabins and jet engines; atop traffic control 

towers; and from departure gate to arrival terminal at 

airports everywhere, the DNA of the entire aviation 

industry is infused with technology that has its roots in 

NASA research.

NASA is with you when you fly.

About the AAPT

The American Association of Physics Teachers was 

established in 1930 with the fundamental goal of ensuring 

the “dissemination of knowledge of physics, particularly 

by way of teaching.” Today, that vision is supported 

by members around the world. The AAPT is a strong 

professional physics science society dedicated to the pursuit 

of excellence in physical science education.

National meetings are held each winter and summer, and are 

opportunities for members, colleagues, and future physicists 

from around the world to:

• Participate in physics workshops

• Meet and greet other physics educators

• Form networks nationally and locally

• Engage exhibitors and learn about the latest physics 

resources

• Discuss innovations in teaching methods

• Share the results of research about teaching and learning.

The AAPT also hosts or supports smaller workshops and 

conferences and symposia throughout the year to provide 

further opportunities for professional development and 

knowledge sharing. The AAPT also supports physics 

teachers through peer-reviewed journals, teacher resources, 

competitions and contests, and awards and grants.

Learn more about NASA Aeronautics at:

http://www.nasa.gov/aeronautics

Learn more about the AAPT at:

http://www.aapt.org/
6 Aeronautics for Introductory Physics: Standards and Best Practices

Standards and Best Practices

Standards

Next Generation Science Standards

Many of the activities in this guide provide an opportunity to teach about standard physics Core Disciplinary Ideas (CDI) in 

mechanics and beyond. The specific referenced standards are listed below. Further, most Cross-Cutting Concepts (CCC) and 

many Science and Engineering Practices (SEP) can be explicitly addressed through these activities. 

Core Disciplinary Ideas

HS.PS2 - Forces and Interactions: http://goo.gl/Z8KDSG

HS.PS3 - Energy: http://goo.gl/TeiZH5

Science and Engineering Practices

Appendix F: http://goo.gl/ucX7iY

1. Asking questions (for science) and defining problems (for engineering) 

2. Developing and using models 

3. Planning and carrying out investigations 

4. Analyzing and interpreting data

5. Using mathematics and computational thinking 

6. Constructing explanations (for science) and designing solutions (for engineering) 

7. Engaging in argument from evidence 

8. Obtaining, evaluating, and communicating information 

Cross Cutting Concepts

Appendix G: http://goo.gl/WhEEjr

1. Patterns

2. Cause and effect

3. Scale, proportion, and quantity

4. Systems and system models

5. Energy and matter

6. Structure and function

7. Stability and change

Common Core State Standards 

English Language Arts Reading in Science and Technical Subjects

This educators’ guide provides a number of opportunities for reading technical work. Students are encouraged to use The 

Simple Science of Flight, by Tennekes, for the guided literary and data analysis passages. Further, most activities have a 

“NASA Connection” that is directly linked to a brief article about current NASA Aeronautics Research.

Grades 9-10: http://goo.gl/n3KObX

Grades 11-12: http://goo.gl/gFceem

English Language Arts Writing in Science and Technical Subjects

By extending a number of the laboratory activities, students can be asked to defend their results in writing through formal 

reports.

Grades 9-10: http://goo.gl/SUqw8Y

Grades 11-12: http://goo.gl/PPg5rM

*Registered trademark of Achieve. Neither Achieve 

nor the lead states and partners that developed the 

NGSS was involved in the product of, and does not 

endorse, this product.
Aeronautics for Introductory Physics: Standards and Best Practices 7

Theoretical Foundations

Building Conceptual Models of Flight Physics

Aeronautics for Introductory Physics’ approach to contextual physics teaching and learning does not rely upon building 

mental models that are unique to flight – rather, it is founded on the ideas of traditional physics instruction, paralleling the 

structures used in the Modeling Method of Instruction in Physics. For all of the inquiry labs in this document, students are 

expected to build graphical and mathematical models to describe relationships between variables in the system.

Within mechanics, there is ample room for students to build mental sub-models to describe flight because of the chaotic 

nature of microscopic fluid particles. Some of these mental models can be built around the following concepts. 

Models of Drag (as opposed to simple surface friction) – Teacher Resource

Models of Lift – Teacher Resource

Although students without a solid background in chemistry might not be able to build these models on their own, teachers can 

offer supports. The following models are not discussed in depth within this document, but teachers and students can find very 

helpful supporting information at NASA’s Beginner’s Guide to Aeronautics.

Both of the above models can be highly complex, and require some understanding of molecular kinetic theory (viscosity, 

intermolecular forces, chaos, etc.), as well as Newton’s Third Law. In essence, students can work their way to understanding 

that no matter the dominating mental model (simple or complex), it is the mechanical conservation laws (mass, energy, 

momentum) that fully describe flight.

Additionally, aeronautics offers the opportunity for students to see that a single model produced for an object moving through 

a fluid might be very specific – unlike many of the “universal” mental models typically developed in the physics classroom.
8 Aeronautics for Introductory Physics: Standards and Best Practices

Modeling Method of Instruction in Physics

For those familiar with the Modeling Method of Instruction...

Many of the labs presented in this document are an opportunity to have students model relationships mathematically and 

graphically, and to explain those relationships by building their own conceptual models for the interaction of objects 

flowing through a fluid. A number of the labs could be used as paradigm labs for initial development of conceptual models. 

Additionally, the nature of aerodynamics introduces students to the complexities of modeling associated with dimensionless 

constants. Further, this study of flight allows students to realize that models might not be universally applicable, and that 

relationships that cannot be easily represented by a single simple equation. For example, coefficients of lift and drag are often 

measured experimentally for a single aircraft in a very specific set of environmental conditions – modifications in the system 

might not result in a predictable change.

For those not familiar with the Modeling Method of Instruction... 

The National Science Foundation funded Arizona State University (ASU) Modeling Instruction Program for 16 years, from 

1990 to 2005. The American Modeling Teachers Association was founded at the expiration of NSF funding in 2005, and over 

the past 10 years has reached about 5000 more teachers with Modeling workshops. At present a little over 7500 teachers have 

attended one or more Modeling workshops since 1990. In 2015, about 1000 more teachers will attend workshops. AMTA 

coordinates 60 or more Modeling workshops each summer in physics, chemistry, biology and middle school science (in 

2013 and 2014 there were ~80 workshops each summer). At current capacity, AMTA is on track to serve 1000 teachers every 

summer with Modeling workshops across the disciplines (physics, chemistry, biology, and physical science), and is working to 

build capacity as demand for Middle School Modeling workshops exceeds the capacity to provide trained workshop leaders.

From: http://modeling.asu.edu/modeling/WhatModInstructionIs-2014.htm

Modeling instruction was developed at ASU over two decades to integrate insights from physics education research with 

classroom experience of exceptional in-service teachers. In a series of intensive three-week workshops over two years, 

teachers improve their physics, chemistry, or physical science content knowledge and are equipped with a robust teaching 

methodology for developing student abilities to:

• Make sense of physical experience,Understand scientific claims,

• Articulate coherent opinions of their own and defend them with cogent arguments,

• Evaluate evidence in support of justified belief.

 More specifically, teachers learn to:

• Ground their teaching in a well-defined pedagogical framework (Modeling Theory), rather than following rules of 

thumb;

• Organize course content around scientific models as coherent units of structured knowledge;

• Engage students collaboratively in making and using models to describe, to explain, to predict, to design and control 

physical phenomena;

• Involve students in using computers as scientific tools for collecting, organizing, analyzing, visualizing, and 

modeling real data;

• Assess student understanding in more meaningful ways and experiment with more authentic means of assessment;

• Continuously improve and update instruction with new software, curriculum materials and insights from 

educational research;

• Work collaboratively in action research teams to mutually improve their teaching practice.

For more information on the Modeling Method of Instruction, please see Modeling Instruction: An Effective Model of 

Science Education, by Drs. Jane Jackson, Larry Dukerich, and David Hestenes. 

http://www.nsela.org/images/stories/scienceeducator/17article7.pdf

Learn more about the AMTA at:

http://modelinginstruction.org
Aeronautics for Introductory Physics: Standards and Best Practices 9

Levels of Inquiry

Each activity in this document is also classified by its level of inquiry; 

the complexity of the skill expected of students in order to investigate 

the science. Historically, Herron’s levels of inquiry (0) Confirmation/

Verification, (1) Structured, (2) Guided, and (3) Open focus exclusively 

on three parts of an inquiry process: problem, procedure, and 

solution. However, little attention is given in that model to understand 

the varying levels of complexity of the problem to be studied, the 

procedure to be undertaken, or the solution to be comprehended. The 

Levels of Inquiry Model by Dr. Carl J. Wenning of Illinois State 

University is used throughout this document. These levels are also 

supplemented by Literary/Data Analysis and Model Building, both 

which can occur during or external to inquiry activities.

Discovery 

Learning

Interactive 

Demonstration

Inquiry Lesson Inquiry Lab Real-World ApplicationsHypothetical 

Explanations

Low Intellectual Sophistication High

Teacher Locus of Control Student

Discovery Learning – Students experience a variety of phenomena, are asked to identify simple trends, relationships, or 

similarities/differences among them.

Interactive Demonstration – The teacher provides a guided demonstration to identify preconceptions, cause cognitive 

dissonance, or help students identify causation.

Inquiry Lesson/Lab – The teacher guides students in an inquiry-based discussion (lesson) or allows students to work on 

their own (lab) to determine a quantitative relationship between variables. Because an experience in the laboratory can be 

easily modified to be more teacher- or student-directed, these activities are grouped together in this document.

Real-World Application – Students use a known relationship to apply conceptual understanding to a new situation either 

through a paper-pencil or discussion lesson, or through a hands-on lab.

Hypothetical Explanations – Students generate, develop, and test hypotheses. The focus during these activities is on the 

cause and explanation behind phenomena – not on describing them. (Note: Although predictions are a key part in developing 

working explanations, hypotheses are the focus here. Hypotheses have explanatory power. Predictions are simply anticipatory 

statements about what will happen to something under a given set of circumstances.) Although some activities in this 

document can be used to create hypothetical explanations, no activities in this document bear this level of complexity.

Literary and/or Data Analysis Activities – These activities might not directly involve any scientific inquiry, but allow 

students to apply what they have learned to examine technical passages.

Model Building - Students and teachers work together to elicit, confront, and resolve misconceptions while building up 

an underlying conceptual foundation by organizing course content around coherent units of structure knowledge (see the 

Modeling Method of Instruction, above). In its full form, model building occurs at every stage of scientific inquiry, and 

is explicitly addressed when students are asked to synthesize what they have learned based upon evidence from their 

investigations.

For more information on the levels of inquiry, see: Levels of Inquiry: Hierarchies of Pedagogical Practices and Inquiry 

Processes by Dr. Carl J. Wenning. http://www.dlsu.edu.ph/offices/asist/documents/Levels_of_Inquiry.pdf
10 Aeronautics for Introductory Physics: Standards and Best Practices

Learning Cycle

Within each of the levels of inquiry, teachers can effectively employ 

a self-regulated learning cycle to structure their individual lessons 

and support student interaction. There are many learning cycles 

that have been identified throughout educational history, including 

experiential learning cycles presented by John Dewey, Hans Aebli 

and Eric Mazur, among others. However, attention in the learning 

cycle often fails to take into consideration the collaborative nature of 

student learning, self-regulation in the classroom, and the importance 

of informed decision-making. In this document, each of the inquiry 

and application lessons and labs in this document are broken into 

key parts based upon the Self-Regulated Learning Cycle / Cycle of 

Action proposed by Dr. André Bresges of the University of Cologne, 

Germany.

1. Aim: 

In this phase all participants of the action should get a vision of what they want to reach. Criteria for a 

successful fulfillment of the task should be formulated to clarify the end goal.

2. Plan: 

In this phase all necessary information is collected that helps to reach the →aim, including the search for 

already existing solutions. While structuring the gathered information, ideas for different solution paths 

should be gathered.

3. Decide: 

Now a consensus should be reached among all participants of the action. One of the solution paths 

developed and gathered in →planning must be chosen. (Alternative solution paths may be kept in mind 

if a new decision may become necessary.) Important: Latest in this phase, quantitative/measurable criteria 

should be noted for measuring the success of the cycle of action (CoA).

4. Act: 

All conscious actions should be performed according to the →decision. (If unplanned problems occur, 

another CoA can be started, but not before finishing and documenting the ongoing CoA).

5. Verify: 

The Verification is performed with the criteria created during phase 1 and 3. (→aim and →decision). 

Important: A clear separation of →verification and →evaluation helps to promote scientific thinking. 

Otherwise, students often have great difficulties distinguishing between results and conclusions in their 

lab reports.

6. Evaluate: 

The participants of the action estimate how successful the cycle of action was and may look for 

improvements for another CoA. This is the highlight of the learning process. With the results from the first 

CoA the →aim for further CoAs can be formulated much better.

For more information about the Self-Regulated Learning Cycle, please see: 

http://www.physikdidaktik.uni-koeln.de/11287.html?&L=1
Aeronautics for Introductory Physics: Standards and Best Practices 11

Aeronautical Dictionary

The following are a number of terms that, without explicit definition, tend to prove problematic to students. These terms are 

not necessarily meant to be learned by students at the outset, but this dictionary should serve as a reference point as their use 

becomes necessary. The following definitions have been drawn and modified from the NASA Aeronautical Dictionary, by 

Frank Davis Adams, (1959).

Aeronautics, noun: The science and art of design, construction, and operation of aircraft.

Angle of Attack, noun: (1) The angle between the direction of the flow and the airfoil’s chord line (from the leading edge of 

the air foil to the tailing edge). (2) A zero degree angle of attack is defined as the angle at which a body generates zero lift in 

a flow. The angle of attack is the angle between the zero lift axis and the direction of the air flow. 

Drag, noun: (1) A retarding force acting upon a body in opposite direction to the motion of the body. (2) A force acting upon 

a body in direction of a flowing fluid (gas or liquid).

*Although overall drag occurs opposite the direction of the total motion of the aircraft, the actual direction for a 

given surface will depend upon the movement of the various surfaces of the aircraft.

Lift, noun: The component of the total aerodynamic force acting on a body perpendicular to the direction of the undisturbed 

airflow relative to the body. This lift, sometimes called “aerodynamic lift,” acts on any body or system of bodies such as an 

airfoil, a fuselage, an airplane, an airship, a rotor, etc., at a suitable angle of attack in the airflow. 

*Lift is usually though of as a force acting in an upward direction, giving sustentation to the aircraft. By definition, 

however, lift can, and does, act in any direction: downward, as with a horizontal tail when required for longitudinal 

trim; sideward, as with a vertical tail when an aircraft is turned. When lift in a direction other than upwards is under 

discussion, it may be specified in the expressions “negative (downward) lift,” or “horizontal lift,” although the latter 

expression is usually avoided.

This phenomenon of aerodynamic lift is typically explained by either or both of two laws or theorems: (1) By 

Newton’s third law of motion, in which a body, causing air to flow in one direction, obtains a force upon itself acting 

in the other direction. (2) By Bernoulli’s law, in which an increase of air velocity over the body gives a pressure 

decrease resulting in lift. However, using Bernoulli’s law as the sole explanation for lift is not fully true for an 

open system, and can lead to misconceptions. Lift can also be a result of buoyant force, although this is generally 

negligible for heavier-than-air aircraft. 

Thrust, noun: The forward-directed pushing or pulling force developed by an aircraft engine or a rocket engine.

*Although thrust generally pushes an aircraft forward, a more accurate definition is to say that it acts parallel to the 

direction of the engine. Some aircraft, such as vertical-lift aircraft, can apply a thrust in an upward direction. Other 

airplanes with variable-angle engines might apply thrust in such a way that only a component actually causes the 

airplane to move forward.

Weight, noun: The gravitational force acting on an object by the earth. An object with weight experiences a force that is 

directed downward toward the center of mass of the earth. 

*In all cases with respect to weight, “downward” is in reference to the center of the earth.
12 Aeronautics for Introductory Physics: Scientific Thinking

Scientific Thinking

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
Aeronautics for Introductory Physics: Scientific Thinking 13

Inquiry Lesson/Lab: Scaling and Paper Airplanes

Robert H. Johns. The Physics Teacher, 9, 541 (1971).

Paper airplanes are a classic children’s toy, easy to make, and easy to use for inquiry 

investigations. In this activity, students construct paper airplanes. Technically, paper airplanes 

are actually “gliders,” as they do not provide themselves any thrust throughout their flight. 

Students are asked to compare the wing’s top surface area with the weight of scaled gliders 

of various sizes.

This lesson can be an opportunity for students 

to perform a controlled experiment, and to 

begin learning about related variables that might 

influence flight, such as drag and shape. The 

end purpose of this activity is much less to 

derive a specific equation, but, rather, to learn to 

identify and describe meaningful relationships 

between variables. This activity also provides a 

broad overview for many foundational concepts 

associated with flight.

Objective: Determine the graphical and mathematical relationship between the surface 

area and the weight of scaled gliders (“paper airplanes”). 

Objectives 

Determine the relationship 

between the surface area and 

weight of scaled gliders.

Materials 

Paper of various thicknesses 

Scissors 

Electronic balance (optional)

NGSS 

SEP 2, 4, 7

CCC 1, 3

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Scientific Thinking

ACTIVITY

Scaling and Paper 

Airplanes
14 Aeronautics for Introductory Physics: Scientific Thinking

Ask students to build a number of scaled, but otherwise identical, gliders. 

Using some simple geometry, students can determine the approximate surface 

area of the wings, and plot this against the measured or calculated weight of the 

gliders. (Hint: Most packaged paper lists the mass in g/m2

. Students can 

determine weight from proportional reasoning and known starting surface areas 

of their papers before folding the airplanes). The graph to the right displays an 

example best-fit line representative of the data produced from this lab.

• Encourage students to design an airplane for which the surface area 

of the wings can be easily estimated. Wings composed of right triangles and rectangles are best.

• Ensure that the students are only measuring the surface area for the 

“top” of the wings.

• Ensure that the airplane design actually results in flight that produces “lift.” It should be a good glider.

• Ensure that students use papers with the same starting ratios. A way to simplify this is to ask students to always 

begin with square – not rectangular – pieces of paper.

• Students should find that this relationship between weight and surface 

area is approximately linear. However, it is possible that the smaller 

airplanes do not quite fit on the straight line, as their relative size in 

comparison to the paper used to actually make the folds results in a 

smaller surface area.

Comparing the gliding ability of gliders can be difficult to do in a consistent 

way, because students vary the amount of force with which they push the glider, 

and the release angle and height can change dramatically from throw to throw.

To launch gliders consistently, build a launcher from a piece of scrap wood, two 

nails, and a rubber band. A paperclip can be easily attached to any plan as a 

“hook,” and the glider can be launched consistently by pulling back to the same 

stretch point each time. Force can be varied by changing the type of rubber band 

or pulling back more or less before launch. Alternatively, commercial paper 

airplane launchers can be purchased from multiple science education companies.

Lastly, consider the implications that weight and surface area of gliders might 

have for the anatomy of birds and the design of airplanes. The next activity 

allows students to investigate these relationships in much more detail.
Aeronautics for Introductory Physics: Scientific Thinking 15

Teacher Answer Key: Scaling and Paper Airplanes

 Objective: Determine the graphical and mathematical relationship between the weight and the 

surface area of scaled gliders (“paper airplanes”). 

Early engineers found that building vehicles for powered flight 

was a difficult task. Before building propelled airplanes, they first 

attempted to make gliders that would remain in the air for a while 

after being launched. Leonardo da Vinci is well known for his 

visualizations of an “ornithopter.” Perhaps less well known, the 

Englishman George Cayley developed a variety of successful 

gliders in the 1800s, followed by the German Otto Lilienthal.

One of the great challenges of the times was to develop a glider that 

had a very small weight-to-surface area ratio. In fact, some of the 

best gliders actually have the same weight-to-surface area ratios as 

the best gliding birds. (Hence, the Wright brothers’ experience with 

lightweight materials for bicycle design naturally gave them some 

advantage in their design for gliders and early airplanes.)

In this lab activity, you will determine the relationship between weight and surface area of the wings of a 

glider. Plan how you will collect the data, and what kinds of materials you will use that are available to you. 

Begin by folding a piece of paper into a successful glider. 

Once you have found a glider that glides well, decide what data you need to collect. Before you fold your 

gliders, ensure that the pieces of paper you use have proportional dimensions: 

Dimension 

Ratios

Glider #1 Glider #2 Glider #3 Glider #4 Glider #5 Glider #6

Length

Width
16 Aeronautics for Introductory Physics: Scientific Thinking

Estimate the surface area of the wings by identifying the simple geometric shapes that make up the wings. 

Draw a top-view of your paper airplane, identifying the shapes. Describe – in words – how you calculate 

the total surface area of the tops of the wings each time.

Description of calculation area:

Drawing:

Collect data for total weight and surface area for each of your gliders. Record your data on your own.

Plot your data on a graph, as shown to the lower right, and answer the following questions:

1. What is the general equation that describes this graph?

The standard equation for a linear graph is y = mx +b. Substituting the variable names, and slope

and intercept (with units), this results in a general equation of “Weight = (____ N/m2) Surface

Area + ____ N.”

2. What are the meanings of the slope(s) and intercepts on this graph?

The slope is the ratio of force per unit area. For every increase of unit area, the weight increases by a certain amount.

The intercept should be zero, because a glider with no surface

area has no weight! If there is a non-zero value, this is likely

the result of error (lack of precision or accuracy in measurements), or is because the relationship is not entirely linear.

3. Conduct multiple experiments to compare the surface area/

weight ratios for various materials

More dense materials should have a greater slope value. The y-intercept should still be zero.

4. Conduct multiple experiments to compare the surface area/weight ratios for various glider styles.

If the same material is used, the slopes should be greater for glider styles that have smaller wings.

5. Compare the surface area/weight ratio to qualitative gliding ability (i.e. the speed required for the

glider to maintain lift, and/or how far the gliders go when launched in a consistent way).

Generally, gliders with small wings will have to have a higher speed in order to launch and glide

properly.

Look at your group’s data – how reliable is it? Share the data you collected with your classmates. Do you 

results compare in a similar way to your classmates? If not, why not?
Aeronautics for Introductory Physics: Scientific Thinking 17

Develop a visual, verbal, or physical model to explain why you think that surface area affects the gliding 

ability of a glider.

Considering only Newton’s laws, a greater surface area results in more collisions with air particles and a 

greater deflection of both air particles that hit the wings of the glider, as well as the nearby air particles 

that also get pushed downward by other particles surrounding them. As the air particles hit the wing, they 

reflect off the wing. The air particles, that initially had a horizontal momentum, are now pushed forward 

(with respect to the direction of the glider) and downward. As a result, the airplane is pushed backward 

(drag), but also provided with an upward push (lift). More collisions from a greater surface area result 

in more total force applied to the wings (and a greater change in momentum on the airplane). However, 

greater wing surface area typically results in greater mass of the glider overall, which means that the glider 

is actually less likely to accelerate. Overall, this relationship can be very difficult for students to observe. 

The key is to help students to think about air particles and collisions.

Hypothesize why some gliders fly better than others. 

Many factors come into play here: center of gravity (which impacts the angle of attack), glide speed (the 

speed of the air over/under the wings in order for the glider to glide most efficiently), mass, angle of wings 

(with respect to the horizontal plane of the wings from side-to-side), etc. 
18 Aeronautics for Introductory Physics: Scientific Thinking

Student Worksheet: Scaling and Paper Airplanes

Objective: Determine the graphical and mathematical relationship between the weight and the 

surface area of scaled gliders (“paper airplanes”). 

Early engineers found that building vehicles for powered flight was 

a difficult task. Before building propelled airplanes, they first 

attempted to make gliders that would remain in the air for a while 

after being launched. Leonardo da Vinci is well known for his 

visualizations of an “ornithopter.” Perhaps less well known, the 

Englishman George Cayley developed a variety of successful 

gliders in the 1800s, followed by the German Otto Lillenthal.

One of the great challenges of the times was to develop a glider that 

had a very small weight-to-surface area ratio. In fact, some of the 

best gliders actually have the same weight-to-surface area ratios as 

the best gliding birds. (Hence, the Wright brothers’ experience with 

lightweight materials for bicycle design naturally gave them some 

advantage in their design for gliders and early airplanes.)

In this lab activity, you will determine the relationship between 

weight and surface area of the wings of a glider. Plan how you will 

collect the data, and what kinds of materials you will use that are 

available to you. Begin by folding a piece of paper into a 

successful glider. 

Once you have found a glider that glides well, decide what data you 

need to collect. Before you fold your gliders, ensure that the pieces 

of paper you use have proportional dimensions: 

Dimension 

Ratios

Glider #1 Glider #2 Glider #3 Glider #4 Glider #5 Glider #6

Length

Width

Estimate the surface area of the wings by identifying the simple geometric shapes that make up the wings. 

Draw a top-view of your paper airplane, identifying the shapes. Describe – in words – how you calculate 

the total surface area of the tops of the wings each time.

Description of calculation of area:

Drawing:
Aeronautics for Introductory Physics: Scientific Thinking 19

Collect data for total weight and surface area for each of your gliders. Record your data on your own.

Plot your data on a graph, as shown to the lower right, and answer the following questions:

1. What is the general equation that describes 

this graph?

2. What are the meanings of the slope(s) and 

intercepts on this graph?

3. Conduct multiple experiments to compare the

surface area/weight ratios for various materials.

How do the graphs of the data compare?

4. Conduct multiple experiments to compare the surface area/weight ratios for various glider styles. 

How do the graphs of the data compare?

5. Compare the surface area/weight ratio to qualitative gliding ability (i.e. the speed required for the 

glider to maintain lift, and/or how far the gliders go when launched in a consistent way).

6. Look at your group’s data – how reliable is it? Share the data you collected with your classmates. 

Do you results compare in a similar way to your classmates? If not, why not?

7. Develop a visual, verbal, or physical model to explain why you think that surface area affects the 

gliding ability of a glider.

8. Hypothesize why some gliders fly better than others.
20 Aeronautics for Introductory Physics: Scientific Thinking

Literary/Data Analysis: Wings According to Size

Henk Tennekes. The Simple Science of Flight. 

The MIT Press: Cambridge, Massachusetts. Ch. 1 Free Sample: 

http://mitpress.mit.edu/sites/default/files/titles/content/9780262513135_sch_0001.pdf

The relationship between paper gliders, birds, and airplanes 

is surprisingly similar. Yet, when air speed around the 

wings necessary to sustain level flight is considered, these 

relationships can become quite complex. A quantitative 

analysis of bird anatomy and their comparison to airplanes 

can be found by reading Henk Tennekes’ The Simple Science 

of Flight. Chapter 1, used for this activity, is available as a 

free sample from MIT Press at: 

http://mitpress.mit.edu/sites/default/files/titles/

content/9780262513135_sch_0001.pdf. 

This activity allows students to investigate data tables 

and graphs that display non-linear relationships and their 

corresponding equations. The purpose of this activity is 

primarily to introduce students to variables that influence lift, 

and to recognize that airplanes follow many of the same rules of proportion as do birds.

* Cover image and graphics from The Simple Science of Flight are reproduced here with 

permissions from The MIT Press.

Objectives 

Compare and relate variables 

that influence the flight of 

birds and planes.

Materials 

The Simple Science of Flight, 

Ch. 1

NGSS 

CCC 1, 2, 3, 6

CCSS 

Reading in Science 

Writing in Science

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Scientific Thinking

ACTIVITY

Wings According to 

Size
Aeronautics for Introductory Physics: Scientific Thinking 21

Teacher Answer Key: Wings According to Size 

Read The Simple Science of Flight, chapter 1: Wings According to Size. (A free copy of this chapter can be viewed and 

downloaded from http://goo.gl/7dIYB0). As you read through the passage, focus on the relationships of variables that 

influence flight.

1. Based on the first two pages of the chapter, how is “carrying capacity” of an airplane (or any other object) 

calculated? 

Carrying capacity = weight / surface area that carries the weight

2. Performance can also be measured by fuel consumption. What variables do you think influence fuel consumption of 

any moving object?

Weight of the aircraft / animal, metabolism / fuel efficiency, presence of drag (resulting from the aerodynamics as well 

as the weather conditions), etc.

Lift, Weight, and Speed

3. The author identifies four factors that influence the lift of an airplane or flying bird. Complete the chart below. 

(Note: You will need to choose a symbol for the angle of wings.)

Factor Symbol Unit Rationale (Why does it influence lift?)

Wing size S m2 More surface area increases carrying capacity, more 

change in momentum

Air speed V m/s More wind results in more collisions with air particles more 

often.

Air density D kg/m3 Air particles can transfer a greater amount of momentum to 

the flying object.

Angle of wings α (inferred) Degree With more tilt of the wings, the air particles are influenced 

more greatly.

4. How must lift (L) compare to weight (W) 

in order for an object to cruise at a constant 

altitude? Why?

They must be equal, otherwise the object 

will accelerate upward or downward if there 

is a net (uneven) force. This is a result of 

Newton’s second law of motion.

Wing Loading 

Look at Table 1, reproduced here to the right, to 

answer the following questions:

5. What is the specific trend between a bird’s 

weight and the speed it needs to be able to 

cruise? (Is this a direct, linear 

relationship?) Explain.

In general, larger birds have to fly faster. 

However, the relationship does not appear to 

be linear. A doubling in weight typically only 

requires a small increase in velocity.
22 Aeronautics for Introductory Physics: Scientific Thinking

6. This table shows data for various seabirds. How

would you expect the data in each column to 

change if this table showed, instead, 

non-migrating backyard songbirds? Why?

Non-migrating backyard songbirds do not need to 

soar like sea birds. As a result, it is reasonable to 

expect that non-migrating backyard songbirds are 

not as efficient as their sea bird counterparts, and 

likely have to sustain much higher velocities per 

unit weight in order to “soar” simply because they 

do not have to do it that often.

7. How would the data from your lab experience 

with the paper planes compare? 

It is unlikely to be very similar. Paper gliders tend 

to demonstrate a very linear relationship between 

W and S, while birds do not. Hence, the required 

velocities for gliding are also likely to also be 

different.

Look at Figure 1, reproduced on the next page, to 

answer the following questions:

8. Describe the relationship between weight and 

wing loading for these seabirds. (Careful – this is 

not a standard linear graph!) If necessary, re-plot 

the data on a linearly-scaled graph. Why do you 

think this trend is present?

When the weight increases by 100 times, the W/S (carrying capacity) only increases by five times. This difference might 

have something to do with the fact that the increase in weight is not in muscle mass only, but also includes bones and 

fat, which do not contribute to the ability to carry.

9. What is the general meaning of equation (3)? Compare equations (1), (2) and (3) below.

Equation What does it mean? How is it different from the others?

W = 0.3dV2

S Weight is lifted by a flying object is dependent 

upon angle of attack, density, air speed, and 

surface area.

Raw equation. 0.3 comes from average 

angle of attack of 6 degrees.

W/S = 0.38V2 Carrying capacity is dependent upon a constant 

and air speed.

d has been substituted with value at sea 

level.

W/S = c W0.33 Wing loading is proportional to the cube root of 

weight.

Quantifies Figure 1.
Aeronautics for Introductory Physics: Scientific Thinking 23

Great Flight Diagram

Look at Figure 2, reproduced to the right, to 

answer the following questions:

10. What is the meaning of the vertical 

line on the graph? What can you infer 

about birds/planes to the left or to the 

right of the vertical line?

The line represents 22 miles per hour 

(mi/hr) winds. Birds to the left of the 

line are slow flyers, and might not be 

able to overcome strong winds if they 

go out over the ocean.

11. Where is Equation (3) visibly represented on the graph?

The vertical and top horizontal axis 

represent the variables found in 

equation 3.
24 Aeronautics for Introductory Physics: Scientific Thinking

12. Given what you know about the vertical line and the angled line on the graph, compare the following flying objects’

capabilities. How are they different?

Boeing 747 Highest weight, wing loading, and cruising speed.

Pteradon Heavy, slow flyer.

Human-powered Airplane Heavier than pteradon, but nearly same wing loading and cruising speed.

13. Likewise, compare a crane fly, spotted sandpiper, and Beech King airplane. How are they different? 

(Compare weight, wing loading, and cruising speed). How are they similar?

All three of these fall on the same trend line. A crane fly is small and slow, a spotted sandpiper is heavier and faster, 

and a Beech King airplane is even heavier and faster than the sandpiper.

14. Based upon the graph and the information presented in the text, explain one of the reasons why the Concorde is now 

out of commission.

Unlike all of the aircraft in the “group,” the Concorde fell slightly to the left of the trend line. This means, that for its 

group, it tended to be a little bit heavier for its ideal cruising speed, and had a slightly lower wing loading value, than 

the rest of its counterparts. Most importantly, the Concorde did not fly at its ideal cruising speed. As the fastest 

commercial jet then in service in the world, it flew much faster than the ideal speed listed on the diagram. Its ideal 

speed was much lower because the Concorde’s wings had to be big enough for it to take off and land at lower speeds. 

As a result, the Concorde had to “drag along” the larger wings during its high-speed flight.

Use Figure 3 (below to the right) and Figure 4 to 

answer the following questions.

15. What do birds do to modify their wings for various stages of flight? (Takeoff, cruising, landing). 

Why is this so important?

As birds take off, their wings are fully expanded. 

A larger surface area is needed in order to attain 

lift at a slow speed. As birds go faster and faster 

as they cruise, they can decrease the surface 

area of their wings. This is important in order to 

reduce drag and increase efficiency.

16. Standard passenger airplanes DO modify their 

wings during takeoff and landing. Watch the 

following video of an airplane wing release its 

extender flaps here: http://goo.gl/pd1bDM
Aeronautics for Introductory Physics: Scientific Thinking 25

Student Worksheet: Wings According to Size

Read The Simple Science of Flight, chapter 1: Wings According to Size. As you read through the passage, focus on the relationships of variables that influence flight.

1. Based on the first two pages of the chapter, how is “carrying capacity” of an airplane (or any other object) calculated? 

2. Performance can also be measured by fuel consumption. What variables do you think influence fuel consumption of 

any moving object?

Lift, Weight, and Speed

3. The author identifies four factors that influence the lift of an airplane or flying bird. Complete the chart below. (Note: 

You will need to choose a symbol for the angle of wings.)

Factor Symbol Unit MY Rationale

Why does it influence lift?

OTHER’s Rationale

Interview Your Partner

4. How must lift (L) compare to weight (W) in order for an object to cruise at a constant altitude? Why?

Wing Loading

Look at Table 1, reproduced here to the right, to answer the 

following questions:

5. What is the general trend between a bird’s weight 

and the speed it needs to be able to cruise?

6. This table shows data for various seabirds. How 

would you expect the data in each column to change 

if this table showed, instead, non-migrating backyard 

songbirds? Why?

7. How would the data from your lab experience with 

the paper planes compare? 
26 Aeronautics for Introductory Physics: Scientific Thinking

Look at Figure 1 to answer the following questions:

8. Describe the relationship between weight and wing loading for these seabirds. (Careful – this is not a standard linear 

graph!) If necessary, re-plot the data on a linearly-scaled graph. Why do you think this trend is present?

9. What is the general meaning of equation (3)? Compare equations (1), (2) and (3) below.

Equation What does it mean? How is it different from the others?
Aeronautics for Introductory Physics: Scientific Thinking 27

Great Flight Diagram

This Great Flight Diagram displays the OPTIMAL cruising speeds for a variety of flying things. Look at Figure 2, 

reproduced to the right, to answer the following questions:

10. What is the meaning of the vertical line on the graph? What can you infer about birds/planes to the left or to the right 

of the vertical line?

11. Where is Equation (3) visibly represented on the graph?

12. Given what you know about the vertical line and the angled line on the graph, compare the following flying objects’

capabilities. How are they different?

Boeing 747

Pteradon

Human-powered Airplane
28 Aeronautics for Introductory Physics: Scientific Thinking

13. Likewise, compare a crane fly, spotted sandpiper, and Beech King airplane. How are they different? (Compare weight, 

wing loading, and cruising speed). How are they similar?

14. Based upon the graph and the information presented in the text, explain one of the reasons why the Concorde is now 

out of commission.

Use Figure 3 (below to the right) and Figure 4 to 

answer the following questions. 

15. What do birds do to modify their wings for various stages of flight? (Takeoff, cruising, landing). 

Why is this so 

important?

16. Standard passenger airplanes DO modify their wings during takeoff and landing. Watch the following video of an airplane wing release its extender flaps here: http://goo.gl/pd1bDM
Aeronautics for Introductory Physics: Scientific Thinking 29

Successful Flight Test of Shape-Changing Wing Surface

http://www.nasa.gov/content/successful-flight-test-of-shape-changing-wing-surface/#.VHOino29-Rh

Analysis Questions

Explain two ways that shape-changing surfaces are different from typical airplane wings.

Explain two potential benefits of shape-changing surfaces over typical airplane wings.

Click on the “More” link immediately below the text to see an image of the “corrugated” trailing edge 

of a wing. Why do you think this is beneficial? (Hint: Perform an Internet search to find out how owls 

can fly so silently).

What do you think potential designs of the future might look like? Consider what you know about birds 

as you consider possible designs.
30 Aeronautics for Introductory Physics: Constant Velocity

Constant Velocity

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
Aeronautics for Introductory Physics: Constant Velocity 31

Real-World Applications: Flying Speed

In this activity, students use an equation that they can previously derive through inquiry 

(Distance = Speed * Time). They use a real-time plane finder to watch a plane as it travels 

from one recognizable point to another while at cruising speed (far away from its departure or 

arrival airport) in order to estimate its speed.

Direct students to go to the 

following website: 

http://planefinder.net/

Ask students to browse the 

page and to note the greatest 

concentration of air flights 

around the world. Students may 

be provided a map, and asked 

to mark the countries and areas 

where there is the most flight 

activity.

Discuss potential reasons for 

air flight concentrations being 

where they are. Consider:

• Economics

• Population density

• Geography (deserts, mountain ranges, “remote” areas)

• Weather (locations of storm systems, hurricanes)

• Time of day (consider time zones)

• Cultural holidays

Objectives 

Estimate the average speed 

of an airplane as it travels 

between two points.

Materials 

Internet access 

Timer 

Planefinder.net

Google.com/maps

NGSS 

CDI HS-PS2 

SEP 2,7

CCC 1

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Flying Speed
32 Aeronautics for Introductory Physics: Constant Velocity

Ask students to zoom in to a particular area where 

there is a regular amount of air traffic. The goal 

at this point is to identify an aircraft that is 

moving from one notable location to another 

(such as a city, interstate intersection, lake, etc.), 

so that speed may be determined. (Note: If 

students zoom in very close to an aircraft and 

select Settings → Plane, they can get the airline, 

flight number, origin and destination airports, as 

well as the speed, in knots (KTS). However, most 

students will not recognize KTS. This value can 

be used to evaluate the calculated speed, but 

should not be used directly for this activity).

Choose one aircraft, and time it as it goes from 

one point of interest to the other. Students should 

choose a range of time somewhere between one 

and five minutes to ensure that enough time is collected for accurate calculations. (Caution: If the 

goal is for students to determine the cruising speed of aircraft, they should look at an aircraft once it 

is well along its flight path. Students should not choose planes travelling in significantly curved 

paths, or near airports, where speeds change. However, looking at speeds near takeoff and landing is 

a great extension activity).

Make sure that students choose recognizable points of interest, 

because students will next need to determine the distance crossed by 

re-locating and measuring the distance between the two points using 

Google Maps.

To measure the distance between the points, follow the example 

procedure below:

• Go to www.google.com/maps to get to Google Maps

• As an example, a student might have observed a plane go 

from Holtwood, Pa. to Whitehall, Md. in a period of 2.387 

seconds.

• Students should find, on Google Maps, the initial point of 

the plane’s timing. 

• Right-click on the initial location to get a menu. Choose 

“Measure distance.” The second point of interest can be 

clicked, and an estimate given for the distance between the 

two points. In this example, the distance is 22.58 miles.

• To make speeds more relevant, it is suggested that students convert time in seconds to hours. Using a simple 

conversion, student can find that 2.387 s is equivalent to 

0.03978 hr.

• Using the equation Distance = (Speed) Time, and re-arranging for speed, students can get Speed = Distance/Time. In 

this example, Speed = 22.58 mi / 0.03978 hr, which is equal 

to 567.6 mi/hr. This is a very reasonable estimate! Most 

commercial airplanes cruise at about 550 mi/hr. 
Aeronautics for Introductory Physics: Constant Velocity 33

Take a broader look at the flight paths across the United States. 

• Do you notice any patterns for eastward or westward flights?

• How do flight paths change upon entering or leaving airports? 
34 Aeronautics for Introductory Physics: Constant Velocity

A Day in the Life of Air Traffic Control

With the massive increase in passenger and air cargo planes, the skies have become a very busy place. 

NASA Aeronautics works hard to solve the problem of air traffic congestion. Modeling air traffic over 

a span of time to determine problems points is integral to managing the network of highways in the 

sky. Help students to see how busy the airspace over the US really is by watching the video, “A Day in 

the Life of Air Traffic Control over the US.”

1. Have students to to: http://www.aviationsystemsdivision.arc.nasa.gov/research/modeling/

facet.shtml

2. Ask students to read the article about NASA’s work with FACET to manage air traffic across the 

nation.

a. What is FACET?

b. How does it benefit the environment, economy, and education?

3. Ask students to play the linked video (click view Facet Animation) showing 24 hours of air 

traffic over the United States. http://www.aeronautics.nasa.gov/videos/facet24.mov and answer 

the following questions:

a. From which direction doe most of the flights leave / arrive? (SE, NE, NW) Why don’t they 

go directly N, S, E, and W?

b. The simulation begins at 0 UTC (Coordinated Universal Time), which corresponds to 7 p.m. 

Eastern Standard Time. Describe in words, what happens to the density of flight traffic, on 

average during a 24 hour period?

c. Draw a simple graph showing the average flight density versus time.

d. What are the three or four busiest airports in the US?

4. What are two or three practical solutions to decreasing traffic congestion in the US?
Aeronautics for Introductory Physics: Constant Velocity 35

Student Worksheet: Flying Speed

Objective: Estimate the average speed of an airplane as it travels between two points.

There is a significant amount of aircraft in the skies every day of the year. Because these aircraft must be 

monitored to help pilots get to their destination efficiently, and to avoid in-air collisions, it is possible to 

track any commercial aircraft around the globe. There are many websites and apps available to help people 

track specific flights or to browse flights around the world. For this activity, go to http://planefinder.net/

1. Which parts of the world experience the least air traffic? 

2. Choose one country or region surrounded by little air traffic, and explain why you think it does not 

have a busy airspace. (Consider the economics of these countries, population density, geography, 

weather, time of day, and the observance of any cultural holidays).

3. Which parts of the world experience the most air traffic? 

4. Choose one country or region surrounded by lots of air traffic, and explain why you think it has 

a very busy airspace. (Consider the economics of these countries, population density, geography, 

weather, time of day, and the observance of any cultural holidays).

5. How do you anticipate that the airspace of the future will change? Do you think the skies will get 

busier or less busy? Why?

6. The goal of this activity is for you to determine 

the average speed of an aircraft during its cruise 

portion of its flight. Explain what information you 

will need to determine average speed.

7. Zoom in to a particular area where there is a 

regular amount of air traffic. The goal at this point 

is to identify an aircraft that is moving from one 

notable location to another (such as a city, interstate intersection, lake, etc.), so that speed may 

be determined. Avoid starting or ending at an 

airport, as aircraft speed changes significantly 

at those points.

8. Choose one aircraft, and record the time it as it flies from one point of interest to the other in a 

straight line. Choose a range of time somewhere between one and five minutes to ensure that enough 

time is collected for accurate calculations. If your airplane significantly changes direction or follows 

an indiscernible path, consider choosing a different aircraft. Record the time here: 
36 Aeronautics for Introductory Physics: Constant Velocity

9. Measure the distance between the two points 

of interest by following the example 

procedure below:

• Go to www.google.com/maps to get to 

Google Maps

• On Google Maps, find the initial point 

of the plane’s timing. 

• Right-click on the initial location to get 

a menu. Choose “Measure distance.” 

The second point of interest can be 

clicked, and an estimate given for the 

distance between the two points. 

10. Show your work, and, if appropriate, explain 

how you can calculate the average velocity 

of the airplane.

11. For a typical passenger jet, the airplane will cruise at around 500-550 miles per hour. How does your 

answer compare? You might need to convert your velocity units to make a fair comparison.

12. What aspects of this activity might lead to error in your measurements and calculations? Provide 

three specific examples, and explain if this would increase or decrease your calculated value.

13. Compare your results to that of your peers. Why might your peers have calculated different values?

14. Take a broader look at the flight paths across the United States. 

• What patterns do you notice for eastward or westward flights?

• How do flight paths change upon entering or leaving airports? Why do you think this is the case?
Aeronautics for Introductory Physics: Constant Velocity 37

Problem Set: Comparative Velocities

Background

Airplanes were developed with a primary goal: to take their occupants across the skies at speeds never before imagined. For 

thousands of years, before the invention of the wheel, humans were limited by the speed of their legs. Using a mathematical 

model (equation) you developed in your regular class activities to represent the relationship between position and time for an 

object moving at a constant velocity, perform these very simple calculations to get a sense for how fast and far we can now 

easily travel because of aircraft.

Problem Set

1. Determine your average walking speed. Explain how you accomplished this:

2. Convert your speed in meters per second (m/s) to kilometers per hour (km/hr), then use that speed to solve the 

following problems. How much time would it take you to walk the following routes, assuming you never stopped along 

the way? 

Land Routes Distance to Walk Time to Walk 

(Show work)

Time to Fly Non-Stop 

on Commercial Aircraft

Oregon Trail 

(MO to OR)

3,219 km

(2,000 miles)

Immigration route used from 

1830s to 1860s for settlers heading west for new land.

3 hours

Silk Road 

(Western China to 

Constantinople)

6,437 km

(4,000 miles)

Cultural and commercial trade 

route used from 200 BCE to 

1450s CE to connect culture and 

trade between the East and West.

8 hours

Early Land Migration 

(Northern Russia to 

Tierra del Fuego)

17,703 km

(11,000 miles)

Possible route taken by early 

civilizations from the Asian continent, beginning about 30,000 

years ago to their arrival at the 

southernmost tip of South America about 15,000 years ago.

19 hours
38 Aeronautics for Introductory Physics: Constant Velocity

3. How long would it take each of the following NASA planes to travel the exact same routes above, assuming they were 

at their maximum speed for the entire duration of the flight?

Land Routes Distance to Walk NASA Aircraft Speed Time to Flight

(show work)

Oregon Trail 

(MO to OR)

3,219 km

(2,000 miles)

Immigration route used from 

1830s to 1860s for settlers heading west for new land.

X-1: 1,127 km/hr

Chuck Yeager was the first 

person to break the speed 

of sound in 1947 using 

rockets in conjunction with 

traditional jet engines.

Silk Road 

(Western China to 

Constantinople)

6,437 km

(4,000 miles)

Cultural and commercial trade 

route used from 200 BCE to 

1450s CE to connect culture and 

trade between the East and West.

D-558: 2,124 km/hr

Scott Crossfield was the 

first person to fly more than 

twice the speed of sound 

(Mach 2) in 1953.

Early Land Migration 

(Northern Russia to 

Tierra del Fuego)

17,703 km

(11,000 miles)

Possible route taken by early 

civilizations from the Asian continent, beginning about 30,000 

years ago to their arrival at the 

southernmost tip of South America about 15,000 years ago.

X-15: 7,274 km/hr

William J. “Pete” Knight 

broke the record for the 

fastes piloted airplane flight 

speed ever recorded, at 6.7 

times the speed of sound 

(Mach 6.7) in 1967.
Aeronautics for Introductory Physics: Constant Velocity 39

Problem Set: Vectors in Relative Velocities Problems

from Beginner’s Guide to Aeronautics

BACKGROUND: (Note: This background is most appropriate after students have been introduced to these concepts through 

constructivist and/or inquiry activities.)

One of the most confusing concepts for young scientists 

is the relative velocity between objects. Aerodynamic 

forces are generated by an object moving through a 

fluid (liquid or gas). A fixed object in a static fluid does 

not generate aerodynamic forces. Hot air balloons “lift” 

because of buoyancy forces and some aircraft - like 

Harrier jets - use thrust to enable the vehicle to hover, 

but these are not examples of aerodynamic lift. To 

generate lift, an object must move through the air, or air 

must move past the object. Aerodynamic lift depends 

on the square of the velocity between the object and 

the air. Now things get confusing because not only can 

the object be moved through the air, but the air itself 

can move. To properly define the relative velocity, it is 

necessary to pick a fixed reference point and measure 

velocities relative to the fixed point. In this slide, the 

reference point is fixed to the ground, but it could just as easily be fixed to the aircraft itself. It is important to understand the 

relationships of wind speed to ground speed and airspeed.

Wind Speed

For a reference point picked on the ground, the air moves relative to the reference point at the wind speed. Notice that the 

wind speed is a vector quantity and has both a magnitude and a direction. Direction is important. A 20 mi/hr wind from the 

west is different from a 20 mi/hr wind from the east. The wind has components in all three primary directions (north-south, 

east-west, and up-down). In this figure, we are considering only velocities along the aircraft’s flight path. A positive velocity 

is defined to be in the direction of the aircraft’s motion. We are neglecting cross winds, which occur perpendicular to the 

flight path but parallel to the ground, and updrafts and downdrafts, which occur perpendicular to the ground.

Ground Speed

For a reference point picked on the ground, the aircraft moves relative to the reference point at the ground speed. Ground 

speed is also a vector quantity so a comparison of the ground speed to the wind speed must be done according to rules for 

vector comparisons.

Airspeed

The important quantity in the generation of lift is the relative velocity between the object and the air, which is called the 

airspeed. Airspeed cannot be directly measured from a ground position, but must be computed from the ground speed and 

the wind speed. Airspeed is the vector difference between the ground speed and the wind speed.

Airspeed = Ground Speed - Wind Speed

On a perfectly still day, the airspeed is equal to the ground speed. But if the wind is blowing in the same direction that the 

aircraft is moving, the airspeed will be less than the ground speed.
40 Aeronautics for Introductory Physics: Constant Velocity

Examples

Suppose we had an airplane that could take off on a windless day at 100 mi/hr (liftoff airspeed is 100 mi/hr). We are at an 

airport with an east-west runway that is one mile long. The wind is blowing 20 mi/hr towards the west and the airplane takes 

off going east. The wind is blowing towards the aircraft which we call a headwind. Since we have defined a positive velocity 

to be in the direction of the aircraft’s motion, a headwind is a negative velocity. While the plane is sitting still on the runway, 

it has a ground speed of 0 and an airspeed of 20 mi/hr:

Airspeed = Ground Speed (0) - Wind Speed (-20) = 20 mi/hr

The airplane starts its take off roll and has a constant acceleration a. From Newton’s second law of motion, the ground speed 

v at any time t is: 

Δv = a t

and the distance Δx down the runway at any time is:

Δx = 1/2 a t2 + vi t

For a fixed length runway, this specifies the time to be used in the velocity equation. Let’s assume that at 5000 feet down the 

runway, the velocity is 80 mi/hr. Then the airspeed is given by

Airspeed = Ground Speed (80) - Wind Speed (-20) = 100 mi/hr

and the airplane begins to fly. Now another pilot, with exactly the same airplane decides to take off to the west. The wind 

is now in the same direction as the motion and this is called a tailwind. The sign on the wind speed is now positive, not 

negative as with the headwind. The acceleration along the ground is the same, so at 5000 feet down the runway, the ground 

speed is again 80 mi/hr. The airspeed is then given by:

Airspeed = Ground Speed (80) - Wind Speed (20) = 60 mi/hr

This airplane doesn’t have enough airspeed to fly. It runs off the end of the runway!

Significance of Understanding Relative Velocity

The importance of the relative velocity explains why airplanes take off and land on different runways on different days. 

Airplanes always try to take off and land into the wind. This requires a lower ground speed to become airborne, which means 

the plane can take off or land in the shortest distance traveled along the ground. Since runways have a fixed length, you want 

to get airborne as fast as possible on takeoff and stopped as soon as possible on landing. In the old days, a large conical tube 

known as a windsock was hung near the runway for pilots to see which way the wind was blowing to adjust their takeoff and 

landing directions. Now mechanical or electronic devices provide the information that is radioed to the cockpit.

The relationship between airspeed, wind speed, and ground speed explains why wind tunnel testing is possible and how kites fly.

• In the wind tunnel, the ground speed is zero because the model is fixed to the walls of the tunnel. The airspeed is 

then the negative of the wind speed that is generated in the tunnel. Whether the object moves through the air, or the 

air moves over the object, the forces are the same.

• A kite usually has no ground speed because a kite is held on the end of a string. But the kite still has an airspeed that 

is equal to the wind speed. You can fly a kite only with the wind at your back.
Aeronautics for Introductory Physics: Constant Velocity 41

Problem Set from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Key

A pilot is about to take off from a Nasaville airport that has two runways. Runway A runs east-west and Runway B runs 30 

degrees west of north. There is a wind of 25 mi/hr out of the northwest (see figure above).

1. Which runway should the pilot choose?

2. What is the magnitude of the velocity vector of the wind along the desired runway?

3. If an airspeed of 100 mi/hr is required for takeoff, what is the ground speed required?

4. Before the pilot is able to take off, the wind shifts. The wind is now directly out of the north at 25 mi/hr. 

Will the pilot choose the same runway?

What is the ground speed required now?
42 Aeronautics for Introductory Physics: Constant Velocity

Problem Set from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Key

In airport takeoff situations, the ground speed of the aircraft and the wind speed are the only factors in airspeed. The U.S. 

Navy has an additional variable. Its mobile airports - aircraft carriers - move on the ocean at speeds that can be in excess of 

35 mi/hr. Airspeed in this unique situation is a sum of the ship’s speed through the water, wind speed, and the speed of the 

aircraft over the deck of the ship provided by the ship’s catapult.

A catapult is like a huge rubber band and is used 

to propel an aircraft to higher speeds over a short 

distance in a few seconds. For more information 

on aircraft carriers and catapults, click on Aircraft 

Carriers. Assume that a typical catapult can propel an 

aircraft from zero to 150 mi/hr in two seconds. The 

aircraft carrier problem is pictured graphically in the 

following illustration. (Note: See the airspeed equation 

at the bottom.)

You are the captain of the Aircraft Carrier USS 

Enterprise (CVN-65). Assume that the required 

airspeed for takeoff of an F-14 is 170 mi/hr. Using the 

airspeed equation and information provided, complete 

the following table and answer the questions that 

follow. Assume that a positive wind velocity is in the 

direction of travel of the ship.

Catapult Speed (vc) mi/hr Ship Speed (vs) mi/hr Wind Speed (vw) mi/hr Airspeed Go/No Go (>170 mi/hr)

150 20 -10

150 20 10

150 30 -30

150 0 20

Airspeed = Ship Speed (vs

) + Catapult Speed (vc) - Wind Speed (vw)

1. With no wind, what should the speed of the ship be to launch an F-14? 

2. On launch, any aviator wants to maximize lift. What would you do as captain of the ship any time you launch aircraft?

3. Are wind direction and ship speed factors in landing as well? Explain.
Aeronautics for Introductory Physics: Constant Velocity 43

Real-World Applications: Estimating the Speed and Distance of an 

Airplane

Laurence I. Gould & Charles Waiveris. The Physics Teacher, 29, 108 (1991).

There are many ways to estimate the distance and speed of an airplane, and one of the 

simplest ways is to compare the location of a plane by sight with the location of its sound, 

similar to the process of estimating the distance of a lightning bolt by using the time it takes 

for the thunder to arrive. In this activity, the speed of sound is assumed to be a standard 343 

meters per second (m/s). The value of the speed of sound can be determined experimentally in 

advance using simple mechanics (with a long tube and two microphones to determine speed 

from transmission to reception of a sound pulse).

This activity best takes place at a location between two to six kilometers away from a regular 

plane flight path. It is advisable to find a public viewing location near an airport. Smaller 

distances result in less time delay for the sound, and greater distances make it difficult to 

accurately measure locations. Using simple trigonometry (law of cosines) and an estimate for 

the speed of sound (343 m/s, adjusted for climate conditions), reasonable values for airplane 

speed and distance can be obtained. 

Two approaches are provided for this activity – as a more open-ended inquiry lab as well 

as through direct instruction. 

Inquiry Approach – Simplified and Updated by Florian Genz, University of Cologne/

Germany

Objectives 

Estimate the speed and 

distance of an airplane using 

the speed of sound.

Materials 

Meter stick 

Lap timer

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 3,4

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Estimating the Speed 

and Distance of an 

Airplane
44 Aeronautics for Introductory Physics: Constant Velocity

The inquiry approach is ideal because of its more holistic, conceptual approach. This approach requires observers to be 

directly underneath the flight path of aircraft. This approach uses a sound intensity meter, and provides a good opportunity 

for data analysis through the GeoGebra interactive online worksheet. Students can also complete this activity using a readily 

available YouTube video.

Why? There are many ways to estimate the distance and speed of an 

airplane but few labs connect everyday technology, such as smartphone 

sensors, with motivating problems in physics.

Where? This activity best takes place at a location 2-6 km in front of 

the runway and under a regular plane approach path. Alternative: Watch 

youtu.be/i0bd9UQKLG0

Setup:

1. Position the camera exactly facing up (the internal G-Sensors or a 

plumb-bob may help).

2. Fix a long ruler in front of the video camera.

3. Determine the distance between the ruler and the camera.

4. Film an airplane at approach from below.

Determining the plane’s height

A plane is usually loudest when directly above an observer. Observe 

the video and note the time when the plane passed exactly overhead. 

Then analyze the soundtrack of the video clip and determine at which 

timestamp the plane was loudest. By noting the time difference in 

seconds the height can be determined:

In this activity, the speed of sound is assumed to be:

Determining the plane’s speed:

Using simple trigonometry (similar triangles) delivers:

Students may find these relations on their own or with little hints.

With the Geogebra interactive worksheet “Application Lab: Estimating Height and Speed of Airplanes” young learners can 

get a feeling for the scales. It visualizes their measurements and it allows students to vary and compare their results fast: 

http://ggbtu.be/m646385 
Aeronautics for Introductory Physics: Constant Velocity 45

Assumptions

These calculations assume horizontal flight. Typical path angles are α=3° at approach (NOT at takeoff!). Advanced students 

may click “error estimation” and find that small approach angles are negligible.

For estimating the height horizontal flight was assumed. Typical sink rates at approach are 700 ft/min (= 3.6 m/s). This 

results in the loudest (=closest) position being not exactly above head. For h > 1000 m and Δt < 3 s students may find it 

negligible, too.

Further assumptions:

• No wind.

• Infinite speed of light.

• Constant speed of sound.

• No lens errors (e.g. barrel distortion).

Didactic plea: Students may come up with better solutions – or none. Keep in mind: Our aim is to promote scientific 

thinking, and not simply to produce a solution!

The following student worksheet is structured by the concept of Hacker, Volpert & Dörner’s “Vollständige selbstregulierte 

Lernhandlung” (“Complete self-regulated learning action”) modified by A. Bresges (2008).
46 Aeronautics for Introductory Physics: Constant Velocity

Student Worksheet: Estimating the Speed and Distance of an Airplane

INQUIRY APPROACH

Objective: Estimate the speed and height of an airplane using the speed of sound.

Why? Approaching planes near 

airports can look very close and 

slow. Children living near airports 

sometimes have nightmares that a 

plane above them will “fall” from the 

sky into their bedroom. Can you help 

them by proving to them how fast and 

high the planes still really are?

To do this activity, you will need 

to find a location not far from an 

airport, where you can see airplanes 

fly overhead as they are just about 

to land. Try to find a location that is 

between 2 to 6 kilometers away from 

airplanes as they fly.

You may use:

• A long ruler (for challenge B)

• A tablet or smartphone with

• a movie cutting app

• a noise analysis app

Challenge A: Measuring a plane’s travelling height

1. You might already have found out that the speed of sound (vsound) is somewhere around 343 m/s. 

Also you know that the speed of the sound multiplied with the time it travels is the travel height of 

the plane. Make a sketch or find a formula for this! ____________

2. Collect ideas how to measure or calculate the missing physical values. There is no right and wrong 

here! Which tools can you use? Which measuring points or positions are suitable? Which physical 

units are useful?

3. Now listen to the ideas of your partner(s) and present your ideas. Make a team decision on how to 

continue. Deciding to ask for help here is also a decision. Try to write down your problem before.

4. Do what you decided. If problems occur, go back to planning and try again.

1912: People looking up to the sky, watching the first aircraft ever 

to fly over Townsville 

(Image Credit: John Oxley Library, State Library of Queensland).

(alone)

(small group/pair)

(small group/pair)
Aeronautics for Introductory Physics: Constant Velocity 47

5. Estimate how reliable your measurements are. Can you repeat them exactly? What is your measuring 

tolerance?

Finally discuss WHAT you actually measured and present your findings to the whole class. 

Listen carefully to the findings of the other groups and try to find differences and similarities.

Now that you already have successfully completed circle of a self-regulated learning action try to do it 

again, but this time with an even harder challenge: 

Challenge B: Measuring the plane’s speed

You know the travelling height (hplane) of the plane now. Maybe this triangle 

sketch will give you ideas on how to first measure the plane’s traveling distance 

(dplane).

1. My ideas are...

2. We decided...

3. This we had to change:

Because...

This worked as planed:

4. Verification 

Quantity →

Unit →

#1

#2

#3

#4

 ...

Average:

Our measuring results seem to...

5. What if the plane doesn’t fly horizontally? Would it influence the measurements? Discuss with the 

class. Is it possible with your method to record the speed and height of a plane from other directions?

* Ultimate challenge: Can you persuade a much younger child of your findings with simple words?

(small group/pair)

(class)

(alone)

(small group/pair)

(small group/pair)

(small group/pair)

(class)
48 Aeronautics for Introductory Physics: Constant Velocity

Direct Instruction Approach

The direct instructional approach closely models the method referenced in The Physics Teacher, and might provide greater 

clarity to the teacher or student who needs more scaffolding or who appreciates mathematical puzzles. Additionally, the 

approach suggested does not require students to be directly underneath the flight path – only near it.

The more direct approach, outlined in The Physics Teacher article, uses the following method:

Once an airplane begins to pass overhead, hold out the meter stick perpendicular to the line of sight toward the airplane (A). 

Once the position is noted (a), simultaneously begin the lap timer (T0). A helper should also measure the distance of the 

sighter’s eye to point a (x). When the sound of the airplane also appears to come from point A, record the sighted position (b) 

of the plane’s current location (B), and again press the lap timer (T1). Again, wait for the sound to appear to come from point 

B, and record the sighted position (c) of the actual plane (C), and press the lap timer again (T2).

The change in time for the intervals can be defined as: 

and 

The actual speed and distance of the airplane can then be estimated 

using the law of cosines. The view of the flight path can be denoted by 

triangle AOB, with sides D1, D2, and D12. Most high school students 

will at least have some level of familiarity with the Pythagorean 

theorem’s more general expression for any triangle:

This can be substituted for actual values from the activity, to give

Because distance, D, can be expressed as the product of speed and 

change in time (v t), this equation can be further defined as

and simplified to

The angle can be estimated with the simple form of the 

Pythagorean theorem, as:

If possible, students can also attempt to identify the type 

of plane, and to compare their resulting speeds to speeds 

typical of planes as they take off or land.

Sample data is reproduced here in Table 1.
Aeronautics for Introductory Physics: Constant Velocity 49

Student Worksheet: Estimating the Speed and Distance of an Airplane

DIRECT INSTRUCTION APPROACH

Objective: Estimate the speed and distance of an airplane using the speed of sound.

From your previous investigations, you might have found out that 

the speed of sound is somewhere around 343 m/s, and perhaps a 

bit higher or lower depending upon atmospheric conditions. In 

this activity, you will use trigonometry and the difference in time 

between the location of an airplane and the apparent location of 

its sound, in order to find out both the speed and distance of an 

airplane.

To complete this activity, you will need to find a location not 

far from an airport, where you can see airplanes fly overhead as 

they are just about to land or takeoff. Try to find a location that is 

between two and six kilometers away from airplanes as they fly.

1. What travels faster – sound or light? Explain how you know. 

(Consider lightning and thunder, or the sights and sounds of 

fireworks).

For this activity, you may assume that light travels instantaneously 

(although it actually doesn’t), and that sounds travels at 343 m/s. 

Similar to the situations identified in #1 above, it is possible to look 

at a plane, and to hear the sound appearing to travel behind it. In 

this activity, you will use a timer to mark the time and location of an airplane at two points on a meter stick, and also time 

how long it takes for the sound to appear to catch up to the plane’s previous positions. 

This task will take some coordination, so be patient, and make sure you are in a location where multiple airplanes pass by 

every hour. Be sure to read the instructions first, as you will not have time to read them once you begin!

2. Hold out a meter stick parallel to your face. Have your partner record the distance between your eyes and the point on 

the meter stick, so that it makes a 90 degree angle (0 – a). Put a piece of tape at this position. 

Record the distance from eyes to meter stick here: 0 – a: m

Record the position of “a” here: a: m

 

3. Watch for the first aircraft to pass by. When the aircraft is located at “a,” call TIME! Your partner should begin the 

stopwatch.
50 Aeronautics for Introductory Physics: Constant Velocity

4. Listen carefully for the airplane’s sound until it also appears to be located at “a.” At that moment, quickly glance at the 

airplane’s location and call out the position on the meter stick. At that moment, your partner should hit the “lap” button 

on the timer and record the position you called out. This position is now known as “b.” See the image below if you need 

help.

Record the first lap time here: T1 – T0 = t1: s

Record the position of “b” here: b: m

5. Listen carefully for the airplane’s sound until it also appears to be located at “b.” At that moment, quickly glance at the 

airplane’s location and call out the position on the meter stick. At that moment, your partner should hit the “lap” button 

on the time and record the position you called out. This position is now known as “c.”

Record the second lap time here: T2 – T1 = t2: s

Record the position of “c” here: c: m

6. Now that you have all of your data, you can estimate the 

speed and distance of the airplane. For simplicity, focus 

only on the first triangle, AOB, with sides D12, D1, and 

D2. Note that this is not a right triangle. However, the 

extended form of the Pythagorean theorem can be used, 

which can be substituted with the symbols for the sides.

7. However, you don’t have the value of any of the sides! 

Instead, how can you represent an expression for D? (Hint: 

Use your equation for distance, speed, and time, and solve 

for D. Substitute this expression for D into the equation, 

ensuring that you are using the proper time). 

Because distance, D, can be expressed as the product of 

speed and change in time (v t), this equation can be further 

defined as

where vs

 is the speed of sound and vp is the speed of the plane.

8. Explain how the equation above can be simplified to: 
Aeronautics for Introductory Physics: Constant Velocity 51

9. Now, all that remains is to solve for the angle near your eye. Use your markings on your meter stick for positions “a” 

and “b,” as well as the value of x, to solve for the angle. Show work.

10. Substitute all values into the equation from #8, and calculate the speed of the plane.

11. Perform the same analysis for triangle AOC. Calculate the speed of the plane using that data.

12. How does the data for your airplane speed compare 

with each other? How does it compare to sample 

data produced by other physics students in Table 1?

13. Would it be possible to perform the same analysis 

with triangle BOC? Why or why not?
52 Aeronautics for Introductory Physics: Constant Velocity

Real-World Applications: Noise Doppler-Shift Measurement of 

Airplane Speed

Ivan F. Costa and Alexandra Mocellin. The Physics Teacher, 45, 356 (2007).

While out measuring the distance and speed of 

airplanes using trigonometry and/or sound intensity, 

a further option is to measure the speed of airplanes 

using Doppler shift. (However, ensure that students 

develop a good conceptual understanding of the 

Doppler effect before pursuing this activity).

Simple fast Fourier transform (FFT) analyzers can 

be obtained freely through mobile device apps or 

probeware companies. Allow students to first hear 

the Doppler effect of aircraft as they pass by. Then, 

use an FFT analyzer to determine the 

average frequency of the plane as it 

comes toward the observer, is near the 

observer, and as it goes away from the 

observer. 

Three primary frequencies should 

be identified in the FFT profile – 

frequency when coming toward, 

frequency when directly above/beside 

the observer, and frequency when 

leaving. Although this pattern can be 

observed in Figure 3 as a kind of stair 

step graph, different FFT analyzers 

will represent these values in different ways. (Additionally, students will note that there is 

a lot of background noise, originating from erratic vibration, turbulence, etc.) Using these 

frequencies, apply the Doppler shift equation to estimate the speed of the source.

Objectives 

Use linear relationships, 

graphs, and body movement to 

simulate air traffic control.

Materials 

Internet access: 

http://smartskies.nasa.gov 

Paper (type rolls)

Paper placards 

Timers

NGSS 

CDI HS-PS4-1

SEP 1-7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Falling and Air 

Resistance
Aeronautics for Introductory Physics: Constant Velocity 53

New Acoustics Techniques Clear Path for Quieter Aviation

http://www.nasa.gov/larc/new-acoustics-techniques-clear-path-for-quieter-aviation/#.VHS3oo29-Rh

Analysis Questions

• What is “auralization”? Why is it important?

• What is “psychoacoustics”? Why is it important to use human test subjects, rather just than 

electronic sound analyzers?

• Click on the hyperlink near the end of the article to listen to a variety of Aircraft Flyover Simulation files. Listen to a number of different scenarios, then comment on at least five variables 

that you think researchers must take into account when doing “auralization” work. If you are 

uncertain, click on the research papers associated with the files to learn more.
54 Aeronautics for Introductory Physics: Constant Velocity

Real-World Applications: Smart Skies

http://smartskies.nasa.gov/

The following activities are a synthesis of currently available Smart Skies and FlyBy Math 

modules produced by NASA. This handout provides an introduction to the tasks, but formal 

teacher guides and student worksheets and resources are located online at the links to the side.

Objective

Model the difficulties and problems associated with air traffic control using mathematics and 

simulations. NASA has developed two comprehensive curricular modules, titled Fly By Math 

and Line Up With Math, as well as a free app, Sector 33, which can be used independently. 

Fly by Math

Fly By Math makes use of five different scenarios 

of airplanes approaching each other. Students 

are provided with each scenario, and asked to 

determine a variety of information about the 

airplanes, including the time it will take for each 

plane to merge onto the main route (problems 

1-4) and the time when the trailing plane will 

catch up to the leading plane (problem 5).

Objectives 

Use linear relationships, 

graphs, and body movement to 

simulate air traffic control.

Materials 

Internet access: 

http://smartskies.nasa.gov 

Paper (type rolls)

Paper placards 

Timers

NGSS 

CDI HS-PS2 

SEP 2, 4, 5, 7

CCC 3, 4, 7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Smart Skies
Aeronautics for Introductory Physics: Constant Velocity 55

Students will solve each of the above problems using up to six different calculation methods:

• Counting feet and seconds using a jet route diagram

• Drawing blocks to make a bar graph

• Plotting points on two vertical number lines

• Plotting points on a Cartesian coordinate system

• Deriving and using the distance-rate-time formula

• Graphing two linear equations

In addition, students can use the Fly By Math Simulator to plot airplane 

distance versus time, and see the effects of changing speed and starting 

distance from the origin by modifying variables in the linear equation.

Line Up With Math

Line Up with Math makes use of six different problems 

about airplanes. Airplanes must be carefully aligned to be 

within three nautical miles of each other upon approach 

into an air sector to ensure that plane distances are both 

safe as well as efficient. Students must find solutions to 

problems involving two-plane and three-plane conflicts 

resolved with route changes, two-plane and three-plane 

conflicts resolved with speed changes, and three-plane, 

four-plane, and five-plane conflicts resolved with speed 

changes or with speed and route changes. Students use an 

online air traffic control simulator, supported by student 

worksheets, to model the movements and placements of 

airplanes, and to change their speeds and/or routes.

Students and the wider community can access the benefits 

of Line Up With Math with the stand-alone Sector 33 app 

for iOS and Android mobile devices. 

Sector 33

Sector 33 is an app that allows users of all backgrounds 

to experience what it is like to manage air traffic control. 

Sector 33 is the actual airspace around the San Francisco 

Bay area.

Sector 33 can be downloaded, for free, from the Apple 

iTunes Store and the Google Play Store.

iOS iTunes Store: https://itunes.

apple.com/us/app/sector-33/

id486953105?mt=8

Android Google Play Store:

https://play.google.com/store/

apps/details?id=gov.nasa.stem
56 Aeronautics for Introductory Physics: Constant Velocity

Sector 33 App Description

It’s a stormy Friday evening in Northern California as the evening rush of air traffic fast approaches the San Francisco Bay 

Area. All flights going to San Francisco airport from the east pass through “Sector 33” – YOUR sector of the airspace.

As the lead air traffic controller for Sector 33, you must merge the arriving planes into a single traffic stream as they pass over 

Modesto, Calif. on the western edge of your sector. The planes must be properly spaced and arrive over Modesto as soon as 

possible. Every minute you delay a plane during the traffic rush, that delay is passed on to ALL the other planes flying behind 

it. Although time is of the essence, to assure safety, the planes must NEVER violate minimum spacing requirements.

Can you handle Sector 33?

Features

• Thirty-five problems

• Two to five airplanes

• Speed and route controls

• Thunderstorm obstacles

• Four levels of controller certification

• Locked levels

• Scoring for each problem

• Scoring for each certification level

• In-game introduction

• In-game hints

• Help section

• Extra videos

• Links to related websites

• Links to social websites

In the US Sector 33 airspace, three main air highways merge together over Modesto. In order to ensure that the planes are 

efficiently trafficked upon arrival at their destination airport – to save fuel and decrease the incidence of delays – all of the planes 

must be within three nautical miles of each other, but no less than two nautical miles, to ensure safety. In the scenario, the user 

must provide instructions to each of the airplanes on their speed and route. The game player must use mathematical strategy, 

ratios, and time sequencing by thinking ahead. Failing to do so results in a loss of points, and, in the real world, possible tragedy. 

As an example, Level 3-g is displayed. Three airplanes, 

AAL12, DAL88, and UAL74 are approaching Modesto at the 

same speed of 600 knots. AAL12 is approximately 19 nautical 

miles from Modesto, while DAL88 is 21 nautical miles 

away, and UAL74 is 23 nautical miles away if it follows 

the southern-most path, or 26 miles away from Modesto if 

it directed to branch off toward position OAL. Because all 

airplanes must achieve a spacing of three nautical miles away 

from each other by the time the last plane reaches Modesto, 

there are a variety of possible solutions, as all of the planes’

speeds are relative to one another. However, not all solutions 

are viable, given weather conditions that block certain 

pathways, and the inability to speed up planes (only decrease 

speed from a maximum of 600 knots). 
Aeronautics for Introductory Physics: Constant Velocity 57

Although this might appear quite complex, students are built up from simple to more complex scenarios along the way, and 

are given a variety of helpful hints:

1. Sending a plane along an intermediary branch (i.e. from positions MINAH to OAL, or LIDAT to OAL) adds an additional three nautical miles to the total distance.

2. Speed options decrease by 60 knots each (i.e. from 600 knots to 540 knots). Because the definition of one knot is “one 

nautical mile per hour,” a decrease in speed by 60 knots decreases the distance covered in one hour by one-sixtieth of 

the distance covered per hour (i.e. the distance covered in one minute).

In reference to the example problem in level 3-g above, one possible solution would require the user to :

1. Decrease DAL88’s speed to 540 knots for one minute, in order to increase distance from AAL12 by one more nautical 

mile.

2. Send UAL74 from LIDAT to MOD, and decrease speed to 540 knots for two minutes, or 480 knots for one minute, in 

order to space UAL74 three nautical miles away from DAL88.

This engaging, educational game demonstrates to students not only the complexities of managing relative speeds and 

distances, but demonstrates, on a small scale, how difficult it can be to safely and efficiently manage systems of hundreds of 

planes that might pass through a single airspace sector each day.
58 Aeronautics for Introductory Physics: Constant Velocity

Air Traffic Operations Lab Answering Big Questions About the Future of Air Travel

http://www.nasa.gov/content/air-traffic-operations-lab-answering-big-questions-about-the-future-ofair-travel/#.VHSn5Y29-Rg

Questions for Analysis

• The article references that “algorithms” are used to model air traffic. What is an algorithm?

• Why is computer modeling of air traffic insufficient? Why must human test subjects be used?

NASA Tool Helps Airliners Minimize Weather Delays

http://www.nasa.gov/aero/nasa-tool-helps-airliners-minimize-weather-delays.html#.VHS1e429-Rg

Questions for Analysis

• At the time the article was written, how much did jet fuel cost per gallon?

• By cutting 6.2 minutes of flight time from each of 538 American Airlines flights, how much 

money was saved per flight? How much money was saved in total?

• What values and interests do you think influence what kinds of research get done, in general?
Aeronautics for Introductory Physics: Constant Velocity 59

Interactive Demonstration: Pitot-Static Tube

Using visual clues or global positioning system (GPS) instrumentation, it can be very easy for 

observers to determine the speed of an aircraft with respect to the ground (groundspeed). 

However, as you have seen earlier, what is really more important to aircraft is its speed with 

respect to the surrounding air (airspeed). 

The way that aircraft determine their speed requires the use of small tubes (pitot tubes) that 

point toward the source of the airflow. This tube measures the total pressure (atmospheric 

pressure and pressure resulting from the airflow directed into the tube). Additionally, other 

tubes are open to the air 90 degrees to the airflow, which measures the static pressure 

(atmospheric pressure at the point of the opening, resulting from flow over the aircraft as well 

as other climatic conditions). The difference in these values (total pressure – static pressure) 

results in dynamic pressure. (Note: Dynamic pressure is actually the difference between the 

pressures at two points, and should be mathematically represented with a delta (Δ) when not 

written in its differential form.)

Objectives 

Build a model Prandtl pressure 

tube. Qualitatively determine 

“Bernoulli’s” principle. 

Measure static, dynamic, and 

total pressure using studentdetermined units.

Materials 

Translucent bendy straws 

Translucent cup 

Colored water 

Ruler 

Tape 

Wind source of variable speed 

(fan, hair dryer)

Fluid barometer (optional)

NGSS 

CDI HS-PS2 

SEP 2, 3, 6

CCC 2, 4, 5, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Pitot Tube
60 Aeronautics for Introductory Physics: Constant Velocity

To demonstrate this, help students to construct a very simple static pressure probe. Place a single straw vertically in a cup 

of colored water. Take another straw (or portion of a straw), and use it to blow directly across the opening of the vertical 

straw. When the students blow, the fluid will rise to a particular level. At high air speeds, the fluid rises to a higher level. 

At very high speeds, students will observe the device behave as an atomizer and sprinkle fluid. (Caution: Food coloring 

might stain fabric).

Ask students to qualitatively determine the relationship between 

static pressure and wind speed above the opening. Students will easily 

recognize that as wind velocity increases, static pressure decreases. 

Pedagogically, however, there is a bit more complexity. To understand 

pitot tubes, it is important to understand pressure difference, a concept 

that is best emphasized through and understanding of reservoirs, 

barometers, and manometer. The fact that the fluid rises up in the straw 

when air is blown over it is not just a matter of the fact that pressure was 

decreased across the top of the straw. In fact, it is the air pressure pushing 

down on the fluid in the reservoir (the liquid in the cup) that is greater 

than the pressure inside of the straw. Hence, the fluid height is actually a 

result of a difference in pressures between the two. (Likewise, consider 

offering students the option to use a straw to drink up a soda – it is not the 

“sucking” action that causes the fluid to flow, but the pressure from the air 

on the surface of the soda that pushes it down and up into the vacuum of 

the mouth).

The fact that atmospheric pressure has an 

effect can be demonstrated with a barometer. 

Barometers are traditionally produced by 

filling a tube completely with a fluid (such as 

mercury, because of its high density), and then 

inverting the tube into a reservoir. A vacuum is 

created inside of the top of the tube, because the 

atmosphere can only push up a certain quantity 

of it in the tube. As atmospheric pressure 

changes, more or less fluid gets pushed into the 

tube. The reading of the relative height is then 

converted into units of atmospheric pressure.

Alternatively, a manometer can be used to 

measure pressure difference as well. Although, 

like barometers, manometers have one (or 

two) ends that are open to the surrounding 

atmosphere, they do not need to be. Manometers 

can be used to measure the pressure of a gas 

relative to another gas that does not need to be 

the atmosphere.
Aeronautics for Introductory Physics: Constant Velocity 61

Pitot-static tubes work on the principle of difference of pressure to determine the relative airspeed. In the case of the straw 

demonstration, speed can be inferred based upon how high the water goes up the straw (which is a result of the difference 

between the pressure inside of the straw and around the straw on the surface of the water). In the case of pitot-static tubes, 

pressure is compared in a tube facing directly into the wind (total pressure), to the pressure at a tube that points 90 degrees 

to the wind (static pressure). If the static pressure is MUCH lower than the total pressure, then the airspeed is very high. If 

the static pressure is a LITTLE BIT lower than the total pressure, then the airspeed is very low. If the two pressure values are 

the same, then the airspeed is zero (and, hopefully, the aircraft is at rest on the ground!)

Next, allow students to fashion a model pitot-static tube using the following worksheet.

.
62 Aeronautics for Introductory Physics: Constant Velocity

Teacher Answer Key: Pitot-Static Tube

Provide students with materials, and encourage them to make a model pitot tube and then pitot-static tube. The simplest 

pitot-static tube can be fashioned out of a cup, water, and some straws. Students might decide to make their pitot-static tubes 

similar to manometers (with a connected reservoir) or with an open reservoir, such as a cup open to the atmosphere. 

Encourage students to demonstrate their understanding by asking them to do the following:

1. Blow air directly over a static pressure tube (vertical tube inserted into 

water, but not resting against the bottom of the cup). What 

happens? Why do you think this happens? 

As air is blown across the tube, the fluid level settles at a level higher 

than it was before. Sometimes, the fluid from the reservoir even leaves 

the top of the tube and sprays all over. This is because the pressure 

outside of the tube around the reservoir (atmospheric pressure) is 

greater than the pressure inside of the vertical tube. Hence, the 

pressure inside of the tube must decrease when the wind blows over it.

2. Demonstrate how relative speed of air 

blowing over the tube can be determined. 

Use a wind source with a high or low 

setting to show how the device works.

As wind blows over the Static tube, its 

fluid level will settle at a higher level 

than before (lower pressure than before). 

As wind blows into the pitot tube, its fluid level will settle at a level lower than 

before (higher pressure than before).

3. Identify the variables that influence the 

actual static pressure. 

Atmospheric pressure and wind speed 

are the greatest contributors. Wind 

speed can also be influenced by regional factors, including 

turbulent flow that results from aircraft shape or climactic changes. 

4. Build a model pitot-static tube with the available materials. Again, blow 

air directly across the Static tube and into the pitot tube. What happens? 

Why?

The fluid in the pitot tube settles lower than the reservoir level (because 

the pressure is higher inside of the tube), and the fluid in the Static tube 

settles higher than the reservoir level (because the pressure is lower inside 

of the tube). If this cannot be seen, either because the straws are opaque 

or it is difficult to see the fluid, then have students listen to the sound produced in the tubes. As students force faster air over the pitot tube, the pitch 

decreases. As they force faster air over the Static tube, the pitch increases. 

The change in pitch is a result of the resonance of tubes of different length, 

which depends upon the level of the fluid in the tube.
Aeronautics for Introductory Physics: Constant Velocity 63

5. Identify the variables that influence the total pressure.

All of the above variables influence total pressure. However, wind speed has a much greater impact on the pitot tube 

because particles are actually being forced into the tube.

6. Explain how the changes in fluid levels could be quantified and interpreted to make meaning of dynamic pressure. 

(Note: Allow students to look at a glass barometer. They will notice that the pressure gradations are linear. Therefore, 

for an incompressible fluid, twice the pressure results in twice the chance in height of the fluid). 

Fluid levels could actually be measured with a small ruler. Large changes in fluid height mean that there has been a 

large change in pressure, and vice versa. Total pressure (at the pitot tube) could be subtracted from the static pressure 

(at the static tube) to determine a relative fluid height representing the dynamic pressure. While the actual value for 

dynamic pressure might not be easily calculated, it should be relatively easy to determine if the dynamic pressure is 

greater or less than the static pressure.

7. Describe the benefits/disadvantages to using a fluid reservoir that is open to the atmosphere, versus connecting the two 

tubes to make a manometer. 

Opening the fluid reservoir to the atmosphere results in a third pressure value that is likely a known value (if an 

atmospheric barometer is available). When using a manometer that is closed to the atmosphere, it is possible to know 

relative pressure (how many times greater the pressure of one gas is compared to the other), but actual pressure cannot 

be known without knowing the actual pressure of at least one of the gasses.

8. Explain why it is vitally important to pilots to know their airspeed (and not just their groundspeed). 

Airspeed is what provides the lift. If an airplane is traveling at exactly the same velocity as the air around it, its relative 

speed is zero, and no lift will be produced.
64 Aeronautics for Introductory Physics: Constant Velocity

Student Worksheet: Pitot-Static Tube

Watch your teacher demonstrate a static pressure tube made from straws.

1. Blow air directly over a static pressure tube (vertical tube inserted into water, but not resting against the bottom of the 

cup). What happens? Why do you think this happens? 

2. Demonstrate how the relative speed of air blowing over the tube can be determined. Use a wind source with a high or 

low setting to show how the device works.

3. Identify the variables that influence the actual static pressure.

4. Build a model pitot-static tube with the available materials. Include a sketch of your design. 

Again, blow air directly across the static tube 

and into the pitot tube. What happens? Why?

5. Identify the variables that influence the total 

pressure.

6. Explain how the changes in fluid levels could 

be quantified and interpreted to make meaning 

of dynamic pressure. (Note: Allow students to look at a glass barometer. They will notice that the pressure gradations 

are linear. Therefore, for an incompressible fluid, twice the pressure results in twice the chance in height of the fluid). 

7. Describe the benefits/disadvantages to using a fluid 

reservoir that is open to the atmosphere, versus 

connecting the two tubes to make a manometer.

8. Explain why it is vitally important to pilots to know 

their airspeed (and not just their groundspeed). 
Aeronautics for Introductory Physics: Constant Velocity 65

Inquiry Lesson/Lab: Bernoulli Effect Lab

Developed by Florian Genz, University of Cologne

Most textbooks (elementary through 

college) attribute a large part, if not all 

of, the force of lift directly to 

Bernoulli’s principle. In the simplest 

terms, most books describe that an 

airplane’s lift originates from the idea 

that air must flow faster over a 

cambered airfoil rather than under it, 

resulting in a decrease of pressure on 

top of the airfoil. While this statement 

is not entirely untrue, it is often based 

upon very faulty assumptions, and a 

disproportionate contribution to the 

total lift force. As student progress, they 

will learn more about these common 

misconceptions in the activity “Models 

of Lift.”

In closed systems of incompressible 

fluid, Bernoulli’s Principle states the 

relationship between differences in 

static pressure, ps, and flow velocity, v:

In the previous activity, students will 

have determined that wind speed above 

a static pressure tube influences the 

pressure in the tube. Faster air currents 

result in a lower pressure than slower air currents.

Objectives 

Build a cambered airfoil. 

Measure the airfoil’s pressure 

foil in varying situations.

Materials 

Thin and transparent straws 

Transparent container 

Colored Water

Ruler 

Clay 

Tape 

Cardboard 

Cord wire 

Suspension material 

Large box fan 

Sharp knife or blade

NGSS 

CDI HS-PS2 

SEP 2, 3, 4, 7

CCC 1, 2, 4, 5, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Bernoulli Effect Lab
66 Aeronautics for Introductory Physics: Constant Velocity

Again however, Bernoulli deals with incompressible fluids (such as most liquids), not gases. Additionally, a closed tube 

system is very different from an airplane’s wing. For example, holding cardboard horizontally over or under the wing to 

simulate a “closed system” can change the pressure profile dramatically. Also winglets (cardboard on the side edges) make 

the pressure profile more Venturi-tube situation-like (https://commons.wikimedia.org/wiki/File%3AVenturiFlow.png By user: 

ComputerGeezer and Geof (Modified from :de:Bild:Venturirohr.jpg) CC-BY-SA-3.0, via Wikimedia Commons).

Objective: Qualitatively or quantitatively determine the pressure profiles of a cambered airfoil.

The following instructions are for a low-tech, qualitative approach to this lab. Using precision laboratory equipment, however, 

it is possible to get a much deeper understanding of the relationship between pressure profiles due to Bernoulli effects, and 

the total impact on lift. Reference Leybold Lesson P1.8.6.3 in the “Additional Resources” at the end of this document for 

an example high-tech version of a similar lab. By observing the force generated from the pressure profile of the airfoil, and 

comparing it to total lift observed on an electronic balance, it becomes possible to see that the Bernoulli-based path-length 

explanation has very little effect on total lift.

For the low-tech version:

To prepare a cambered airfoil, use a firm piece of cardboard and use a hole-puncher to make holes for the straws. To get 

a pressure profile for your cambered airfoil, pressure must be measured both above and below the airfoil at given chord 

thicknesses. (At least four straws should point “up,” and four straws at the same chord thickness should be inserted and then 

inverted back down through the cardboard again. Clay can be used to form the airfoil to the desired shape. See the visual 

instructions below:

Prepare holes in the cardboard for the internal air fluid system. The images below show how the straws will look inside the 

clay airfoil. 
Aeronautics for Introductory Physics: Constant Velocity 67

Form a cambered airfoil with clay. Remove the clay and insert the straws. Break the clay into pieces, as necessary, to form 

around the straws once they have been inserted.

Top, Bottom, and Side Views:

Insert at least two thin rods into the cardboard, and attach string to the front and back ends of the rods so that the airfoil can 

be supported from a ring stand. Cut off excess straws on the bottom and top of the airfoil.
68 Aeronautics for Introductory Physics: Constant Velocity

Prepare the reservoir filled with water and food coloring. Support the straws with an additional piece of cardboard just above 

the reservoir. Adjust the airfoil to the desired angle of attack.

A more high-tech approach to this lab entails the use of an airfoil and precision manometer to measure actual pressures 

differences (in pascals, or Pa) and relative speed at individual points. Both lab materials and an accompanying lab manual 

(http://www.ld-didactic.de/documents/en-US/EXP/P/P1/P1863_e.pdf) can be acquired through Leybold.
Aeronautics for Introductory Physics: Constant Velocity 69

Teacher Worksheet: Bernoulli Effect Lab

Objective: Qualitatively or quantitatively 

determine the pressure profiles of a 

cambered airfoil.

Observe differences in fluid height in each 

pressure tube when the wind source is 

provided at varying speeds.

1. Choose one point along the airfoil’s 

length. Compare the pressure above 

and below the airfoil. Which pressure 

is higher? Why do you think this is the 

case?

Pressures below the airfoil are typically much higher than the points directly 

above them at positive angles of attack. 

This is due to many reasons, but most 

of them track back to the inertia of the air particles. However, the skipping stone analogy oversimplifies as much as the Venturi tube analogy. Most air particles are interacting with each other and very 

few hit the airfoil directly. Interfering effects might be:

• Turbulence (especially for very small scaled models)

• The straws’ holes might be so big that we CANNOT assume laminar flow around them.

2. Prepare to collect data along all points of the airfoil, above and below. Feel free to investigate the 

effect of wind speed, water coloring, etc., on the visibility of pressure difference. Determine how you 

will collect data. Will it be qualitative or quantitative? Provide a rationale for how you will collect the 

most meaningful data.

Student responses will vary. Some differences in pressure could be measured with millimeters, although precision is likely to be low. Students might simply choose to rank pressure values, or use 

non-numeric symbols to represent relative pressures and pressure differences.

3. Decide what kind of set-up you will use to develop a pressure profile. Decide which three different 

variables you will change in the set-up to observe changes in the pressure profiles.

4. Using a single setting on your air source (large box fan, leaf blower, hair dryer, etc.), develop the 

pressure profiles for your airfoil. Be certain to record any constants. 

Trial #1 Constants (list below):

Constants might include angle of attack, camber, air composition, and wind speed.

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference
70 Aeronautics for Introductory Physics: Constant Velocity

5. Choose to modify your setup in any way. (Consider camber thickness, angle of attack, wind speed, 

etc.). How does this change your pressure profile?

Trial #2 Constants (list below):

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference

6. Again, choose to modify your setup in any way. (Consider camber thickness, angle of attack, wind 

speed, etc.). How does this change your pressure profile?

Trial #3 Constants (list below):

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference

7. Share the results of your pressure profiles 

with your classmates. If appropriate, show 

your data as a chart or as a graph. Discuss 

how your results compare. If they are different, determine why.

8. Share your results with your classmates. 

What conclusion(s) can you draw about the pressure differences above and below an airfoil? Provide 

a rationale for your conclusion.

Typically, the pressure below a cambered airfoil is greater than above it. Data should support this. 

However, this is not true if the airfoil is pointed downward, or if the airfoil has an angle of attack of 

zero degrees while being symmetric.

9. The image to the above right shows pressure profiles. Select ONE of your trials from above. Draw a 

sketch of your airfoil, and provide a qualitative sketch of the pressure profile above and below the 

airfoil at the four points you studied.

Student responses will vary.

10. Of your three trials, which airfoil or airfoil setup do you think would be most effective for achieving 

flight? Why?

Student responses will vary.

11. Look at a different kind of vertical pressure profile for the 

entire body of an aircraft. What direction is being measured 

as upward and downward pressure, and how do you know? 

Why do you think that some relative, static pressures are 

positive and why are some negative? Explain.

Pressure near the nose is very high because it is penetrating the air wall. Pressure must be high near the back of the 

plane as well to keep it balanced.
Aeronautics for Introductory Physics: Constant Velocity 71

Student Worksheet: Bernoulli Effect Lab

Objective: Qualitatively or quantitatively 

determine the pressure profiles of a 

cambered airfoil.

Observe differences in fluid height in each 

pressure tube when the wind source is 

provided at varying speeds.

1. Choose one point along the airfoil’s 

length. Compare the pressure above 

and below the airfoil. Which pressure 

is higher? Why do you think this is the 

case?

2. Prepare to collect data along all points of the airfoil, above and below. Feel free to investigate the 

effect of wind speed, water coloring, etc., on the visibility of pressure difference. Determine how you 

will collect data. Will it be qualitative or quantitative? Provide a rationale for how you will collect the 

most meaningful data.

3. Decide what kind of set-up you will use to develop a pressure profile. Decide which three variables 

you will modify in the set-up to observe changes in the pressure profiles.

4. Using a single setting on your air source (large box fan, leaf blower, hair dryer, etc.), develop the 

pressure profiles for your airfoil. Be certain to record any constants. 

Trial #1 Constants (list below):

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference

5. Choose to modify your set-up in any way. (Consider camber thickness, angle of attack, wind speed, 

etc.). How does this change your pressure profile?

Trial #2 Constants (list below):

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference
72 Aeronautics for Introductory Physics: Constant Velocity

6. Again, choose to modify your set-up in any way. (Consider camber thickness, angle of attack, wind 

speed, etc.). How does this change your pressure profile?

Trial #2 Constants (list below):

Point 1 Point 2 Point 3 Point 4

Pressure Above

Pressure Below

Pressure Difference

7. Share the results of your 

pressure profiles with your 

classmates. If appropriate, 

show your data as a chart or 

as a graph. Discuss how your 

results compare. If they are 

different, determine why.

8. What conclusion(s) can you draw about the pressure differences above and below an airfoil? Provide 

a rationale for your conclusion.

9. The image to the above right shows pressure profiles. Select ONE of your trials from above. Draw a 

sketch of your airfoil, and provide a qualitative sketch of the pressure profile above and below the 

airfoil at the four points you studied.

10. Of your three trials, which airfoil or airfoil setup do you think would be most effective for achieving 

flight? Why?

11. Look at a different kind of vertical pressure 

profile for the entire body of an aircraft. What 

direction is being measured as upward and downward pressure, and how do you know? Why do 

you think that some relative, static pressures are 

positive and why are some negative? Explain.
Aeronautics for Introductory Physics: Constant Velocity 73

Problem Set: Bernoulli’s Equation and Pitot-Static Tubes

from Beginner’s Guide to Aeronautics

BACKGROUND

(Note: This background is most appropriate 

after students have been introduced to these 

concepts through constructivist and/or 

inquiry activities.)

In the 1700s, Daniel Bernoulli investigated 

the forces present in a moving fluid. 

This slide shows one of many forms of 

Bernoulli’s equation. The equation appears 

in many physics, fluid mechanics, and 

airplane textbooks. The equation states that 

the static pressure ps in the flow plus the 

dynamic pressure, one half of the density ρ

times the velocity v squared, is equal to a 

constant throughout the flow. We call this 

constant the total pressure PT of the flow.

As discussed on the gas properties page, there are two ways to look at a fluid; from the large, macroscale properties of the 

fluid that we can measure, and from the small, microscale of the molecular motion and interaction. On this page, we will 

consider Bernoulli’s equation from both standpoints.

Macro Scale Derivation

Thermodynamics is the branch of science which describes the macro scale properties of a fluid. One of the principle results 

of the study of thermodynamics is the conservation of energy; within a system, energy is neither created nor destroyed but 

may be converted from one form to another. We shall derive Bernoulli’s equation by starting with the conservation of energy 

equation. The most general form for the conservation of energy is given on the Navier-Stokes equation page. This formula 

includes the effects of unsteady flows and viscous interactions. Assuming a steady, inviscid flow we have a simplified 

conservation of energy equation in terms of the enthalpy of the fluid:

where hT is the total enthalpy of the fluid (hT1 is the total enthalpy of the fluid before, and hT2 is the total enthalpy of the fluid 

after”), q is the heat transfer into the fluid, and Wsh is the useful work done by the fluid.

Assuming no heat transfer into the fluid, and no work done by the fluid, we have:

From the definition of total enthalpy:

where e is the internal energy, P is the pressure, V is the specific volume, and v is the velocity of the fluid. From the first law 

of thermodynamics if there is no work and no heat transfer, the internal energy remains the same:

The specific volume is the inverse of the fluid density ρ:
74 Aeronautics for Introductory Physics: Constant Velocity

Assuming that the flow is incompressible, the density is a constant. Multiplying the energy equation by the constant 

density:

This is the simplest form of Bernoulli’s equation and the one most often quoted in textbooks. If we make different 

assumptions in the derivation, we can derive other forms of the equation.

It is important when applying any equation that you are aware of the restrictions on its use; the restrictions usually arise in 

the derivation of the equation when certain simplifying assumptions about the nature of the problem are made. If you ignore 

the restrictions, you may often get an incorrect answer from the equation. For instance, this form of the equation was derived 

while assuming that the flow was incompressible, which means that the speed of the flow is much less than the speed of sound. 

If you use this form for a supersonic flow, the answer will be wrong.

Molecular Scale Derivation

We can make another interpretation of the equation by considering the motion of the gas molecules. The molecules within 

a fluid are in constant random motion, colliding with each other and with the walls of an object in the fluid. The motion of 

the molecules gives the molecules a linear momentum and the fluid pressure is a measure of this momentum. If a gas is at 

rest, all of the motion of the molecules is random and the pressure that we detect is the total pressure of the gas. If the gas is 

set in motion or flows, some of the random components of velocity are changed in favor of the directed motion. We call the 

directed motion “ordered,” as opposed to the disordered random motion.

We can associate a pressure with the momentum of the ordered motion of the gas. We call this pressure the dynamic pressure. 

The remaining random motion of the molecules still produces a pressure called the static pressure. At the molecular level, 

there is no distinction between random and ordered motion. Each molecule has a velocity in some direction until it collides 

with another molecule and the velocity is changed. But when you sum up all the velocities of all the molecules you will 

detect the ordered motion. From a conservation of energy and momentum standpoint, the static pressure plus the dynamic 

pressure is equal to the original total pressure in a flow (assuming we do not add or subtract energy in the flow). The form of 

the dynamic pressure is the density times the square of the velocity divided by two.

Applications of Bernoulli’s Equation

The fluids problem shown on the image on the prior page is low speed flow through a tube with changing cross-sectional area. 

For a streamline along the center of the tube, the velocity decreases from station one to two. Bernoulli’s equation describes 

the relation between velocity, density, and pressure for this flow problem. Since density is a constant for a low speed problem, 

the equation at the bottom of the slide relates the pressure and velocity at station two to the conditions at station one.

Along a low speed airfoil, the flow is incompressible and the density remains a constant. Bernoulli’s equation then reduces 

to a simple relation between velocity and static pressure. The closest streamlines follow the surface of the airfoil. Since the 

velocity varies along the streamline, Bernoulli’s equation can be used to compute the change in pressure. The static pressure 

integrated along the entire surface of the airfoil gives the total aerodynamic force on the foil. This force can be broken down 

into the lift and drag of the airfoil.

Bernoulli’s equation is also used on aircraft to provide a speedometer called a pitot-static tube. A pressure is quite easy to 

measure with a mechanical device. In a pitot-static tube, we measure the static and total pressure and can then use Bernoulli’s 

equation to compute the velocity.
Aeronautics for Introductory Physics: Constant Velocity 75

Problem Set from Beginner’s Guide to Aeronautics:

Student Worksheet

Teacher Key

Solve each problem. Make sure all units agree! (Note: For practical purposes, velocity is measured in KNOTS. One knot = 

1.69 feet/second.)

1. Find the total air pressure detected by the pitot tube for a Cessna 182 airplane flying at a pressure altitude of 10,000 feet 

(static pressure of 1455.6 lb/ft2 ) and a speed of 120 knots if the air density is 0.00162 slugs/ft3.

2. Find the difference in pressure (pt-ps) for the same conditions described above if the airplane speeds up to 160 knots.

3. If the total air pressure for the same Cessna 182 is measured at 1478.7 lb/ft2, what is the new velocity in knots of the 

airplane?
76 Aeronautics for Introductory Physics: Constant Velocity

Discovery Lab: Airplane Dynamics: Engine Thrust, Braking, 

and Lift

Joe Pizzo. The Physics Teacher 26, 122 (1988).

Joe Pizzo describes the use of a modified two-dimension low-friction fan cart to demonstrate 

the effect of the Bernoulli principle on lift, as well as the effect of engine breaking. Objectives 

Observe and model engine 

thrust, breaking, and lift.

Materials 

Modified fan-cart 

(moveable in three directions), 

or water tray with floating 

surfaces, 

Dry ice 

Balsa / foam airfoil

Aluminum flashing

NGSS 

CDI HS-PS2-1

SEP 2, 3, 6

CCC 2, 5

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Velocity

ACTIVITY

Airplane Dynamics
Aeronautics for Introductory Physics: Constant Velocity 77

A standard fan cart can be used to observe Activity #1 and 

Activity #2. Activity #3, however, requires the significant 

modification of the fan cart so that it can move in two 

dimensions. (This might be a great engineering challenge 

for students). Some options include mounting the fan onto 

a set of low-friction ball-bearings that can slide across the 

table top, placing it onto a piece of floating Styrofoam in 

a pool of water, using floating airfoils on a plate of oil, or 

balancing them on a small slab of dry ice while being 

blown on by a uniform air stream.

With a few simple instructions, students can modify the 

placement of airfoils in front of the fan cart, and allow 

Activity #1 - Thrust: Students observe the effect of 

thrust on a low-friction cart. Students will notice that the 

cart will accelerate due to the constant net force.

Activity #2 – Engine Braking: Aluminum flashing or a 

cheap, thin aluminum cylinder can be split and then bent 

open to create model air brakes. As the air blows into the 

air brakes, the air is directed backwards, causing the cart 

to move in the same direction as the wind from the fan. 

CAUTION: The edges of aluminum flashing are very 

sharp! Use appropriate hand and eye protection when 

cutting or handling aluminum flashing.

Activity #3 – Lift (Bernoulli effects only): Students 

can fashion a cambered airfoil from Styrofoam or balsa 

wood. By placing a cambered airfoil with no angle of 

attack in front of a low-friction fan cart, students should 

observe the cart to move to be a net sideways force on 

the cart, causing it to move at an angle. Bernoulli is 

the primary effect only when angle of attack is zero, 

however. Even then, students will observe that effects 

from Bernoulli are a very small overall contributor to lift.
78 Aeronautics for Introductory Physics: Uniform Acceleration

Uniform Acceleration

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
Aeronautics for Introductory Physics: Uniform Acceleration 79

Literary/Data Analysis: Prepare for Landing

Gregory A. DiLisi and Richard A. Rarick. The Physics Teacher, 45, 300 (2007).

The authors of the article 

Prepare for Landing decided to 

make a quick analysis of their 

plane’s motion on their descent 

into their destination airport 

on an Airbus A320. A crude 

accelerometer was constructed 

by hanging a homemade 

pendulum bob on the surface 

of the window. (Note: This 

activity requires the use of a 

worksheet based upon an in-air 

problem while flying. Students 

should not be requested to 

complete a similar in-flight 

experience without the consent 

of the airline attendants).

This activity is a “contextualized verification problem,” in which students are asked to 

compare the actual angle of a pendulum bob during a constant acceleration descent to the 

theoretically calculated angle of a pendulum bob. Two images can be seen superimposed 

upon one another: the angle of the pendulum while in the final stages of descent, exactly 

eight minutes before touchdown, and the angle of the pendulum while at rest on the ground. 

This angle can be fairly easily measured and compared to the estimated angle based upon the 

Airbus A320’s spec sheet for a typical landing.

Objectives 

Given the cruising speed of 

the plane, angle of descent, 

landing speed, and time until 

landing, compare the calculated angle of deflection to 

the actual measured angle of 

deflection.

Materials 

Protractor or computer

NGSS 

CDI HS-PS2 

SEP 1, 2, 4, 5, 7

CCC 3, 4

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Uniform Acceleration

ACTIVITY

Prepare for Landing
80 Aeronautics for Introductory Physics: Uniform Acceleration

Teacher Answer Key: Prepare for Landing 

A passenger in an Airbus A320 takes a photograph of a pendulum bob 

suspended from a window during the constant acceleration of the final 

descent of an airplane, as well as when the airplane is at rest. The pilot of 

the aircraft announces that the final descent has begun, and that the plan will 

land in exactly eight minutes.

Based upon the specifications offered for the Airbus A320 at cruising 

altitude, descent, and landing, answer the following questions in order to 

theoretically estimate the angle measured by the passenger.

Cruising Speed Descent Angle of Attack Landing Speed

Airbus A320 Spec’s 525-575 mi/hr 135-155 mi/hr

Average estimates for this problem 550 mi/hr 3.3° 145 mi/hr

1. What two factors would influence the angle of the pendulum recorded during the constant acceleration and descent 

of the airplane? 

The angle of attack of the airplane, as well as the “downand-forward” acceleration of the airplane’s propulsion 

system.

2. Assuming an average estimate of the cruising speed, calculate the horizontal acceleration of the plane from the 

beginning of its final descent to its landing.

3. Calculate the vertical acceleration of the plane from the beginning of its descent to its landing.

4. Consider that the pendulum bob is inside an airplane that 

is accelerating. Draw a force diagram for the pendulum 

bob, using the airplane as the “flat” frame of reference.

The angle of the tension force is a sum of both the angle 

of attack (which we assume to be 3.3, according to the 

Airbus A320’s specifications), and the “down-and-to-theright” acceleration of the plane’s propulsion system. 
Aeronautics for Introductory Physics: Uniform Acceleration 81

5. Now, draw an acceleration vector diagram for the pendulum bob. Again, use the airplane as the “flat” frame of 

reference, and neglect the angle of attack in your frame of reference. 

6. Break up the plane’s acceleration vector into its components. (Note: The original 

article suggests the use of “pseudoforces” for solving this problem. For younger 

students, an approach with acceleration vectors is likely more appropriate).

7. Using vector addition, graphically demonstrate how these acceleration vectors can be summed. Draw in the resultant 

acceleration.

8. Calculate the angle between the two acceleration vector components caused by the Earth and due to the airplane.

9. Calculate the sum total theoretical angle of deflection of the pendulum due to the angle of attack of the airplane as well 

as the accelerated frame of reference.

10. Compare the theoretical angle of deflection of the pendulum to the 

actual angle of deflection from the image below:

The angle cited by the author, using MS WORD’s “Format Autoshape” 

function to create a right triangle and to measure the lengths, resulted 

in a measured deflection of 5.9° ± 0.9°.
82 Aeronautics for Introductory Physics: Uniform Acceleration

Prepare for Landing: Student Worksheet

A passenger in an Airbus A320 takes a photograph of a pendulum bob 

suspended from a window during the constant acceleration of the final 

descent of an airplane, as well as when the airplane is at rest. The pilot of 

the aircraft announces that the final descent has begun, and that the plan will 

land in exactly eight minutes.

Based upon the specifications offered for the Airbus A320 at cruising 

altitude, descent, and landing, answer the following questions to 

theoretically estimate the angle of the pendulum eight minutes before 

landing, as measured by the passenger.

Cruising Speed Descent Angle of Attack Landing Speed

Airbus A320 Spec’s 525-575 mi/hr 135-155 mi/hr

Average estimates for this problem 550 mi/hr 3.3° 145 mi/hr

1. What two factors would influence the angle of the 

pendulum recorded during the constant acceleration and 

descent of the airplane?

2. Assuming an average estimate of the cruising speed, calculate the horizontal acceleration of the plane from the beginning of its final descent to its landing.

3. Calculate the vertical acceleration of the plane from the beginning of its descent to its landing.

4. Consider that the pendulum bob is inside an airplane that is accelerating. Draw a force diagram for the pendulum bob, using the 

airplane as the “flat” frame of reference. 

The angle of the tension force is a sum of both the angle of attack 

(which we assume to be 3.3, according to the Airbus A320’s specifications), and the “down-and-to-the-right” acceleration of the plane’s 

propulsion system. 
Aeronautics for Introductory Physics: Uniform Acceleration 83

5. Now, draw an acceleration vector diagram for the pendulum bob. Again, use the airplane as the “flat” frame of reference, and neglect the angle of attack in your frame of reference. 

6. Break up the plane’s acceleration vector into its components.

7. Using vector addition, graphically demonstrate how these acceleration vectors can be summed. Draw in the resultant 

acceleration.

8. Calculate the angle between the two acceleration vector components caused by the Earth and due to the airplane.

9. Calculate the sum total theoretical angle of deflection of the pendulum due to the tilt of the airplane as well as the 

accelerated frame of reference.

10. Compare the theoretical angle of deflection of the pendulum to the 

actual angle of deflection from the image below:
84 Aeronautics for Introductory Physics: Uniform Acceleration

Shhhh! Keep it Down Please!

http://www.nasa.gov/topics/aeronautics/features/aircraft_noise.html#.VHOhCI29-Rg

Analysis Questions:

• Takeoff and landing are often the noisiest part of flight. Based on the article you read, why is 

this the case?

• Why is aircraft noise a problem?

• The article mentioned N+1, and N+2 aircraft. How will N+1 aircraft be different from what 

we have today? How will N+2 aircraft be even further improved?

• What are some additional ideas about ways you think noise reduction can be improved?
Aeronautics for Introductory Physics: Uniform Acceleration 85

Application Lesson/Lab: Mobile Accelerometers

The advent of mobile devices, such as tablets and smartphones, allow for a wide array of data 

collection and analysis opportunities. Ideally, students can collect their own data during an 

airplane ride in order to make a variety of qualitative and quantitative analyses. (Note: Please 

remind students that they must always follow FAA and local regulations regarding the use 

of mobile devices in airplanes, especially during takeoff and landing.) This overview will 

provide the teacher with some sample data and questions that could be proposed to students.

There are a number of apps from iOS and Android marketplaces that allow mobile devices to 

display and record linear acceleration and/or g-force. (g-force is the ratio of the strength of 

acceleration felt compared to acceleration due to gravity, also represented as a/g or FN/Fg. A 

thorough discussion of this concept should be undertaken before attempting to use such data). 

In order to understand how accelerometers in mobile devices work, have students complete 

the following activities.

Objectives 

Analyze data for acceleration 

of an airplane.

Materials 

Mobile devices with accelerometer apps, example data

NGSS 

CDI HS-PS2-1

SEP 1-7

CCC 2, 7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Uniform Acceleration

ACTIVITY

Mobile 

Accelerometers
86 Aeronautics for Introductory Physics: Uniform Acceleration

Teacher Answer Key: Mobile Accelerometers

1. Download and open an accelerometer app on your device. For the purposes 

of this activity, Android’s “Physics Toolbox Accelerometer” is used, but there 

are many available apps for iPhone and Android that display acceleration and 

g-forces. 

2. Click on the “Acceleration” button in the top menu -- this will change your 

graph reading from g-force to linear acceleration. You should now have a 

graph that reads acceleration versus time.

3. Lay your device flat on the table. What value do all of the x, y, and z accelerations read? Why?

They all read zero. If the device is not moving (no matter where it is placed), 

then accelerations in all dimensions will read zero.

4. Try shaking your device in all three dimensions. Identify which dimensions are x, y, and z, and label them as such. Also, 

identify which directions are positive and negative.

5. Pick up your device, still holding it perpendicular to the floor so you can see the screen. Choose ONE dimension, and 

jump! Record what the acceleration graph looks like for that dimension only. Label the MAX and MIN acceleration for 

each peak. Avoid putting any data that is irrelevant to the jump.
Aeronautics for Introductory Physics: Uniform Acceleration 87

6. You should have 2 maxima, and 1 minima. Explain what is happening at each of these points:

1st Max: This is the acceleration of the initial jump. This first peak is being recorded as the person is pushing off of the 

ground (while in contact with it).

Min: This is the acceleration due to gravity – and it should be somewhere around -9.8 (m/s)/s, which occurs while the 

person is in free fall (from the moment the feet come off of the floor, and then just before touching the ground again).

2nd Max: This is the acceleration of the “land.”

7. Determine the mass of your mobile device (or of your body) to find out the net force that is acting to cause these accelerations...if you are already familiar with Newton’s second law of motion.

1st Max: Fnet = ma. Once students determine the mass of the object of their choosing (device or body, in kg), they can 

simply multiply it times the max and min values of the acceleration displayed on the graph.

Min:

2nd Max:

8. Now, go back and forth between the “accelerometer” and “g-force” settings. G-force is a measure of how many times 

stronger you feel a force acting on you, as compared to the pull of gravity on your body. Hence, a g-force of one is 

equal to your weight, while a g-force of two means that you feel twice as heavy. A zero g-force means that you feel 

weightless. (However, near the surface of the earth, your weight doesn’t actually change at all).

Now that you know how to understand accelerometer graphs, try to make sense of the following set of data, recorded 

while in an aircraft.

9. Which way was the face of the mobile device oriented in the recorder’s hands? 

How do you know?

The device must have been lying flat with the face up, because the g-force is 

positive in the z-direction.

10. What is the average g-force acting in each dimension? (Why does one dimension always appear to hover around 1g?)

Initially, the g-force in the z-direction was +1, which would be expected, 

because we feel our weight in the vertical direction. The other two directions 

have a g-force around zero, initially, because there is no “weight” experienced in either of those directions.

11. Assuming that the recorder placed the device on his/her lap in such a way that 

it could be correctly viewed, is this a graph of takeoff or landing? How do you 

know?

Takeoff. The recorder is experiencing a force from back to front, or “forward-moving.” Also, there appears to be a “blip” later in the z-direction just 

above +1 g-force, suggesting that the recorder feels “heavy.” This would only 

be expected as the plane accelerates upward.
88 Aeronautics for Introductory Physics: Uniform Acceleration

Student Worksheet: Mobile Accelerometers

Objective: Analyze data for acceleration of an airplane.

1. Download and open an accelerometer app on your device. For the purposes of 

this activity, we will look at Android’s “Physics Toolbox Accelerometer,” but 

there are many available apps for iPhone and Android that display acceleration 

and g-forces. 

2. Click on the “Acceleration” button in the top menu – this will change your 

graph reading from g-force to linear acceleration. You should now have a graph 

that reads acceleration versus time.

3. Lay your device flat on the table. What value do all of the x, y, and z accelerations read? Why?

4. Try shaking your device in all three dimensions. Identify which dimensions are x, y, and z, and label them as such. Also, 

identify which directions are positive and negative.

5. Choose ONE dimension, and jump! Record what the acceleration graph looks like for that dimension only. Label the 

MAX and MIN acceleration for each peak. Avoid putting any data that is irrelevant to the jump.
Aeronautics for Introductory Physics: Uniform Acceleration 89

6. You should have 2 maxima, and 1 minima. Explain what is happening at each of these points:

1st Max: 

Min: 

2nd Max: 

7. Determine the mass of your mobile device (or of your body) to find out the net force that is acting to cause these accelerations...if you are already familiar with Newton’s second law of motion.

1st Max: 

Min: 

2nd Max: 

8. Now, go back and forth between the “accelerometer” and “g-force” settings. G-force is a measure of how many times 

stronger you feel a force acting on you, as compared to the pull of gravity on your body. Hence, a g-force of one is 

equal to your weight, while a g-force of two means that you feel twice as heavy. A zero g-force means that you feel 

weightless. (However, near the surface of the Earth, your weight doesn’t actually change at all).

Now that you know how to understand accelerometer graphs, try to make sense of the following set of data, recorded 

while in an aircraft.

9. Which way was the face of the mobile device oriented in the recorder’s hands? 

How do you know?

10. What is the average g-force acting in each dimension? (Why does one dimension always appear to hover around one g?)

11. Assuming that the recorded placed the device on his/her lap in such a way that 

it could be correctly viewed, is this a graph of takeoff or landing? How do you 

know?
90 Aeronautics for Introductory Physics: Uniform Acceleration

NASA-Pioneered Automatic Ground-Collision Avoidance System Operational

http://www.nasa.gov/centers/armstrong/Features/Auto-GCAS_Installed_in_USAF_F-16s.html#.

VHS6So29-Rg

Analysis Questions:

• What is the purpose of Automatic Ground-Collision Avoidance systems?

• Considering your understanding of acceleration, propose an hypothesis for how an airplane’s 

computer determines the time before impact with the ground.

• In this particular bit of NASA research, an Android smartphone was used. What kinds of sensors on an Android smartphone might provide information about altitude and location? 
Aeronautics for Introductory Physics: Terminal Velocity 91

Terminal Velocity

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
92 Aeronautics for Introductory Physics: Terminal Velocity

Discovery Lab (Raw): Drag Stations Lab

Terminal velocity is something that most students associate with parachutes and other falling 

objects – not airplanes. However, a study of terminal velocity helps students to understand 

the difference between objects in idealized, frictionless environments, and objects that are 

significantly influenced by drag, as is the case of flight. The majority of students believe that, 

under all circumstances, heavier objects fall more quickly than lighter objects. In contrast, 

students are often erroneously taught in school that all objects fall at the same rate. 

This discovery lab should first be initiated with a classic interactive demonstration dealing 

with two similarly-shaped and similarly-sized, aerodynamic objects (such as balls), but with 

significantly different masses. Ask students to predict which object will hit the ground first. 

Demonstrate that the difference is negligible, and ask them to keep the demonstration in mind 

as they work through the Drag Stations Lab.

Following the stations lab, ask students to again return to the demonstration of the 

two aerodynamic objects (the balls) that appeared to land at the same time. Students 

should quickly recognize that this is not a “law” that holds true under all circumstances. 

Aerodynamic effects, such as fluid viscosity, shape, size, and mass all influence the rate of fall 

when there is drag. In contrast, when drag is negligible, none of these factors have any effect 

on the fall times, and objects only continue to keep falling faster. 

This activity is particular important with regard to a discussion of air friction present in 

the atmosphere. Air density, pressure, and temperature all change in various layers of the 

atmosphere. As a result, the design of flying aircraft must be different when considering flight 

at differing altitudes or climatic conditions. In particular, airplane design becomes critical 

when considering hypersonic or supersonic flight near the edge of the Earth’s atmosphere, or 

in planes that actually enter into space.

Objectives 

Identify the variables that 

influence the drag and time of 

fall on various objects.

Materials 

Electronic balance 

Steel balls 

Small metal BBS

Large ball bearings 

Carpet 

Coins of different masses, 

Baseball,

Clay or dough, 

Meter stick, 

Motion detector, 

Photogate, 

Picket fence, 

Regular coffee filters,

Industrial-sized coffee filters 

(or muffin/candy cups), 

Test tubes or graduated 

cylinder filled with water, 

soap, etc., small masses, 

Pipette (optional), 

Cup of water (optional)

NGSS 

CDI HS-PS2 

SEP 3, 4, 6, 7

CCC 2, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

Drag Stations Lab
Aeronautics for Introductory Physics: Terminal Velocity 93

Teacher Answer Key: Drag Stations Lab

Station 1: Mass

Provide two aerodynamic objects of the same size and shape, but of significantly different densities. If possible, acquire a 

large steel ball bearing and an equivalent-sized sports ball, bouncy ball, ball of cork, etc. Ensure that both of the objects 

accelerate at about the same rate. If heavy steel balls are used, designate a “fall area” on the ground and ensure that 

students keep their body parts away from that area, and remind students that even heavy balls can rebound and cause injury. 

Also, protect the floor from collision by padding it with a piece of carpet or a towel.

Alternatively, have students use coins of significantly different masses and drop them instead. Students can do this more 

efficiently by using a ruler to push the coins off of the edge of a table at the same time.

1. Have the same person hold a steel ball and the baseball in one hand. Drop them carefully onto a protected, padded 

surface. What do you notice about their fall times?

Students should notice that their fall times are approximately the same. (Encourage students to do this with both of 

their hands, as using two people to do this activity often results in different fall times).

2. Does your observation agree with what you have experienced outside of the classroom? Explain.

Not necessarily. Really lightweight like feathers or leaves don’t fall at the same rate as heavy objects.

Station 2: Velocity and Time

Place a large piece of modeling clay or dough on the table. Have students flatten the dough into a thick pancake so that it 

can accurately record the impressions from ball bearings dropped onto it. Provide students with at least one ball bearing and 

a meter stick. 

1. Hold a steel ball 20 centimeters (cm) above the clay and drop it. Do the same thing again, but at a height of 80 centimeters. Which height resulted in the deepest dent?

The ball at 20 cm made a much smaller dent than the ball dropped from 80 cm. (For more advanced students, consider 

having students measure the volume of the dents. Using non-permeable clay, have students fill the dents with water, 

counting the drops as an approximately uniform unit volume. Place a drop of soap into the water to release surface 

tension. Ask students to compare: How does quadrupling the height affect the volume of the dent? Why?)

2. What factors influence the depth of the dents? (Consider that the balls used each time had the same mass). Explain why 

the dents had different depths.

Balls released from a higher point end up having a greater final velocity than balls released from a lower point. This is 

because the higher ball has more time to speed up on its way down.

Station 3: Motion Detector

Provide students with a computer-based motion detector, 

graphical analysis program, and a large ball (ideally, 

basketballs or similarly-sized balls are best for motion detection). 

Ask students to collect velocity-time graph data and to determine 

the acceleration due to gravity. Students can accomplish this by 

taking the slope of the velocity-time graph. Caution students to 

always place the motion detector above the ball (so as to avoid 

crushing it), and to only get the slope of the relevant portion of 

the graph.

1. Hold the motion detector above the suspended basketball. 

Determine the acceleration of the basketball as it falls using 

the velocity graph.

Acc = (The value should be somewhere around 9.8 m/s/s, although it might be less 

for balls with great surface areas).
94 Aeronautics for Introductory Physics: Terminal Velocity

2. Explain how you accomplished this.

Students should find the linear portion of the velocity graph, and use the computer program to get the slope of that 

region.

Station 4: Photogate

Provide students with a photogate, graphical analysis program, 

and traditional “picket fence” made of alternating clear and 

opaque segments. Typical graphical analysis programs with 

photogate sensors will automatically recognize picket fence data 

(ensure that the settings on the program for the spacings between 

the opaque segments are correct). This will provide students 

with a velocity-time graph, which can be used to get a slope to 

determine acceleration.

1. Determine the acceleration of the picket fence as it falls 

through the photogate.

Acc = (The value should be somewhere around 9.8 (m/s)/s).

2. Explain how you accomplished this.

Students should find the linear portion of the velocity graph, and use the computer program to get the slope of that 

region.

Station 5: Drag Test: Shape

Provide students with multiple coffee filters that can be formed into a 

variety of shapes.

1. Drop a single coffee filter from shoulder height. What do you 

notice about its speed on the way down?

The coffee filter appears to reach a terminal (steady) velocity.

2. Take another coffee filter and change its shape. Explain how its 

falling speed compares to a regular coffee filter.

The coffee filter that has been modified will likely either continue to accelerate, or will achieve a higher terminal velocity than the regular coffee filter.

Station 6: Drag Test: Size

Provide students with regular and industrial sized coffee filters. If industrial sized coffee filters are not available, consider 

having students use similarly-shaped objects (such as peanut butter cup papers, muffin cup papers, etc.), or allowing 

students to modify regular coffee filters to get new filters with the different sizes but same shape. Caution students that no 

matter how size is modified for objects of the same shape, that all mass must be accounted for and equal between the two 

objects in question. Provide students with an electronic balance, if appropriate. For example, if using an industrialized 

coffee filter, students should compare it to a stack of joined coffee filters that make up the same mass as the industrialized 

coffee filter.

1. Drop a set of regular coffee filters (having the same mass as the industrial sized coffee filter) and one industrial sized 

coffee filter. (If you are using muffin or candy paper cups instead of an industrial coffee filter, make sure you use several of the smaller objects stacked together to equal the mass of the larger filter). What is the relationship between size 

and falling speed?

The industrial sized (larger) coffee filter falls much more slowly than the smaller coffee filters of equivalent mass.

2. Explain why their falling speeds are different.

The larger coffee filter has a greater surface area, so it comes into contact with more air particles, resulting in more 

drag.
Aeronautics for Introductory Physics: Terminal Velocity 95

Station 7: Drag Test: Mass

Provide students with sets of coffee filters in groups of 1, 3, and 9.

1. Drop the following: 1 coffee filter, 3 coffee filters, and 9 coffee 

filters. What do you notice about their falling speeds?

The more massive sets of coffee filters fall much faster than the 

single coffee filter.

2. Explain how this situation is different from the two balls at 

Station 1.

The two balls are aerodynamic (it is easy for the air particles to move around them, because of their shape), while the 

coffee filters have a large surface area and tend to “capture” the air more, resulting in greater friction.

Station 8: Drag Test: Viscosity

Provide students with at least two large test tubes filled separately with water and soap (or, different concentrations of 

water-soap mixture). Place a single metal BB/ small ball bearing into each test tube, and seal with a rubber stopper. 

Allow students to observe the ball bearing as it falls through each test tube as they invert it. If necessary, use adjust the 

concentrations of water/soap mixture, as too much water might result in a ball bearing that falls too quickly to observe, and 

pure soap might result in a ball bearing that falls too slowly to observe. In addition, if you are using glass test tubes, caution 

your students to handle the tubes and ball bearings carefully so they don’t break the glass with the ball bearing. Ideally, use 

metal BBs with as small a diameter as possible.

1. Watch the metal ball as it falls down the length of the test tubes filled with water or soap. Compare the time it takes for 

a ball bearing to fall through each liquid.

The ball bearing takes much less time to fall through thin liquids than through thick liquids.

2. How does this demonstration relate to objects falling through the air?

The air also has a “thickness” associated with it. Especially at high speeds, drag becomes a significant factor and 

impacts the object’s motion.

Summary Questions

1. When objects fall, what happens to their velocities?

Most of the time, velocities will increase – at least initially – and then either continue to increase, in the absence of 

drag, or attain its maximum speed – terminal velocity.

2. Under what atmospheric circumstances do size, shape, and mass have an impact on falling speed? Explain.

Whenever there is air friction.

3. Under what atmospheric circumstances do size, shape, and mass NOT have an impact on falling speed? Explain.

Whenever there is NO air friction, and, as a result, no drag.
96 Aeronautics for Introductory Physics: Terminal Velocity

Student Worksheet: Drag Stations Lab

Station 1: Mass

1. Have the same person hold a steel ball and the baseball in one hand. Drop them carefully onto a protected, padded 

surface. What do you notice about their fall times?

2. Does your observation agree with what you have experienced outside of the classroom? Explain.

Station 2: Velocity and Time

1. Hold a steel ball 20 centimeters above the clay and drop it. 

Do the same thing again, but at a height of 80 centimeters. 

Which height resulted in the deepest dent?

2. What factors influence the depth of the dents? (Consider 

that the balls used each time had the same mass). Explain 

why the dents had different depths.

Station 3: Motion Detector

1. Hold the motion detector above the suspended basketball. 

Determine the acceleration of the basketball as it falls using 

the velocity graph. 

Acc = 

2. Explain how you accomplished this.
Aeronautics for Introductory Physics: Terminal Velocity 97

Station 4: Photogate

1. Determine the acceleration of the picket fence as it falls through the photogate. 

Acc = 

2. Explain how you accomplished this.

Station 5: Drag Test: Shape

1. Drop a single coffee filter from shoulder height. What do you 

notice about its speed on the way down?

2. Take another coffee filter and change its shape. Explain how its 

falling speed compares to a regular coffee filter.

Station 6: Drag Test: Size

1. Drop a set of regular coffee filters (having the same mass as the industrial sized coffee filter) and one industrial size 

coffee filter. If you are using muffin or candy cups instead of the industrial coffee filter, make sure you use several of 

the smaller objects stacked together to equal the mass of the larger filter. What is the relationship between size and 

falling speed?

2. Explain why their falling speeds are different.
98 Aeronautics for Introductory Physics: Terminal Velocity

Station 7: Drag Test: Mass

1. Drop the following: 1 coffee filter, 3 coffee filters, and 9 coffee 

filters. What do you notice about their falling speeds?

2. Explain how this situation is different from the two balls at 

Station 1.

Station 8: Drag Test: Viscosity

1. Watch the metal ball as it falls down the length of the test tubes filled with water or soap. Compare the time it takes for 

a ball bearing to fall through each liquid.

2. How does this demonstration relate to objects falling through the air?

Summary Questions

1. When objects fall, what happens to their velocities?

2. Under what atmospheric circumstances do size, shape, and mass have an impact on falling speed? Explain.

3. Under what atmospheric circumstances do size, shape, and mass NOT have an impact on falling speed? Explain.
Aeronautics for Introductory Physics: Terminal Velocity 99

Discovery Lab (Contextual): Drag and Aircraft Design

NASA Aeronautics works to solve the 

many challenges that exist in the nation’s 

airspace system. In fact, every U.S. aircraft 

flying today and every U.S. air traffic 

control tower uses NASA-developed 

technology in some way.

This document includes four very simple 

“station” activities that can be used to teach 

students about some of NASA Aeronautics’ 

themes of ongoing research: namely, 

advancing green aviation, reducing flight 

delays, revisiting supersonic flight, and 

designing future aircraft. Learn more about 

these themes at http://www.nasa.gov/aeronautics

Nearly all of these research themes share a single commonality – an understanding of drag 

and effects on aircraft. In its simplest form, drag is defined as the resistive force on an object 

as the object attempts to move through a fluid (such as air or water). Drag acts opposite to the 

direction of motion of the object that is moving. 

The purpose of these activities is to help students develop a conceptual understanding of drag, 

its causes, and its effect on the motion of an object. Students are also asked to consider, very 

simply, how engineers might design an aircraft to perform a given task (such as fast or slow 

flight, travel at high or low altitudes, or gliding with high or low finesse). 

In fact, NASA engineers are currently working on future aircraft designs that meet many of 

these needs. Because aircraft experience a variety of conditions in flight, engineers are now 

also considering how an aircraft might respond, changing its shape so it can fly in the best 

way possible. To see an example of NASA-designed future aircraft, visit the Down to Earth 

Future Aircraft Gallery: http://www.nasa.gov/content/down-to-earth-future-aircraft-0

Objectives 

Identify the variables that 

influence the drag and time of 

fall on various objects.

Materials 

Four station cards 

Electronic balance, 

Ping-pong balls, 

Golf balls, 

Regular coffee filters,

Industrial-sized coffee filters,

(and/or muffin cups),

Long translucent containers, 

Corn syrup, 

Water, 

Stapler, 

Marbles

NGSS 

CDI HS-PS2 

SEP 3, 4, 6, 7

CCC 2, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

Air Friction Stations 

Lab
100 Aeronautics for Introductory Physics: Terminal Velocity

Objectives

• Define drag.

• Describe the effect of drag on the motion of various objects of different mass, shape, size, and movement 

through fluids of different viscosity.

• Hypothesize some reasons for the cause of drag.

Activities

(Note: In advance of all activities, ensure that all stations are set up as described in the lesson.)

Define Drag

Before beginning with these activities, it is important that students first develop a conceptual understanding of force. Most 

students understand forces to be either a “push” or a “pull,” and while sufficient for this activity, it can be beneficial to first 

provide students with a thorough understanding of force vectors. 

To help students understand drag, provide students with a demonstration that will result in cognitive dissonance – a surprise 

that, for most, will elicit misconceptions about their understanding of objects falling through a fluid (such as air).

Begin by showing students a golf ball and a ping-pong ball. Ask students 

to compare their mass, shape, size, and the kind of fluid that surrounds 

them (air). Students should find that the only major difference between the 

two is that the golf ball is much heavier than the ping-pong ball. Then, ask 

the students to predict the motion of the two balls if they are released from 

the same height at the same time. Most students will say that the heavy 

one will fall faster. Before demonstrating, ask students to prepare to listen 

for the balls as they hit the ground, and to watch for the collision with the 

ground as well. Allow the single demonstrator (teacher or student) to drop 

the two objects from the same height at the same time. Students will often 

be shocked to find that, on average, the balls hit the ground at the same time! 

Then, prepare to show students a single regular coffee filter, and a stack 

of multiple coffee filters. Again, the only major difference between the 

two items is that the single coffee filter weighs much less than the stack 

of coffee filters. Ask students to again predict how their motions will 

be different. Upon dropping the single filter and the set of coffee filters, 

students will see that there is a major difference in the time it takes to drop 

the single filter and the set of coffee filters.

In both cases (golf ball vs. ping-pong ball, and single vs. stack of coffee filters), the only difference between them was the 

mass. So, the question remains: why do the coffee filters behave differently than the balls?

Instead of answering this question outright, ask students to explain what could have caused the single coffee filter to fall so 

slowly. Most older children with an understanding of forces will be able to describe that there was some resistance from the 

air (drag) that was pushing opposite the motion of the falling object. Once this definition has been elaborated, encourage 

students to move to the four stations and to determine the effect of drag on objects of various masses, shapes, sizes, and 

movement through fluids of different viscosity.

Describe the effect of drag on the motion of various objects of different mass, shape, size, and movement through 

fluids of different viscosity.

Prepare the following stations in advance of having students split into small groups. Then, allow students to rotate through 

the stations in small groups until they have accomplished each task.
Aeronautics for Introductory Physics: Terminal Velocity 101

Mass

1. Place the station paper with the RED text box that reads, “Drop coffee 

filters of different masses. How does this affect their motions?”

2. Prepare sets of regular coffee filters of different masses (1, 2, 3, 4, and 9 

coffee filters). Ensure that the stacks stay together by stapling them at the 

base. Label the stacks with the number of coffee filters in the stack.

3. Encourage students to work together to drop the coffee filters at the same 

time, and to describe any differences in motion that they perceive.

4. For more advanced students, ask students to determine how much higher a stack with double the mass must be placed 

in order to fall in the same amount of time as a single-unit stack. 

5. After answering the question, ask students to read and discuss in their small groups, “What’s the big idea?” and “How 

do engineers use this idea in their designs?”

Shape

1. Place the station paper with the GREEN text box that reads, “Drop coffee 

filters of different shapes. How does this affect their motions?”

2. Prepare quantities of single coffee filters: some single coffee filters used 

for control tests, and other single coffee filters that have been set aside 

for students to change their shape (without changing mass). Students may 

crumple or fold the test coffee filters as they see fit.

3. Encourage students to work together to drop the coffee filters at the same 

time, and to describe any differences in motion that they perceive.

4. For more advanced students, ask students to determine if they can change 

or drop the coffee filters in such a way that the modified coffee filter (with 

the changed shape) actually takes more time to fall than the unmodified 

coffee filter. 

5. After answering the question, ask students to read and discuss in their small groups, “What’s the big idea?” and “How 

do engineers use this idea in their designs?”

Size

1. Place the station paper with the ORANGE text box that reads, “Drop coffee 

filters of different sizes. How does this affect their motions?”

2. Prepare quantities of 10-gallon coffee filters (optional; these are readily 

available online), standard coffee filters, and paper muffin tins that all 

display different sizes, but same shape and same mass. To ensure that the 

sets have the same mass, use an electronic or triple-beam balance, and label 

the stacks so that students know they have equal mass. Use a staple to keep 

the stacks together (the mass of the staple is likely negligible; or, include its 

mass on the balance when massing the stacks).

3. Encourage students to work together to drop the coffee filters and muffin 

tins at the same time, and to describe any differences in motion that they 

perceive.

4. For more advanced students, allow the students to prepare the stacks of 

regular coffee filters or muffin tins themselves. In this case, provide the 

balance at their disposal.

5. After answering the question, ask students to read and discuss in their small groups, “What’s the big idea?” and “How 

do engineers use this idea in their designs?”
102 Aeronautics for Introductory Physics: Terminal Velocity

Fluid

1. Place the station paper with the BLUE text box that reads, “Drop marbles through 

different fluids. How does this affect their motions?”

2. Prepare four or five long and thin transparent bottles and fill them with fluids of 

different viscosity. (For the bottles, check the store’s travel section for clear shampoo bottles.) To prepare liquids of different viscosities, place different amounts 

of corn syrup and water into the bottles and shake well. Ensure that one of the bottles is filled with 100% corn syrup, and that another is filled with 100% water. The 

remaining bottles should be filled with different percentages of corn syrup and 

water. Drop a marble into each bottle, and then close the bottle tightly.

3. Use clear tape to seal off the bottles so they are not intentionally or inadvertently 

opened.

4. Encourage students to work together to invert the bottles to determine what is different about the fluids in each bottle, 

and how they know. Ask students to compare the marble traveling through the corn syrup and water mixture to aircraft 

travelling through low-altitude or high-altitude atmosphere, where the atmosphere is more and less thick, respectively.

5. For more advanced students, ask them to rank the viscosities of the fluids in the tubes. Ask students to consider a 

method to quantify the viscosity of the fluids in each tube. (Consider providing a stopwatch, and encouraging them to 

observe the fast-moving marbles as they roll down a gentle incline, without fully inverting the bottles).

6. After answering the question, ask students to read and discuss in their small groups, “What’s the big idea?” and “How 

do engineers use this idea in their designs?”

Hypothesize Reasons for Cause of Drag

After all students have had the opportunity to participate in the activities at each station, ask all students to re-convene to 

share their outcomes. In general, students will find:

• As mass increases, coffee filters tend to fall faster (although, this is not generalizable, given what was seen with the balls).

• As shape gets smaller, coffee filters tend to fall faster.

• As size gets smaller, coffee filters tend to fall faster.

• As viscosity of the fluid increases, marbles tend to fall more slowly.

Ask students to consider if any of these statements are generalizable (could be applied nearly to any object moving through a fluid):

• Students might mention that some very large objects (like the 10-gallon coffee filter) are simply not stable, and 

won’t fall in a uniform way.

• Students should mention that coffee filters and the balls did not seem to behave the same. Mass seemed to make a 

difference in the falling rate of the coffee filters, but not the balls.

Encourage students to consider what actually causes drag, and why the same air particles that affect the coffee filters don’t 

seem to interact with the balls in the same way. Consider discussing the meaning of aerodynamic and non-aerodynamic 

shapes, and how scientists visualize these. 

In the case of the coffee filters, air particles are more likely to get trapped and collide with the surface of the coffee filter, 

while they tend to move around the balls because of their shape. Dropping balls on the surface of the Earth is very similar to 

dropping balls on the surface of the moon; although they will accelerate more quickly on Earth, compared to one another they 

will land on both the Earth and on the moon in nearly the same time. Encourage students to develop their own models, given 

what they know about air, and encourage students to visualize their concepts on whiteboards or posters. 

Drag is not a simple concept. Unlike surface friction, air particles often interact in unexpected, chaotic ways, and a single air 

particle does not need to directly touch an object in order to be influenced by an object as it moves through a fluid. 

For more information about drag, please refer to NASA’s Beginner’s Guide to Aerodynamics: 

https://www.grc.nasa.gov/WWW/k-12/airplane/bga.html
Aeronautics for Introductory Physics: Terminal Velocity 103

What’s the big idea?

When there is air drag present, heavier 

items tend to fall more quickly than 

lighter items. Heavier items take 

longer to reach their final speed when 

there is air, because they have more 

momentum.

How do engineers use this 

idea in their designs?

It requires more energy (fuel and 

money) to keep heavier airplanes in the 

sky. NASA engineers work to make 

airplanes both strong and lightweight. 

The X-48B was a NASA test airplane 

comprised of new composite materials 

so that it would be as light as possible.

Drop coffee filters of different masses.

How does this affect their motions?
104 Aeronautics for Introductory Physics: Terminal Velocity

What’s the big idea?

Some shapes experience less drag because 

air particles are able to flow around them 

more easily.

How do engineers use this 

idea in their designs?

Drag can prevent airplanes from 

moving quickly through the air. 

NASA engineers work to make the 

fastest airplanes in the world, like this 

futuristic design in the image. NASA 

has designed airplanes that have gone 

at least 6.7 times the speed of sound.

Drop coffee filters of different shapes.

How does this affect their motions?
Aeronautics for Introductory Physics: Terminal Velocity 105

What’s the big idea?

Objects with large surfaces have more 

area that can collide with air particles 

and cause drag.

How do engineers use this 

idea in their designs?

Drag isn’t always a bad thing in 

aerospace engineering. NASA 

engineers use drag to help them 

slow down fast-moving objects with 

parachutes. Pictured to the right is the 

capsule from the Orion mission as it 

falls from space back to Earth.

Drop coffee filters of different sizes.

How does this affect their motions?
106 Aeronautics for Introductory Physics: Terminal Velocity

What’s the big idea?

Some fluids are more viscous (“stickier”) 

than others. Stickier fluids cause an object 

to experience more drag. Air might not 

seem very thick, but the atmosphere 

near the ground is much thicker than the 

atmosphere near the edge of space.

How do engineers use this 

idea in their designs?

NASA engineers have to design for 

both thick and thin air. The image 

to the right, NASA engineers have 

designed high-altitude aircraft, such as 

ER-2, with long wings to fly in thin air.

Drop marbles through different fluids.

How does this affect their motions?
Aeronautics for Introductory Physics: Terminal Velocity 107

NASA Researchers to Flying Insects: “Bug Off!”

http://www.nasa.gov/aero/bug_off.html#.VHTQqo29-Rg

Analysis Questions:

• What problems do bugs cause the airline industries? 

• When airplanes fly through bug-infested air, what impact does that have on the environment?

• Describe the tests performed by NASA scientists to overcome the “bug guts” problem.
108 Aeronautics for Introductory Physics: Terminal Velocity

Inquiry Lesson/Lab: Viscosity Tubes

In order to study viscosity of a fluid and its effect on the 

motion of objects going through it, students can drop 

small items such as marbles through a long tube filled 

with a viscous fluid. Alternatively, “bubble tubes” can be 

commercially purchased from many science education 

retailers. Bubble tubes usually come in sets with fluids of 

different viscosities in each tube. Students can observe 

and record the motion of air bubbles as they rise through 

the fluid. The derived speed of the bubble rising or object 

falling through the tube can be used as an indicator of 

viscosity (with slow speeds equating to high viscosity, and 

fast speeds equating to low viscosity). 

Provide students with at least one bubble tube, a meter stick, 

and a timer, and ask students determine the relationships 

below for objects traveling through a viscous fluid. (Note: 

While the use of a meter stick and timer is acceptable, more 

efficient data collection and analysis can be done through 

the use of video analysis software).

Part 1: Distance and time

Part 2: Viscosity and terminal velocity

Part 3: Weight and terminal velocity

Part 4: Temperature and terminal velocity

Although the following student worksheets do not explicitly address each of the stages of 

the learning cycle for its repetitive nature in this kind of in-depth study with multiple parts, 

ensure that students address each step.

Objectives 

Investigate the viscosity of 

a fluid by determining the 

relationship between distance 

traveled and time elapsed for a 

bubble. Investigate how other 

variables influence motion 

through a fluid.

Materials 

Bubble tubes

Meter stick 

Timer 

Graduated cylinder 

Viscous fluids (corn syrup, 

molasses, maple syrup, or honey with varying concentrations 

of water, oil)

Density spheres 

Mass balance

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 1, 2, 4, 5, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

Viscosity Tubes
Aeronautics for Introductory Physics: Terminal Velocity 109

For each part of this laboratory experiment:

Aim: Ensure that students are able to internalize and describe 

the objective of the experiment. In particular, ensure that 

students have correctly identified the variables of study.

Plan: Ensure that students work as individuals and then as 

small groups to plan to experiments. Special attention must 

be given to the materials needed to ensure that students 

are modifying the correct variables while holding other 

variables constant. This becomes especially important when 

performing a set of experiments. For example, students 

in Part 1 are asked to determine the relationship between 

distance and time for a variety of fluids – this requires 

multiple controlled experiments with different base 

conditions. Be attentive to safety considerations, and make 

sure that students use only harmless fluids, and that they 

choose to heat the fluids in a responsible matter, especially in 

Part 4.

Decide: Ensure that students commit to an experimental plan before beginning. Decisions must be made in regard to 

materials used, data collection procedures, number of trials, number of experiments with different baseline conditions, and 

range of data.

Act: Allow students to experiment without much teacher interference.

Verify: Encourage students to perform analysis of the validity (accuracy and precision) and reliability of their data before 

sharing with the whole class. If possible, encourage groups to determine if the relationships they have observed are linear, 

square, inverse, etc. Fluids can sometimes give surprising relationships – encourage students to have an open mind and not 

necessary see relationships that are immediately evident.

Evaluate: Engage students in whole-class discussion to determine the nature of the relationships between all of the variables 

in the system. If conclusions are unclear, have students consider a plan of action to re-create some of the experiments. 

Additionally, encourage students to deepen their research by referring to engineering handbooks on fluids.

Part I

Objective: Determine the graphical and mathematical models relating the 

distance traveled and the time elapsed for a marble falling through a viscous 

fluid. 

Because the marble achieves terminal velocity very quickly through a very 

viscous medium, all derived relationships should be approximately linear. 

Allowing students to compare graphs and equations for the motion of marbles 

through different tubes should result in significantly different slopes.

(Note: Students might note that when the marble is inserted into the fluid the 

starting level of the fluid goes up as fluid is displaced by the volume of the marble. 

Encourage students to measure distances from a consistent point that corresponds 

to time “0.” Remind students that time from the point of the drop above the fluid 

should not be included when determining the average velocity of the marble 

through the fluid. )

General derived relationship from graph:
110 Aeronautics for Introductory Physics: Terminal Velocity

Part II

Objective: Determine the graphical and mathematical models relating the 

viscosity of a fluid and its terminal velocity.

Although it is easy to demonstrate that the same object traveling more slowly 

through one medium than another is the result of differences in viscosity, the 

relationship between terminal velocity and viscosity of a fluid is a bit more 

complex. 

Using the known viscosities of the fluids with which students are working, have 

students plot the viscosity of the fluid versus the terminal velocity. (Known values 

can be retrieved online: ensure that the values are accurate for room temperature). 

For best results, have students collect data on terminal velocity for at least five 

different fluids of known viscosity, using either a very small air bubble rising 

upward or a very small sphere falling downward through the fluid. The use of larger, non-spherical objects will significantly 

distort this relationship.

In general, students should find that the relationship between these two variables is approximately an inverse (as viscosity 

increases, terminal velocity decreases). 

General derived relationship: where η is viscosity (in Pa s or kg / (m s)) and v is terminal velocity (in m/s) through 

the fluid. Naturally, it might be valuable to discuss the meaning of the constant, k. Ask students to consider what variables 

might influence the viscosity of a fluid in contact with a given particle. Variables that might affect perceived viscosity, as 

measured by terminal velocity, might include things such as difference in density (which thereby affects buoyant force and 

the ability of a particle to sink or float in the surrounding fluid), acceleration due to gravity, and radius of the object going 

through it. (It is important to note that viscosity is a term used relevant to a situation. The behavior of a fluid with a given 

viscosity on Earth might not behave the same way on another planet, where its perceived viscosity might be higher or lower 

due to different gravitational fields or temperature!)

In fact, for simple Newtonian fluids at room temperature, viscosity can be calculated as where Δρ is the 

difference in density between the fluid and the object, g is the acceleration due to gravity, and r is the radius of the object 

moving through the fluid.

Although it is not reasonable to assume that students will be able to derive the full meaning of k on their own, the equation 

as given can be used to compare experimentally derived values for viscosity compared to their textbook value. In this case, 

teachers must provide students with a way to measure volume and mass of the fluid and object to determine density, and a 

measuring tool to estimate the radius of the object.

If appropriate, following a discussion about what other variables might influence terminal velocity, students could be briefly 

introduced to Stokes’ law, which defines the drag force experienced by a small particle through a fluid (so that buoyancy is 

negligible), where Fd

 is drag force, μ is the dynamic viscosity of the fluid, R is the radius of the particle, and ν is the velocity 

of the particle. Caution: This law is only truly an inverse relationship between viscosity and terminal velocity if drag force is 

constant between trials with different fluids, which is not the case in this experiment, unless the terminal velocities observed 

are relatively close to one another.

Having developed an understanding of the relationship between terminal velocity and viscosity of materials, provide students 

with an unknown fluid, and ask them to identify the fluid based upon its estimated viscosity.

(Note: Modeling of fluids can be quite tricky, and some models are only applicable for smaller or larger objects, fluids at 

particular temperatures, or fluids with particular behaviors. This activity ties directly into a study of drag force on airplanes. 

For example, at low speeds and small, aerodynamic particles, air drag is directly proportional to speed. 
Aeronautics for Introductory Physics: Terminal Velocity 111

However, for larger particles, such as airplanes,

where C is the coefficient of lift (a constant dependent upon the shape and angle of attack of the object), ρ is the air density, A

is the cross-sectional area of the object, and v is the velocity of the fluid over the surface of the object.)

Part III

Objective: Determine the graphical and mathematical models relating the 

weight of an object and its terminal velocity falling (or rising) through a 

viscous medium.

From the previous discovery lab, students will recognize that, when air resistance 

is present, weight does have a significant impact on terminal velocity. Ideally, 

provide students with density spheres of equal volume. Alternatively, use density 

cubes of the same volume.

Students should find that as velocity increases, weight increases by a quantity 

proportional to terminal velocity or terminal velocity squared (this depends upon 

if the mass is very, very small at low speeds, or larger or moving at higher speeds). 

The following equation(s) could be derived, depending upon the type of object 

dropped:

Students can then be asked to visually model the reason why this might be. Proposals might include items such as that greater 

weight is more able to push air particles out of the way.

(Note: Teachers might feel that “mass” is a more apt term to use than “weight” in this case, which seems to be a more 

complex concept. The emphasis here is on weight (as a force), because, in later experiments, students will recognize that 

weight and frictional drag are equal in magnitude in experiments where objects have achieved terminal velocity.)

Part IV

Determine the graphical and mathematical models relating the temperature of 

a fluid and terminal velocity.

Using a fluid that has viscosity strongly dependent upon temperature (such as 

maple syrup), attempt to qualitatively derive the relationship between terminal 

velocity and temperature. A safe setup can be achieved by filling large test tubes 

with the fluid, adding a ball bearing, sealing them loosely, and placing them in 

water baths of various temperatures from cool to warm – all test tubes should be 

able to be handled without risk of burn. Students can then move from station to 

station to record temperatures of the water bath and terminal velocities of ball 

bearings in the tubes. Remind students to hold the test tube stoppers in place as 

they invert the tubes, and to use caution when working with hot water.

In general, students will find that the relationship is linear, or positively correlated. Students should make an attempt to then 

relate temperature to viscosity, given what they know about velocity’s relationship to viscosity. 
112 Aeronautics for Introductory Physics: Terminal Velocity

Student Worksheet: Viscosity Tubes

The flight of an airplane through the sky is neither free nor easy. Passenger aircraft are often very large and travel very 

quickly through the air, exposing them to a considerable amount of resistance, called air resistance or drag. Air is a fluid, so 

studies of resistance can also take place in other fluids, such as liquids, with similar results. In the following activities, you 

will learn about how to measure how much a fluid resists the motion of a bubble or marble moving through it.

Part I

Fluids that are dense, thick, and strongly resist the motion of objects going through them have high “viscosity.” One way to 

measure viscosity is to determine the speed of objects as they travel through the fluid. In this activity, you will use a marble, 

a cylinder filled with fluid, a meter stick, and a timer to derive a relationship between distance and time as the marble falls 

through the cylinder. 

Objective: Determine the graphical and mathematical models relating the distance traveled and the time elapsed for a 

marble falling through a viscous fluid. 

1. Show collected data below:

2. Plot a sketch of your data on the graph above.

3. Determine the general relationship of your graph:

4. Extension: 

Perform the same experiment as above, but this time using a different fluid. Compare your results.
Aeronautics for Introductory Physics: Terminal Velocity 113

Part II

As you might have guessed, if an object travels more slowly through one fluid than the other, the fluid resulting in the slower 

speed is more viscous. However, it is not necessarily the case that a doubly viscous material will cause an object to fall at half 

the rate as before. Because viscosity can be very difficult to measure, you will instead look for standard viscosity values in a 

chemical or physical handbook, or online. You will compare these values to the terminal velocity of objects falling through 

the fluid.

Objective: Determine the graphical and mathematical models relating the viscosity of a fluid and its terminal velocity.

1. Show collected data below, including your viscosity values source:

2. Plot a sketch of your data on the graph above.

3. Determine the general relationship of your graph:

4. Extension: 

• What kinds of constants in your experiment do you think might factor into the value you calculated for the constant 

in your equation?

• How do you predict that changing those constants in your experiment might change the graph and the equation? 

• Using the equation provided to you by your teacher, determine the experimental value of one of your fluids.
114 Aeronautics for Introductory Physics: Terminal Velocity

Part III

From the previous discovery lab, you will have noticed that the weight of the object makes a significant difference in drag 

experience when friction is present. Now, you will investigate how weight influences terminal velocity in a single type of 

fluid.

Objective: Determine the graphical and mathematical models relating the weight of an object and its terminal velocity 

falling (or rising) through a viscous medium.

1. Show collected data below:

2. Plot a sketch of your data on the graph above.

3. Determine the general relationship of your graph:

4. Extension: 

If weight influences terminal velocity, which, in turn, can be used as an indicator for viscosity, where is “weight” evident in the equation provided by your teacher in the Extension problems for Part III?
Aeronautics for Introductory Physics: Terminal Velocity 115

Part IV

Most equations for viscosity are dependent upon a variety of factors, including fluid density. Fluid density, in turn, is 

influenced by temperature. In this activity, you will travel from station to station to observe a ball bearing as it falls through a 

test tube of fluid that has been heated or cooled in a water bath.

Determine the graphical and mathematical models relating the temperature of a fluid and terminal velocity.

1. Show collected data below:

2. Plot a sketch of your data on the graph above.

3. Determine the general relationship of your graph:

4. Extension: 

Given what you know about the relationship between terminal velocity and viscosity, what would a graph displaying 

viscosity vs. temperature look like? Sketch your prediction below, and provide a rationale.
116 Aeronautics for Introductory Physics: Terminal Velocity

NASA Turns World Cup Into Lesson in Aerodynamics

http://www.nasa.gov/content/nasa-turns-world-cup-into-lesson-in-aerodynamics/#.VHS_-I29-Rg

Analysis Questions:

• Read the article and watch the associated video. Explain how NASA researchers in this article 

visualize drag, and how it is different/similar to what you did to study drag in the previous 

activity.

• From the video, explain the effect of a ball’s spin on the drag experienced around its surfaces.

• Describe how the traditional soccer ball has evolved from Jabulani to Brazuca, and why it has 

been designed differently over time.

• What is a boundary layer? What is knuckling?

• How does knuckling compare for traditional versus modern soccer balls? Why do you think 

this is the case?

• What applications do these studies have on soccer balls have for aeronautics and airplane 

design?
Aeronautics for Introductory Physics: Terminal Velocity 117

Inquiry Lesson/Lab: The Physics of Soaring (and L/D ratios)

Kenneth W. Ford. The Physics Teacher, 38, 8 (2000).

In this activity, students consistently launch paper gliders and make generalizations about 

their lift to drag ratios. Additionally, students use graphs and – if available – text from The 

Physics Teacher article to analyze horizontal and vertical velocity graphs for a gliders to make 

inferences about the gliders’ L/D ratios.

Objective: Measure Lift/Drag ratios. Determine the relationship between surface area 

and Lift/Drag ratios.

Ask students to build a number of paper airplanes and to send them gliding. In light of the 

previous activities dealing with drag, ask them to consider how it would be possible to 

quantify or at least compare the amount of drag experienced by each glider as it moved 

through the air. With a bit of prompting, students will likely determine that, all other things 

being equal, the glider that moves the furthest horizontally has the least drag. However, the 

motion of a glider is a combination of both horizontal and vertical motion, with vertical 

motion being dependent upon the amount of lift generated by the wings of the glider. A glider 

with very high drag and very low lift will fall steeply, while a glider with low drag and high 

lift is more likely to soar for a while. Other variables that influence drag might include the 

shape of the aircraft, the surface area of the wings, and its speed.

Initial launching speed and initial angle of attack of gliders can be controlled for using the 

same method as in “Inquiry Lab: Scaling and Paper Airplanes.” 

Objectives 

Measure Lift / Drag ratios. 

Determine the relationship 

between surface area and 

Lift / Drag ratios.

Materials 

Printer paper 

Ruler 

Meter stick or tape measure

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 1, 3, 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

The Physics of Soaring 

(and L/D Ratios)
118 Aeronautics for Introductory Physics: Terminal Velocity

While aircraft shape is difficult to quantify, it is much easier 

to consider the construction of very simple aircraft in order 

to quantify the surface area of the wings. Again, refer to 

“Inquiry Lab: Scaling and Paper Airplanes” for suggestions. 

The two images below demonstrate how a simple aircraft 

design can be modified to give two dramatically different 

surface areas for wings. 

Help students to understand that the horizontal range and 

vertical drop of their glider can be used not to measure lift or 

drag separately, but, rather, to get an L/D ratio. 

However, help students to understand that for this model to be most accurate, gliders must be launched horizontally. 

Ask students to determine the relationship between surface area, A, of the wings and L/D ratios, plot their data on a graph, 

and determine the relationship between A and L/D. This relationship might not be linear, depending upon the shape, center of 

gravity, and variety of wind speeds encountered by the gliders throughout their flight. However, students should generally see 

that as surface area increase, lift/drag ratios increase, because an increase in thin surface areas parallel to the ground result in 

much greater increase on lift than on drag.

Lift/drag ratios also have significance for coefficients of lift and drag, which will be introduced in later units. For additional 

teacher background, please refer to the following article:
Aeronautics for Introductory Physics: Terminal Velocity 119

BACKGROUND 

from Beginner’s Guide to Aeronautics

There are four forces that act on an aircraft in flight: lift, weight, thrust, and drag. Forces are vector quantities having both 

a magnitude and a direction. The motion of the aircraft through the air depends on the relative magnitude and direction of 

the various forces. The weight of an airplane is determined by the size and materials used in the airplane’s construction and 

on the payload and fuel that the airplane carries. The weight is always directed towards the center of the Earth. The thrust

is determined by the size and type of propulsion system used on the airplane and on the throttle setting selected by the pilot. 

Thrust is normally directed forward along the center-line of the aircraft. Lift and drag are aerodynamic forces that depend on 

the shape and size of the aircraft, air conditions, and the flight velocity. Lift is directed perpendicular to the flight path and 

drag is directed along the flight path.

Because lift and drag are both aerodynamic forces, the ratio of lift to drag is an indication of the aerodynamic efficiency of 

the airplane. Aerodynamicists call the lift to drag ratio the L/D ratio, pronounced “L over D ratio.” An airplane has a high 

L/D ratio if it produces a large amount of lift or a small amount of drag. Under cruise conditions lift is equal to weight. A 

high-lift aircraft can carry a large payload. Under cruise conditions thrust is equal to drag. A low-drag aircraft requires low 

thrust. Thrust is produced by burning a fuel and a low thrust aircraft requires small amounts of fuel be burned. As discussed 

on the maximum flight time page, low fuel usage allows an aircraft to stay aloft for a long time, and that means the aircraft 

can fly long-range missions. So an aircraft with a high L/D ratio can carry a large payload, for a long time, over a long 

distance. For glider aircraft with no engines, a high L/D ratio again produces a long range aircraft by reducing the steady 

state glide angle at which the glider descends.

As shown in the middle of the slide, the L/D ratio is also equal to the ratio of the lift and drag coefficients. The lift equation

indicates that the lift L is equal to one half the air density ρ times the square of the velocity V times the wing area A times 

the lift coefficient CL:

Similarly, the drag equation relates the aircraft drag D to a drag coefficient CD:

Dividing these two equations give:

Lift and drag coefficients are normally determined experimentally using a wind tunnel. But for some simple geometries, they 

can be determined mathematically.
120 Aeronautics for Introductory Physics: Terminal Velocity

Teacher Answer Key: The Physics of Soaring

Pilots who fly gliders (airplanes without engines), and those who do parasailing and other similar activities 

are intensely interested in the lift/drag ratios of their aircraft. Knowing their L/D ratios helps them to 

predict when and how far away they will land. In the case of glider pilots, they must know exactly how far 

away they can get from their landing spot in order to return home safely. In this activity, you will determine 

the L/D of a simple paper glider, then determine the relationship between L/D and the surface area of the 

wings of various similar gliders.

Objective: Measure Lift/Drag ratios. Determine the relationship between surface area and Lift/Drag 

ratios.

1. Build a paper glider that can fly straight. Compare it to a classmate’s glider. Which of your gliders 

has a greater L/D ratio? How do you know?

Answers will vary. Gliders with greater L/D ratios will have a less steep descent.

2. Calculate the L/D ratio of your glider – consider doing this a few times. What is the L/D value? Explain how you accomplished this, and include a sketch to show any measurements you took.

Vertical distance of fall/range

3. What variables do you think influence the L/D ratio of gliders? Consider all of the things you studied 

in this module on terminal velocity, and list them below.

Aircraft shape, surface area of wings, material, weight, fluid viscosity, etc.

4. Next, decide how to study one of these variables: surface area, A. 

Collect data on at least five different gliders. Ensure that you will 

appropriately control for all other variables. 

5. Develop a graphical model for the relationship between surface 

area, A, and L/D ratios. Describe this relationship, in words.

Generally, as surface area increases, lift/drag ratio should increase, although this is likely not usually linear.

6. Determine if your data is reliable. How much variance is there in 

your data? Why?

7. Explain why you think 

that your graph displays 

the relationship you see. 

If it is linear – why? If 

it is not linear – why? 

Compare your graphs 

to other groups’ to see 

if they display the same 

relationship.

As wings get bigger (for 

a paper glider), they 

tend to result in much 

more lift, and only a 

little bit more drag.

If possible, acquire a copy of “The Physics of Soaring” from The Physics Teacher. Otherwise, use 

the data extracted from the article below, and use other available Internet resources to more fully 

understand the importance and meaning of L/D ratios. All of the graphs below are for the same 

typical medium-performance glider. 
Aeronautics for Introductory Physics: Terminal Velocity 121

8. Figure 4 from the article shows a “polar” graph. How is this 

graph different from the graph you generated above?

The polar graph shows vertical and horizontal speeds. 

Each curve is for a single glider moving at different 

horizontal speeds through the air, and their resulting lift 

or fall speeds. The graph generated in the lab is for many 

different gliders. Although L/D relationships are plotted, 

L/D is plotted on a single axis, with surface area on the 

other axes. The two types of graphs are somewhat related, 

but display entirely different relationships.

9. Figure 4 shows the possible conditions for a single type of 

glider. What is the difference between the glider’s circumstances in each of the three diagonal lines?

Each of the gliders are travelling at different horizontal 

speeds. The top-most line (smallest angle with respect to 

the horizontal) is the fastest travelling glider.

10. Figure 5 shows the polar curves for a low-performance, 

medium-performance, and high-performance glider. What 

do you think is meant by “high-performance”? How is that 

evident on the graph?

Higher performance has the highest L/D ratio, because it 

is more likely to stay aloft. 

11. Maximum L/D values are shown for each of the three types 

of gliders. How were they calculated? Show work.

The L/D is actually the inverse of the slope of these graphs. 

The top value of L/D is about 46.5, and it was calculated 

from taking 57 knots/1.23 knots, or change in magnitude of 

x over change in magnitude of y.

12. What is meant by “minimum sink rate”? Why is this value 

important?

Minimum sink rate is the speed at which you achieve the 

maximum L/D. Staying at the associated horizontal speed 

results in the greatest amount of lift, and a longer amount 

of time in the air before landing. 

13. The three curves in Figure 8 are for the same glider in 

sinking air. What would these curves look like for the same 

glider, but in rising air? Explain.

The graphs would simply be displaced upward, because 

the vertical speed is a combination of speed due to lift and 

speed of the surrounding air.
122 Aeronautics for Introductory Physics: Terminal Velocity

Student Worksheet: The Physics of Soaring

Pilots who fly gliders (airplanes without engines), and those who do parasailing and other similar activities are intensely 

interested in the lift/drag ratios of their aircraft. Knowing their L/D ratios helps them to predict when and how far away they 

will land. In the case of glider pilots, they must know exactly how far away they can get from their landing spot in order 

to return home safely. In this activity, you will determine the L/D of a simple paper glider, then determine the relationship 

between L/D and the surface area of the wings of various similar gliders.

Objective: Measure Lift/Drag ratios. Determine the relationship between surface area and Lift/Drag ratios.

1. Build a paper glider that can fly straight. Compare it to a classmate’s glider. Which of your gliders has a greater L/D 

ratio? How do you know?

2. Calculate the L/D ratio of your glider – consider doing this a few times. What is the L/D value? Explain how you accomplished this, and include a sketch to show any measurements you took.

3. What variables do you think influence the L/D ratio of gliders? Consider all of the things you studied in this module on 

terminal velocity, and list them below.

4. Next, decide how to study one of these variables: surface area, A. Collect data 

on at least five different gliders. Ensure that you will appropriately control for 

all other variables. 

5. Develop a graphical model for the relationship between surface area, A, and L/D ratios. Describe this relationship, 

in words.

Generally, as surface area increases, lift/drag ratio should increase, although this is likely not usually linear.

6. Determine if your data is reliable. How much variance is there in your data? Why?
Aeronautics for Introductory Physics: Terminal Velocity 123

7. Explain why you think that your graph 

displays the relationship you see. If it is 

linear – why? If it is not linear – why? 

Compare your graphs to other groups’ to 

see if they display the same relationship.

If possible, acquire a copy of “The Physics of Soaring” from The Physics Teacher. 

Otherwise, use the data extracted from the 

article below, and use other available Internet resources to more fully understand 

the importance and meaning of L/D ratios. 

All of the graphs below are for the same 

typical medium-performance glider. 

8. Figure 4 from the article shows a “polar” graph. How is 

this graph different from the graph you generated above?

9. Figure 4 shows the possible conditions for a single type of 

glider. What is the difference between the glider’s circumstances in each of the three diagonal lines?

10. Figure 5 shows the polar curves for a low-performance, 

medium-performance, and high-performance glider. What 

do you think is meant by “high-performance”? How is that 

evident on the graph?

11. Maximum L/D values are shown for each of the three types of gliders. How were they calculated? Show work.
124 Aeronautics for Introductory Physics: Terminal Velocity

12. What is meant by “minimum sink rate”? Why is this 

value important?

13. The three curves in Figure 8 are for the same glider in

sinking air. What would these curves look like for the

same glider, but in rising air? Explain.
Aeronautics for Introductory Physics: Terminal Velocity 125

Problem Set: Glider Trajectory Problems

Beginner’s Guide to Aeronautics

You will explore the trigonometric relationship tan (a) = h/d where a is the glide angle, h is the altitude of the plane, and d is 

the horizontal distance the plane will travel.

Problems:

1. A full-sized glider has a glide angle of 2 degrees. If it loses 50 meters of altitude, how far would it move horizontally?

2. A model glider moves horizontally 12 meters for every meter of altitude it loses. What is its glide angle?

3. A model glider has a glide angle of 5 degrees. If it flies 23 meters horizontally, how much altitude will it lose? 

4. A full-sized glider has a glide angle of 5 degrees. If the altitude is 300 meters, will the glider make it to an airport 4,000 

meters away?
126 Aeronautics for Introductory Physics: Terminal Velocity

5. A full-sized glider loses 2 meters for every 38 meters it travels horizontally. What is its glide angle?

6. A model glider travels 57 meters horizontally after losing 5 meters of altitude. What is its glide angle?

7. A model glider loses 3 meters of altitude. If it has a glide angle of 4 degrees, how far did the glider travel horizontally?

8. A full-sized glider lands 10,000 meters horizontally from where it began its downward diagonal path. If the glide angle 

is 3 degrees, what was the glider’s altitude?

9. A model glider has an average velocity of 2.2 m/s along its downward diagonal path. After 10 seconds, the glider has 

lost 2 meters of altitude. What is the glider’s glide angle and how far horizontally did it travel?

10. A full-sized glider has a velocity of 36 m/s along its downward diagonal path. If it has a glide angle of 4 degrees, how 

much altitude will it lose in 2 minutes? How far will it travel horizontally in that 2 minutes?

Problems:

Create a spreadsheet that finds the weight of each glider, the measure of each angle in degrees, and the drag on each glider at 

angles of 2, 3, 4, and 5 degrees.

Model Gliders Mass Full-sized Gliders Mass

1 .5 kg 4 500 kg

2 .9 kg 5 900 kg

3 1.4 kg 6 1,400 kg

Questions:

1. Compare the drag between model glider 1 and the full-sized glider 4 for all four angles. What conclusion can you 

draw?

2. Based on your answer in Question 1, would it be valid to use a model glider to study the drag on a full-sized glider?

3. What is the relationship between glide angle and drag?
Aeronautics for Introductory Physics: Terminal Velocity 127

Literary/Data Analysis: Flying with Finesse

Henk Tennekes. The Simple Science of Flight. The MIT Press: 

Cambridge, Massachusetts. Ch. 4

Although the following chapter from The Simple Science of 

Flight must be purchased, the book is well worth the read, 

and is applicable to many physics concepts from mechanics.

Have students read chapter four of The Simple Science of 

Flight using the following guiding questions for literary and 

data analysis.

*Cover image and graphics from The Simple Science of Flight 

are reproduced here with permissions from The MIT Press.

Objectives 

Describe the meaning of glide 

ratio and the variables that 

influence it.

Materials 

The Simple Science of Flight, 

Ch. 4

NGSS 

SEP 4

CCC 1, 6

CCSS 

Reading in Science 

Writing in Science

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

Flying with Finesse
128 Aeronautics for Introductory Physics: Terminal Velocity

Teacher Answer Key: Flying with Finesse

The Art of Flapping

1. Compare the geometry of the forces exerted by people as they skate, birds as they flap, and airplanes as they fly. How 

are they similar? How are they different? Include a diagram or two in your description.

To move forward, skaters push forward and to the side. Birds, on the other hand, push their wings forward and down. 

Bird wings essentially perform the same motions as skaters, but within a different plane.

2. What should the drag/lift (D/L) ratio be in order to maximize flight efficiency? Explain.

The D/L ratio should be as small as possible. A small drag value means that it is easier for the thrust to result in forward motion.

Horizontal Flight

3. What is finesse? Why is it important to both birds and planes?

The L/D ratio. The opposite of D/L, the L/D ratio should be as big as possible. Birds and planes with the greatest 

finesse are more energy efficient; they tend to have thin wings and aerodynamic body shapes.

The Great Gliding Diagram

4. Look at Figure 15, the “Great Gliding 

Diagram.” Do birds demonstrate constant 

values of finesse? Explain why you think 

this is the case.

No. Typically, as flying speed increases, 

finesse initially increases, then decreases. 

This likely has something to do with drag. 

As birds go faster, the drag increases significantly more. There is a “sweet spot” 

for maximum finesse – the best cruising 

speed.

5. According to Figure 15, which birds are 

most fit to be long-term soarers?

Swifts – they do not exceed the maximum 

horizontal speed of 12 m/s, and they 

exceed the soaring limit of 1 m/s vertically. They also have a greater finesse value 

than the cabbage white.

6. Based upon the data, is the Boeing 747 a 

good glider? Why do you suspect this is 

the case?

Moderately so. (Although it has high 

finesse for an airplane, it is moderate in 

comparison to some birds). This is likely because the wings are small in comparison to the weight it carries, so that it 

can also reduce drag at its very high speeds. 

7. Relate Figure 15 to Figure 6 and Figure 7 from chapter 2. Is there a relationship between the data in these graphs? 

Explain.

Figures 6 and 7 demonstrate that there is a “sweet spot” in horizontal speed in terms of the minimum power required 

for flight. This low power requirement seems to correspond with the peaks of each curve on Figure 15 (where the 

slopes are equal to 0 – where an increase in horizontal flight speed does not change the rate of descent.)
Aeronautics for Introductory Physics: Terminal Velocity 129

8. Relate Figure 15 to Figure 2 in Chapter 1. Is there a relationship between the data in these graphs? Explain.

Yes. Heavier birds tend to need to cruise at faster speeds.

Induced Drag

9. What kinds of relationships are inferred by equation (29) on page 73? 

Many parts of this relationship are 

opposite that which we expect for 

frictional drag. Essentially, as flying 

things go faster, induced drag decreases.

10. What is the difference between induced 

and frictional drag? Provide examples of 

each.

Frictional drag is a result of wings going 

through the air, whereas induced drag is 

a result of flapping.

11. How do induced and frictional drag influence optimal cruising speed? Relate back 

to Figure 7. 

As speed goes up, frictional drag goes 

up, but induced drag decreases. Optimal 

cruising speed occurs at the lowest maximum total drag.

12. What information does Table 5 provide 

that is not provided in Figure 15?

S, b, and A
130 Aeronautics for Introductory Physics: Terminal Velocity

Student Worksheet: Flying with Finesse

Reach chapter four of The Simple Science of Flight using the following guiding questions for literary and data analysis.

The Art of Flapping

1. Compare the geometry of the forces exerted by people as they skate, birds as they flap, and airplanes as they fly. How 

are they similar? How are they different? Include a diagram or two in your description.

2. What should the D/L ratio be in order to maximize flight efficiency? Explain.

Horizontal Flight

3. What is finesse? Why is it important to both birds and planes?

The Great Gliding Diagram

4. Look at Figure 15, the “Great Gliding 

Diagram.” Do birds demonstrate constant 

values of finesse? Explain why you think 

this is the case.

5. According to Figure 15, which birds are 

most fit to be long-term soarers?

6. Based upon the data, is the Boeing 747 a 

good glider? Why do you suspect this is 

the case?

7. Relate Figure 15 to Figure 6 and Figure 7. 

Is there a relationship between the data in these graphs? Explain.
Aeronautics for Introductory Physics: Terminal Velocity 131

8. Relate Figure 15 to Figure 2. Is there a relationship between the data in these graphs? Explain. 

Induced Drag

9. What kinds of relationships are inferred by equation (29) on page 73? Rationalize each variable and its placement in the 

equation.

10. What is the difference between 

induced and frictional drag? Provide 

examples of each. 

11. How do induced and frictional drag influence optimal cruising speed? Relate 

back to Figure 7. 

12. What information does Table 5 provide that is not provided in Figure 15?
132 Aeronautics for Introductory Physics: Terminal Velocity

Model Building: Developing a Model for Drag

From the many experiences in this unit, students will understand that drag is a force that is 

dependent upon many different macroscopic variables, including weight, speed, shape, and 

the fluid that surrounds the object. However, it is important to have students also consider 

what is happening microscopically, because drag – and the similar force of lift – are so closely 

related. 

Encourage students to visually and verbally describe their models for drag. Ask students to 

prepare a poster, whiteboard, or some other form of document, and have students retain those 

models for the activity “Developing a Model for Drag” so that they can compare their original 

model for drag with their refined models of lift in the activity “Model: Developing a Model 

for Lift.”

What do you think causes 

drag?

How does it work?

Drawing

How does it work?

Explanation

Objectives 

Develop a micro and macroscopic model for drag.

Materials 

None

NGSS 

CDI HS-PS2 

SEP 2, 6, 7

CCC 4

CCSS 

Reading in Science 

Writing in Science 

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Terminal Velocity

ACTIVITY

Developing a Model 

for Drag
Aeronautics for Introductory Physics: Terminal Velocity 133

Students might immediately consider that drag is a result of the following variables, for the following reasons:

MACROSCOPICALLY:

• Object effects:

• Certain shapes catch, spill, or “slice” through the air.

• Objects with a heavy weight can “slice” through the air more easily.

MICROSCOPICALLY:

• Mental representation of fluid flow:

• Particle model of fluid: As particles bounce against an object, they transfer some of their momentum to the object, 

preventing it from moving forward through the fluid.

• Stream or sheet model of fluid: Fluid particles flow around objects. As these streams of sheets of fluid get separated more from their original orientation, it is harder to get through the fluid.

• Fluid variables

• Density: It is harder to move through more dense fluid, because it’s harder to overcome the inertia of heavy particles, or because it’s harder to move through so many closely spaced particles. Fluid atomic composition, pressure, 

and temperature all affect its density.

• Stickiness: It is harder to move through stickier fluid, because the fluid properties (intermolecular forces) make the 

particles want to stay together more.

While none of the above ideas are wrong, they are almost all very incomplete. See the following articles for some more 

background on drag.
134 Aeronautics for Introductory Physics: Terminal Velocity

BACKGROUND

from Beginner’s Guide to Aeronautics

Drag is the aerodynamic force that opposes an aircraft’s motion through the air. Drag is generated by every part of the 

airplane (even the engines!). How is drag generated?

Drag is a mechanical force. It is generated by the interaction and contact of a solid body with a fluid (liquid or gas). It is 

not generated by a force field, in the sense of a gravitational field or an electromagnetic field, where one object can affect 

another object without being in physical contact. For drag to be generated, the solid body must be in contact with the fluid. If 

there is no fluid, there is no drag. Drag is generated by the difference in velocity between the solid object and the fluid. There 

must be motion between the object and the fluid. If there is no motion, there is no drag. It makes no difference whether the 

object moves through a static fluid or whether the fluid moves past a static solid object.

Drag is a force and is therefore a vector quantity having both a magnitude and a direction. Drag acts in a direction that is 

opposite to the motion of the aircraft. Lift acts perpendicular to the motion. There are many factors that affect the magnitude 

of the drag. Many of the factors also affect lift but there are some factors that are unique to aircraft drag.

We can think of drag as aerodynamic friction, and one of the sources of drag is the skin friction between the molecules of the 

air and the solid surface of the aircraft. Because the skin friction is an interaction between a solid and a gas, the magnitude of 

the skin friction depends on properties of both solid and gas. For the solid, a smooth, waxed surface produces less skin 

friction than a roughened surface. For the gas, the magnitude depends on the viscosity of the air and the relative magnitude of 

the viscous forces to the motion of the flow, expressed as the Reynolds number. Along the solid surface, a boundary layer of 

low energy flow is generated and the magnitude of the skin friction depends on conditions in the boundary layer.

We can also think of drag as aerodynamic resistance to the motion of the object through the fluid. This source of drag depends 

on the shape of the aircraft and is called form drag. As air flows around a body, the local velocity and pressure are changed. 

Since pressure is a measure of the momentum of the gas molecules and a change in momentum produces a force, a varying 

pressure distribution will produce a force on the body. We can determine the magnitude of the force by integrating (or adding 

up) the local pressure times the surface area around the entire body. The component of the aerodynamic force that is opposed 

to the motion is the drag; the component perpendicular to the motion is the lift. Both the lift and drag force act through the 

center of pressure of the object.

There is an additional drag component caused by the generation of lift. Aerodynamicists have named this component the 

induced drag. It is also called “drag due to lift” because it only occurs on finite, lifting wings. Induced drag occurs because 

the distribution of lift is not uniform on a wing, but varies from root to tip. For a lifting wing, there is a pressure difference 

between the upper and lower surfaces of the wing. Vortices are formed at the wing tips, which produce a swirling flow that 

is very strong near the wing tips and decreases toward the wing root. The local angle of attack of the wing is increased by 

the induced flow of the tip vortex, giving an additional, downstream-facing, component to the aerodynamic force acting 

on the wing. The force is called induced drag because it has been induced by the action of the tip vortices. The magnitude 

of induced drag depends on the amount of lift being generated by the wing and on the distribution of lift across the span. 

Long, thin (chordwise) wings have low induced drag; short wings with a large chord have high induced drag. Wings with an 

elliptical distribution of lift have the minimum induced drag. Modern airliners use winglets to reduce the induced drag of the 

wing.

Two additional sources of drag are wave drag and ram drag. As an aircraft approaches the speed of sound, shock waves are 

generated along the surface. The shock waves produce a change in static pressure and a loss of total pressure. Wave drag is 

associated with the formation of the shock waves. The magnitude of the wave drag depends on the Mach number of the flow. 

Ram drag is produced when free stream air is brought inside the aircraft. Jet engines bring air on board, mix the air with fuel, 

burn the fuel, then exhausts the combustion products to produce thrust. If we look at the basic thrust equation, there is a mass 

flow times entrance velocity term that is subtracted from the gross thrust. This “negative thrust” term is the ram drag. Cooling 

inlets on the aircraft are also sources of ram drag.

As students consider a model for drag, they are beginning to consider how objects interact with fluid that flows around it. The 

Navier-Stokes equations describe how fluids interact with their environment, and are the foundation for computational fluid 

dynamics that is so important in modeling flow around a given aircraft or object. While students of introductory physics are 

likely unprepared to fully understand differential equations, such equations allow a better understanding of how the many 

variables identified within their model work together. 
Aeronautics for Introductory Physics: Terminal Velocity 135

BACKGROUND

from Beginner’s Guide to Aeronautics

The Navier-Stokes equations describe how the velocity, pressure, temperature, and density of a moving fluid are related. 

The equations were derived independently by G.G. Stokes, in England, and M. Navier, in France, in the early 1800s. The 

equations are extensions of the Euler equations and include the effects of viscosity on the flow. These equations are very 

complex, yet undergraduate engineering students are taught how to derive them in a process very similar to the derivation 

that we present on the conservation of momentum webpage.

The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using 

methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made 

further approximations and simplifications to the equation set until they had a group of equations that they could solve. 

Recently, high-speed computers have been used to solve approximations to the equations using a variety of techniques 

like finite difference, finite volume, finite element, and spectral methods. This area of study is called computational fluid 

dynamics, or CFD.

The Navier-Stokes equations consist of a time-dependent continuity equation for conservation of mass, three time-dependent 

conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent 

variables in the problem: the x, y, and z spatial coordinates of some domain, and the time t. There are six dependent 

variables; the pressure p, density ρ, and temperature T (which is contained in the energy equation through the total energy 

ET) and three components of the velocity vector; the u component is in the x direction, the v component is in the y direction, 

and the w component is in the z direction, All of the dependent variables are functions of all four independent variables. The 

differential equations are therefore partial differential equations and not the ordinary differential equations that one studies 

in a beginning calculus class.
136 Aeronautics for Introductory Physics: Terminal Velocity

Re is the Reynolds number is a similarity parameter that is the ratio of the scaling of the inertia of the flow to the viscous 

forces in the flow. The q variables are the heat flux components and Pr is the Prandtl number which is a similarity parameter 

that is the ratio of the viscous stresses to the thermal stresses. The tau variables are components of the stress tensor. A tensor 

is generated when you multiply two vectors in a certain way. Our velocity vector has three components; the stress tensor has 

nine components. Each component of the stress tensor is itself a second derivative of the velocity components.

The terms on the left-hand side of the momentum equations are called the convection terms of the equations. Convection is 

a physical process that occurs in a flow of gas in which some property is transported by the ordered motion of the flow. The 

terms on the right hand side of the momentum equations that are multiplied by the inverse Reynolds number are called the 

diffusion terms. Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the 

random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Turbulence, 

and the generation of boundary layers, are the result of diffusion in the flow. The Euler equations contain only the convection 

terms of the Navier-Stokes equations and cannot, therefore, model boundary layers. There is a special simplification of the 

Navier-Stokes equations that describe boundary layer flows.

Notice that all of the dependent variables appear in each equation. To solve a flow problem, you have to solve all five 

equations simultaneously; that is why we call this a coupled system of equations. There are actually some other equation that 

are required to solve this system. We only show five equations for six unknowns. An equation of state relates the pressure, 

temperature, and density of the gas. And we need to specify all of the terms of the stress tensor. In CFD the stress tensor 

terms are often approximated by a turbulence model.
Aeronautics for Introductory Physics: Force Particle 137

Force Particle – Inertia

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
138 Aeronautics for Introductory Physics: Force Particle

Inquiry Lesson/Lab: Falling and Air Resistance (Qualitative)

In previous activities, students will have investigated a wide variety of variables that influence 

the speed of a rising bubble or a falling object through fluids. However, the focus was on the 

motion as well as the viscosity, both of which are related to the drag force on the object.

In this inquiry lab, with their understanding of force diagrams and balanced-force systems, 

students will investigate how the drag force due to air resistance influences the terminal 

velocity of a coffee filter. Because an object moving at constant velocity is in a balanced-force 

system, a coffee filter falling at terminal velocity experiences a drag equal in magnitude to 

its weight. (Hence, this activity is not terribly different from a study relating the mass of an 

object and its terminal velocity.) However, to relate this study more closely to theoretical 

kinematics, the use of a motion detector and timer will not be permitted for the final portion 

of the activity. See the theoretical discussion further on.

Objective: Determine the relationship between air drag and terminal velocity.

The linear model of drag versus terminal velocity is limited only to objects moving at 

relatively slow speeds. For higher speeds, drag force becomes incrementally greater, and 

reaches a point at which drag is proportional to the square of the speed of the moving object 

relative to the motion of the surrounding fluid. The purpose of this activity is to engage 

students in collecting a wide range of data to see if coffee filters have a drag that is more 

dependent upon a direct relationship with speed or with speed squared. 

(Note: To acquire more data points, consider placing half or portions of coffee filters inside of 

the bottom filter!)

Objectives 

Determine the relationship 

between air drag and terminal 

velocity.

Materials 

Motion detector 

Graphical analysis tool 

Coffee filters (of a variety 

of sizes, including industrial 

coffee filters and / or muffin or 

candy cups)

Meter stick

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 1, 2, 4, 5, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle–Inertia

ACTIVITY

Falling and Air 

Resistance
Aeronautics for Introductory Physics: Force Particle 139

Teacher Answer Key: Falling and Air Resistance (Qualitative) 

Objective: Determine the relationship between air drag and terminal velocity.

1. Holding a motion detector as high as possible, drop a single coffee filter below it. Create a qualitative sketch of position 

vs. time and velocity vs. time. On both graphs, label the point where terminal velocity begins.

2. Draw a force diagram for the coffee filter before it has been released from your hand.

Depending upon how the coffee filter is held, there should be a force of gravity downward, and either a normal force or 

tension force suspending it upward, with equal magnitude.

3. Draw a force diagram for the coffee filter after it has been released from your hand, but before it achieves terminal 

velocity.

The force of gravity is the same as before, but the upward force is smaller, and is now a drag (friction) force.

4. Draw a force diagram that represents a coffee filter falling at terminal velocity through the air.

Constant velocity implies balanced forces, so the drag force pointing up should be equal in magnitude and opposite in 

direction to the weight.

5. From prior experience, it appears as though small objects display a relationship in which weight (equal to Fair, the 

friction force of air) is directly proportional to terminal velocity. Fair = W = k vterminal, where k is a constant. From 

W = k vterminal, and assuming that the coffee filter is at terminal velocity essentially from the moment of its release, 

derive an expression for the displacement of the coffee filter relating to weight.

6. Based upon the derived expression above, if drag force is directly proportional to v, what does it say about the distance 

fallen when the weight is doubled? Tripled?

Double weight would result in double the distance fallen, and triple the weight would result in triple the distance fallen 

in the same amount of time. 

7. In contrast, at high speeds, experiments show that Fair could be proportional to velocity squared, leading to 

Fair = W = kv2. Derive an alternate expression for Δx in relationship to weight, if this is the case.

8. Based upon the derived expression above, if drag force is proportional to v2, what does it say about the distance fallen 

when the weight is doubled? Tripled?

Doubling the weight would increase the displacement by 1.4 times (the square root of 2), while tripling the weight 

would increase the displacement by 1.7 times (the square root of 3).
140 Aeronautics for Introductory Physics: Force Particle

Goal: Determine the graphical and general mathematical model relating air friction drag and terminal velocity for 

falling coffee filters. Limitation: You must achieve this without the use of a motion detector or any devices that record time.

Hint: This can be accomplished by estimating the distances 

at which two or more sets of coffee filters of varying weights 

fall in the same amount of time. This requires some team 

effort and consistency!

9. Looking at your data, which theoretical model does 

your data support? Create a presentation and/or report. 

Be sure to include all parts of the learning cycle, from 

your experimental planning to the evaluation of your 

data. Provide an argument for the accuracy and precision of your data collection process as well.
Aeronautics for Introductory Physics: Force Particle 141

Student Worksheet: Falling and Air Resistance (Qualitative)

In many aspects, physics is beautiful because of its simplicity. Many simple rules govern the universe. However, there are 

some phenomena that simply cannot be easily modeled by a single relationship – fluids is one of them. Typically, physics 

textbooks report that drag is proportional to speed squared. Hence, if you run twice as fast into the wind, you will feel 

four times the resistance. However, this relationship doesn’t always hold true for smaller objects. In this activity, you will 

investigate coffee filters as they fall at terminal velocity. Because weight is equal to drag at terminal velocity, you will use 

weight to determine if drag is proportional to the coffee filter’s speed, v, or speed squared, v2.

1. Holding a motion detector as high as possible, drop a single coffee filter below it. Create a qualitative sketch of position 

vs. time and velocity vs. time. On both graphs, label the point where terminal velocity begins.

2. Draw a force diagram for the coffee filter before it has been released from your hand.

3. Draw a force diagram for the coffee filter after it has been released from your hand, but before it achieves terminal 

velocity.

4. Draw a force diagram that represents a coffee filter falling at terminal velocity through the air.

5. From prior experience, it appears as though small objects display a relationship in which weight (equal to Fair, the 

friction force of air) is directly proportional to terminal velocity. Fair = W = k vterminal, where k is a constant. From 

W = k vterminal, and assuming that the coffee filter is at terminal velocity essentially from the moment of its release, 

derive an expression for the displacement of the coffee filter relating to weight.

6. Based upon the derived expression above, if drag force is directly proportional to v, what does it say about the distance 

fallen when the weight is doubled? Tripled?
142 Aeronautics for Introductory Physics: Force Particle

7. In contrast, at high speeds, experiments show that Fair could be proportional to velocity squared, leading to 

Fair = W = kv2. Derive an alternate expression for Δx in relationship to weight, if this is the case.

8. Based upon the derived expression above, if drag force is proportional to v2, what does it say about the distance fallen 

when the weight is doubled? Tripled?

Goal: Determine the graphical and general mathematical model relating air friction drag and terminal velocity for 

falling coffee filters. Limitation: You must achieve this without the use of a motion detector or any devices that records time.

Hint: This can be accomplished by estimating the distances 

at which two or more sets of coffee filters of varying weights 

fall in the same amount of time. This requires some team 

effort and consistency!

9. Looking at your data, which theoretical model does 

your data support? Create a presentation and/or report. 

Be sure to include all parts of the learning cycle, from 

your experimental planning to the evaluation of your 

data. Provide an argument for the accuracy and precision of your data collection process as well.
Aeronautics for Introductory Physics: Force Particle 143

Skyray 48 Takes Flight

http://www.nasa.gov/vision/earth/improvingflight/skyray_48.html#.VHOb5Y29-Rg

Analysis Questions:

• The X-48B is a blended wing body aircraft. What is the purpose behind this kind of design?

• The X-48B is also a remotely controlled aircraft. Why do you think that NASA is pursuing 

research for remote-controlled and autonomous vehicles?
144 Aeronautics for Introductory Physics: Force Particle

Real-World Applications: Hanging an Airplane: A Case Study in 

Static Equilibrium

Debora M. Katz. The Physics Teacher, 47, 516 (2009). 

A wonderful way to help students understand concepts of static equilibrium is to “hang an 

airplane,” much in the same way as an artifact airplane would be suspended in a museum. 

Because planes are generally suspended in such a way as to give the perception of flight, a 

specific angle might be desired. The following is an activity to help students measure tensile 

strength of a string, measure the center of mass of the airplane to be suspended, and calculate 

the necessary lengths and forces in each string to support the plane at a given angle.

Objectives 

Determine tensile strength 

needed to hang an airplane, 

identify center of mass, and 

suspend the airplane at a given 

angle.

Materials 

Thread 

Meter sticks 

Masses 

Tape 

Ring stand 

Protractor 

Force meter (optional)

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 3, 6, 7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle - Inertia

ACTIVITY

Hanging an Airplane
Aeronautics for Introductory Physics: Force Particle 145

Teacher Answer Key: Hanging an Airplane

Objective: Determine the minimum number of strings required to 

support the aircraft by measuring tensile strength.

For this activity, it is best to provide students with a thread with a low 

tensile strength. A string that might break when supporting a model 

airplane should be provided to them, so that they feel compelled to 

determine the minimum number of strands necessary to support the 

aircraft from a single point.

In order to determine the required number of threads to support any object, 

the tensile strength should be determined, where

and Fbreak is the force at which the string breaks, and A is the crosssectional area of the string. Fbreak can be measured by pulling on the string 

with a digital force meter, or adding masses to a string suspended from a 

point until it breaks, and recording the final force. The cross-sectional area 

can be measured by placing threads side by side on a piece of tape, and 

using a ruler to estimate the diameter of a single thread to calculate its 

cross-sectional columnar area.

If a model aircraft is not available, the teacher or students can 

construct a mock aircraft by joining together four meter sticks 

for the wings, a single meter stick for the fuselage, and one or 

two meter sticks for the tail. The difficulty level across the class 

can be modified by giving each group a different assembly.

Objective: Support the airplane from a single point, with the 

fuselage and wings parallel to the ground, by first 

identifying the center of mass.

Students can estimate or calculate the center of mass of the 

airplane. Most simply, students can find the point at which the 

plane balances. Alternatively, students could weigh the airplane 

by placing at least three different points on three balances, and, 

using their understanding of torque, estimate the center of mass.
146 Aeronautics for Introductory Physics: Force Particle

Objective: Using two vertical strings, determine where two

strings must be attached to the wings to make the plane 

incline with 10 degrees from the ground.

This final task can be considerably more challenging. Because 

the airplane must be in static equilibrium, the airplane can be 

considered, along the fuselage, to act much like a lever pivoting 

around its center of mass. The placement of the two support 

points, as a result, must balance each other. The goal is to place 

the fuselage horizontal to the ground, but the wings at a 10 

degree incline to the ground. In this particular case, net force in 

the z-axis must equal zero, and the torque in the x-axis and 

y-axis must add up to zero as well, respectively.

Using this system, knowing the weight of the aircraft, students 

can solve for T1

, T2, and r1. Students should be reminded that 

there are multiple correct solutions to this problem, and that 

they can choose one variable to see the resulting variable, and 

determine if the values are within the appropriate range (i.e. 

wing span and length).

Again, this lab includes three parts with three different objectives. 

Encourage students to work through the full learning cycle for 

each part of the laboratory experience, either formally (in the 

form of a written notebook or report and explicit discussion) or 

informally (through small group and whole class discussion). 
Aeronautics for Introductory Physics: Force Particle 147

Student Worksheet: Hanging an Airplane

Objective: Determine the minimum number of strings required to support 

an aircraft by measuring tensile strength.

Tensile strength (σ) is the maximum force (Fbreak) supported by a string per unit 

of cross-sectional area (A), and is defined mathematically as

1. Explain how you determined tensile strength of the string. Include 

sketches and/or images to demonstrate how you know that your value for 

tensile strength is accurate and precise. 

2. Your teacher has provided a model aircraft to you that must be suspended from a single point. How many strands of 

string must be used to support the airplane from a single point? Explain how you know.

Objective: Support the airplane from a single point, with 

the fuselage and wings parallel to the ground, by first 

identifying the center of mass.

3.Using any method you prefer (without yet suspending the 

airplane from the string), identify the center of mass of your 

airplane. Explain how you accomplished this.
148 Aeronautics for Introductory Physics: Force Particle

4. Attach the string to your predicted center of mass of the plane to determine if, in fact, the plane balances. Get your 

teacher’s approval. Teacher signature: 

Objective: Using two vertical strings, determine where 

two strings must be attached to the wings make the 

plane incline with 10 degrees from the ground. 

Unlike in the picture displayed below, the goal of this 

task is to make the airplane’s fuselage parallel to the 

ground, with the wings titled 10 degrees with respect to 

the ground. 

This activity requires that the plane be in static 

equilibrium. As a result, answer the following questions:

5. What will be the sum of all net forces on the plane? Explain.

6. Considering that the plane will be supported by two strings (with tension force T1 and T2), write an expression for the 

sum of all vertical forces on the plane in the box below:

Using a force meter, measure Fg and record it here: 

7. Consider the two tension forces. Look at the airplane from the 

top perspective, where the dotted lines meet at the center of 

gravity of the airplane. Draw two dots to show in which quadrants the vertical strings must be placed in order to achieve a 

horizontal fuselage and titled wings. Provide a rationale for 

your answer. (There are multiple correct answers – consistency 

is key!)
Aeronautics for Introductory Physics: Force Particle 149

8. Consider the fuselage only. The goal is to have it parallel to 

the ground. Because we can think about rotation around 

any point as a result of torque, think about the airplane 

dipping forward and back about its center of gravity. Draw 

in lines showing the distance from the two dots normal to 

the wing (side-to-side axis), and label them as x1 and x2. 

In this way, you can derive an expression for the torque 

exerted on each point, given the tension in the string and 

its distance from the wing (side-to-side) axis.

9. Using T1 and T2, and x1 and x2, write an expression for the sum of the torques along the fuselage’s (tip-to-tail) axis. 

(Use the Side View image to help!)

10. There are multiple correct placements for x1 and x2. Working only to get the fuselage horizontal to the ground, without 

concern for wing support, work to experimentally measure where those two locations are. 

Record them here: and 

 

11. Using the two expressions you developed in the two boxes above, solve for the value of T1 (first), and then T2.

12. Check your actual tension values using a force meter. Get teacher approval to move on: 

13. Now consider the 10 degree tilt of the wings. Using the 

front view diagram as a guide, write an expression for the 

torque of the wings along the fuselage’s (front-to-back) 

axis. Recall that torque only considers the forces that are 

perpendicular to the surface.
150 Aeronautics for Introductory Physics: Force Particle

14. Work the strings around the plane until you have achieved a horizontal fuselage and a wing tilt of 10 degrees. Using 

your experimental values for r1 and r2, x1 and x2, now solve for the new values of T1 and T2.

15. Check your answers with the use of a force meter, and get teacher approval: 

NASA Model Flies at Air and Space

http://www.nasa.gov/topics/aeronautics/features/bwb_smithsonian.html#.VHOfy429-Rg

Analysis Questions

• The X-48B has no tail rudder, like typical aircraft. How does it make up for this loss of rudder 

control?

• The model was built for actual testing in a wind tunnel. Provide three reasons why scientists 

begin their work with a model before producing a full-sized aircraft.
Aeronautics for Introductory Physics: Force Particle 151

Problem Set: Trimmed Aircraft Activity

Beginner’s Guide to Aeronautics

BACKGROUND

An airplane in flight can be maneuvered by the pilot using the aerodynamic control surfaces; the elevator, rudder, or ailerons. 

As the control surfaces change the amount of force that each surface generates, the aircraft rotates about a point called the 

center of gravity. The center of gravity is the average location of the weight of the aircraft. The weight is actually distributed 

throughout the airplane, and for some problems it is important to know the distribution. But for total aircraft maneuvering, 

we need to be concerned with only the total weight and the location of the center of gravity.

How do engineers determine the location of the center of 

gravity for an airplane which they are designing?

An airplane is a combination of many parts; the wings, 

engines, fuselage, and tail, plus the payload and the fuel. 

Each part has a weight associated with it which the engineer 

can estimate, or calculate, using Newton’s weight equation:

where Fg

 is the weight, m is the mass, and g is the 

gravitational constant which is 32.2 (ft/s)/s in English 

units and 9.8 (m/s)/s or N/kg in metric units. To determine 

the center of gravity cg, we choose a reference location, 

or reference line. The cg is determined relative to this 

reference location. The total weight of the aircraft is simply 

the sum of all the individual weights of the components. 

Since the center of gravity is an average location of the weight, we can say that the weight of the entire aircraft Fg times 

the location cg of the center of gravity is equal to the sum of the weight Fg of each component times the distance d of that 

component from the reference location:

The center of gravity is the mass-weighted average of the component locations.

We can generalize the technique discussed above. If we had a total of “n” discrete components, the center of gravity cg of 

the aircraft times the weight Fg of the aircraft would be the sum of the individual i component weight times the distance 

d from the reference line (Fg * d) with the index i going from 1 to n. Mathematicians use the Greek letter sigma to denote 

this addition. (Sigma is a zig-zag symbol with the index designation being placed below the bottom bar, the total number 

of additions placed over the top bar, and the variable to be summed placed to the right of the sigma with each component 

designated by the index.)

This equation says that the center of gravity times the sum of “n” parts’ weight is equal to the sum of “n” parts’ weight times 

their distance. The discrete equation works for “n” discrete parts.
152 Aeronautics for Introductory Physics: Force Particle

Problem Set 

from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Answer Key

A Boeing 747-400 domestic airplane is soaring the friendly skies (shown below). Use data from the Boeing 747 Wikipedia 

website to complete Table 1. Estimate values for data you are unable to locate in a reasonable amount of time. You will have 

to select an engine to go on your aircraft. You should be able to find the length of the airplane (for the reference distances 

requested), the mass of the engine, and the fuel capacity. The distances in Table 1 should be measured from a reference line 

that starts at the nose (front) of the 747. Use the BACK key to return to this page.

Problem 1 will guide you in the calculation to find the mass of the fuel.

All problems are expressed in metric units.

Mass (kg) Distance from Reference Line (m)

payload d1

engine d2

wings d3

fuselage d4

fuel d5

vertical tail d6

horizontal tail d7

Table 1 : Airplane component masses and distances from reference line

1. What is the fuel capacity of the 747?

Fuel capacity = Volume of fuel = ______________

2. Assuming the density of an average jet fuel is 0.75 g/ml, use the density equation,

density = mass / volume,

to calculate the mass of the fuel. (Hint: You must convert from liters to milliliters.)

Mass of fuel = ______________

Record this value for the mass of the fuel in Table 1.

3. In Table 2, record the weights of the components (parts) listed in Table 1.

(Hint: Remember, Fg = mg = W. The gravitational constant is 9.8 N/kg.)

Mass (kg) Distance from Reference Line (m)

Payload w1

Engine w2

Wings w3

Fuselage w4

Fuel w5

Vertical Tail w6

Horizontal Tail w7

 Table 2 : Airplane component weights
Aeronautics for Introductory Physics: Force Particle 153

4. What does n equal?

[Hint: The number n is the number of quantities being added together.]

n = _________ 

5. What are the values for i?

i = _________________________

6. What is the total weight W of the airplane?

7. What is the value of the sum of the component weights times their distances from the reference line (see the equation 

below)?

8. Using your answers from Problems 6 and 7, calculate the center of gravity.

9. After a long flight, the amount of fuel left in the tanks is 20% of the initial amount. What is the mass and weight of the 

fuel that is left? Record your answers in Table 3.

Mass (kg) Weight (N)

Remaining Fuel

 Table 3 : Remaining Fuel Data

10. Recalculate the airplane’s center of gravity with the reduced fuel weight.

11. Did the center of gravity change? 

a. If yes, by how many meters did the center of gravity move?

b. If yes, did the center of gravity move toward or away from the nose of the airplane?

c. Do you think the pilot would notice such a change in the center of gravity while flying? Why?
154 Aeronautics for Introductory Physics: Force Particle

Problem Set

from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Answer Key

The supply company you work for has just purchased a DC 8-62 cargo transport airplane. You will find the specifications for 

the new company “toy” below. The company has decided to promote you to ground crew supervisor. You are responsible for 

loading the DC 8-62 so that the center of gravity of the airplane is maintained at 75 feet ( 22.9 meters).

1. For the first flight, the cargo airplane is loaded with 2 Igloo cargo shells. The first fiberglass shell contains 4,550 pounds 

(lbs) of cargo and is located 40 feet from the reference point. The second aluminum shell contains 7,000 lbs of cargo 

and is located 110 feet from the reference point. The empty mass (see specifications) is located 75 feet from the reference point. Calculate the center of gravity of your aircraft. (HINT: Don’t forget to add in the weight of the Igloo shell 

to the mass of the cargo.)

2. The second flight is loaded with 2 aluminum Igloo cargo shells each containing 9,590 lbs. Your ground crew has placed 

one container at 120 ft. from the reference point and the other at 90 ft. from the reference point. The empty mass is still 

located at 75 ft. Calculate the center of gravity of your aircraft . (HINT: Don’t forget to add in the weight of the Igloo 

shell to the mass of the cargo.)

3. Did your first flight meet your required center of gravity specifications? If you answered no, go to Question 5.

4. Did your second flight meet your required center of gravity specifications? If you answered no, go to Question 5.

5. What correction would the pilot of the airplane have to make during flight to fly the airplane as it is loaded? Where 

would you move your cargo shells to get the center of gravity to equal the required 75 ft.? 

Forward Belly (bulk) - 800 cu. ft.

Aft Belly (bulk) - 815 cu. ft.

Main Cargo Door (A)- 85” high by 140” wide

Forward Belly Door (B) - 36” high by 44” wide

Aft Belly Door (C) - 36” high by 44” wide

Payload maximum - 90,000 lbs.

Empty mass - 136,600 lbs

Main deck cargo volume (containerized) - 6,160 cu. ft.

[thirteen (13) - 88” x 125” containers or pallets]
Aeronautics for Introductory Physics: Force Particle 155

Emery Worldwide aircraft are units loaded utilizing Type A-3 containers and pallets. The

Igloo illustrated, is an enclosed fiberglass or aluminum shell, open on one side and closed

after loading with a fabric cover and straps. Pallets are essentially the same size as the

Igloo, but without the permanent shell. Freight loaded on pallets is restrained with nets.

Inside dimensions - 121” long, 84” wide, 76” high

Pallet dimensions - 88” x 125”

Volume capacity - 450 cu. ft.

Maximum gross weight - 10,500 lbs.

Approximate tare weights: 410 lbs. - Aluminum shell; 520 lbs. - Fiberglass shell
156 Aeronautics for Introductory Physics: Force Particle

Problem Set

from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Answer Key

Suppose an airplane has a center of pressure on the wings that provides a wing lift force of W. The force W is located a 

distance dw from the airplane’s center of gravity (cg). The center of pressure on the tail provides a lift force of T. The force T 

is located a distance dt from the airplane’s center of gravity (cg). The lift force W will always be positive, which indicates an 

upward force. However, the lift force T can be positive for some airplanes (upward force) and negative (downward force) for 

other airplanes.

The plane is traveling to the left; the tail is on the right.

1. An airplane has a wing lift of 30,000 lbs. centered 20 feet to the right of the cg. The tail lift is centered 100 feet to the 

right of the cg. How much force must the tail lift supply in order to keep the plane from rotating? Is this force upward 

or downward?

2. Continuing with Problem 1, suppose the lift force is located 20 feet to the left of the cg. How much force must the tail 

lift supply in order to keep the plane from rotating ? Is this force upward or downward?

(Hint: Distances measured to the right of the cg are positive while distances to the left are negative.)

Explain why the tail lift force is different in Problems 1 and 2 while the wing lift force is the same.

3. Consider Problem 1 again. Suppose the fuel tanks become significantly lighter after a long flight. This causes the cg to 

move toward the front of the plane (to the left). Will a change in the position of the cg of 10 feet to the left affect the 

previously trimmed (balanced) airplane? (Mathematically support your answer.)

4. If the movement of the cg causes the airplane to become unbalanced, what do you think the pilot will do to regain 

balance?

5. Consider Problem 2 again. Suppose the same thing that happened in Problem 4 happens in Problem 2. Will a change in 

the position of the cg of 10 feet to the left affect the previously trimmed (balanced) airplane? (Mathematically support 

your answer.)
Aeronautics for Introductory Physics: Force Particle 157

Inquiry Lesson/Lab: Investigating Flight with a Toy Helicopter

Michael Liebl. The Physics Teacher, 48, 458 (2010).

Although lift has traditionally been measured 

with the use of a wind tunnel and force 

meters, there is an alternative way to measure 

lift produced by a small remote-controlled 

helicopter. An electronic balance can be used to 

measure mass, which can then be converted into 

weight. Any decrease in “apparent mass” on the 

electronic balance is the result of a net upward 

force, called lift. 

A small helicopter can be loaded onto an 

electronic balance and mounted with the help 

of bits of duct tape. By taring the balance and 

starting at “0,” any lift produced by the helicopter 

(up to the lift that is equal to its weight plus the 

platform to which it has been affixed, unless 

the helicopter itself cannot lift that mass) will 

be represented in the measurement unit with 

a negative value. Students can also measure 

the related frequencies of rotation of the 

propellers by using a stroboscope to perfectly 

match the frequency of the propellers until they 

appear stationary. (Note: Paint or mark ONE 

tip of a single rotor blade with a bright color 

or correction fluid in order to ensure that the 

same propeller wing is being matched with the 

frequency of the stroboscope). Once students 

understand how they can collect this information, 

they can be prompted to develop a model.

Objectives 

Determine the relationship 

between frequency of 

helicopter propellers and lift.

Materials 

RC helicopter 

Electronic balance 

Tape 

Stroboscope 

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 2, 4

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Investigating Flight 

with a Toy Helicopter
158 Aeronautics for Introductory Physics: Force Particle

(Note: It is possible that teachers and students alike would expect the force reading on the electronic balance to read the same 

value, whether or not the helicopter is spinning. However, unlike a fan-cart and card scenario, in which internal forces cancel 

each other out, helicopters tend to push the column of air down and around them, not entirely on the plate below it. A variety 

of techniques can be used to minimize this “ground effect” on the balance reading. Consider mounting the helicopter on a tall 

stand a foot or so above the plate – this can be easily achieved by building a Lego or K’nex tower to support the helicopter. 

Alternatively, consider mounting the helicopter on the end of a meter stick lever suspended over the edge of a table, and 

measuring force with a force meter on the opposite side of the lever, as shown below.)

Image and idea credit: 

Blanco, C. (2013) , “Prop It Up! Static Thrust and Efficiency of Small Aircraft Propellers”, California State Science Fair 

Project J0105, online at http://www.usc.edu/CSSF/History/2013/Projects/J0105.pdf
Aeronautics for Introductory Physics: Force Particle 159

Teacher Background and Key: Investigating Flight with a Toy Helicopter

Achieving lift is a difficult task for heavy objects. As such, it is important to understand what factors influence lift – such as 

the frequency of propeller rotation – and how lift can be achieved in the most efficient way possible. 

First, allow students to qualitatively determine the relationship between frequency and lift. Help them to note that the scale 

maxes out its “lift” reading once the lift equals the weight of the helicopter, and that data collection near at this high end of 

the data range might result in inaccurate values. 

Next, help students learn how to use the 

stroboscope in order to determine the 

frequency of the propellers. (Caution: Check 

student medical records and ask students 

about any sensitivities to bright, flashing 

lights, as these are known triggers for many 

seizure disorders and some emotional 

traumas). Darken the room as much as 

possible (or, for observation by a single 

student, you can place the apparatus inside of 

a box or container), and flash the stroboscope 

as the helicopter propellers rotate. By holding 

the frequency of the propellers as steady as 

possible with the remote controller, adjust the 

stroboscope frequency until the propellers 

appear to be stationary. Once the frequency 

of the propellers and the frequency of the 

stroboscope match, students can record the 

value on the stroboscope. (Caution students 

to always use the lowest frequency possible – 

students might find variable harmonics if the 

frequency increases too much.)

Goal: Determine the graphical and mathematical model relating frequency and lift.

Ask students to collect as many data points as possible within a wide range 

of frequencies. Students should find that the relationship is a square-root 

curve. The data can be effectively linearized to result in a graph such as that 

developed by the article’s author and seen below.

Generally, students should find that

Deriving a relationship from this particular data, students should get
160 Aeronautics for Introductory Physics: Force Particle

Students can then hypothesize significance and meaning of slope and 

intercept. In this particular case, the slope likely has some relationship to 

the mechanics and physical design of the helicopter and its propellers. An 

interesting engineering challenge might be to attempt to minimize the 

slope of this graph by modifying the surface area and shape of the 

propellers.

An opportunity to test this theoretical data is to actually load the 

helicopter with a given mass, and to see at which minimum propeller 

frequency the helicopter is able to hover. Theoretically, aeronautical 

research regularly uses the principle of “ground effect” to maximize lift 

near the ground. Ground effect is an interruption of wingtip vortices and 

downwash that occurs behind a wing, causing drag on an aircraft. When 

this is interrupted and there is less drag, lift increases as a result. Hence, 

actual lift on the scale’s platform, derived from propeller rotations, is 

likely lower in the air, where ground effect has little impact. 

The above mathematical model for the relationship between frequency 

and lifted mass can be derived from downward concentrations of airflow 

through the propellers. The complete derivation can be found in the full 

article from TPT. However, one interesting point to note is that helicopter 

blades are curved, with the greatest angle of attach nearest to the rotor. 

Because tangential velocity significantly increases near the edge of the 

propeller tip, less “twist” is needed to achieve the same lift.
Aeronautics for Introductory Physics: Force Particle 161

Student Worksheet: Investigating Flight with a Toy Helicopter

Achieving lift is a difficult task for heavy objects. As such, it is important to understand what factors 

influence it, and how it can be achieved in the most efficient way possible. One of the primary factors that 

influences lift of rotocraft is the frequency of rotation of the propellers.

Goal: Determine the graphical and mathematical model relating frequency and lift.

1. Get to know your helicopter – play with it and learn how to lift it off of the ground remotely. Plan 

how you will collect data on frequency and lift. If you are uncertain of how to measure frequency, 

ask your teacher for help! (Hint: Use a stroboscope!) To measure lift, consider using an electronic 

balance. With your small group, discuss how lift can be inferred from the reading on the balance.

2. Decide what data you will collect, and how you will go about doing it. Confer with your teacher to 

make sure that your plan is safe.

3. Collect your data below, plot it using a graphical analysis tool, and then sketch the general graph to 

the right:

4. Generally, what kind of relationship is displayed by your graph?

5. If appropriate, perform a linearization or curve fit. Write the general form of the equation below:

6. If you have any constants in your equation, what do they mean? What constants (or error) in your 

experiment might account for them?
162 Aeronautics for Introductory Physics: Force Particle

7. Share your results with the class. What conclusions can you draw about the relationship you studied? 

Do you think they are any limitations to your study? Why or why not?

8. Most propellers (on RC helicopters and on standard aircraft) are “twisted.” Explain why! (Consider 

angular velocity). 

Future Helicopters Get SMART

http://www.nasa.gov/topics/aeronautics/features/smart_rotor.html#.VHOgJ429-Rg 

Analysis Questions:

• Why are rotocraft currently not used for typical transportation, like airplanes?

• What would be a benefit of using rotocraft instead of airplanes, as a primary means for transportation in the future?

• What are piezoelectric materials, and what benefits could they offer rotocraft?
Aeronautics for Introductory Physics: Force Particle 163

Interactive Demonstration: Lift Demonstrations

Students are often incorrectly taught 

that the primary means for heavierthan-air flight is a result of Bernoulli’s 

principle. While it is true that the 

cambered shape of an aircraft wing 

results in a lower pressure above the 

wing, and an increased pressure below 

the wing, the resulting upward force 

is only partially the cause for the 

overall lift of an aircraft. Additionally, 

some aircraft do not have cambered 

wings at all. Further, even those who 

speak of Bernoulli often attribute the 

effect to the theory of equal transit 

times of particles above and below the 

wing, which is a misconception and 

simply not true. Lift can be achieved 

on a variety of wing shapes, including 

simple airfoils made from paper. 

Objectives 

Determine the relationship 

between the surface area and 

lift.

Materials 

Fan 

Pencils / Sticks 

Notecards 

Glue 

Tape 

NGSS 

CDI HS-PS2 

SEP 6

CCC 1, 2, 3, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Lift Demonstrations
164 Aeronautics for Introductory Physics: Force Particle

A discussion of Newton’s third law of motion more accurately represents how lift is achieved in this case. Because of the 

way wings obstruct air particles, air is deflected downward, resulting in an upward force on the wing. However, even this 

perspective can be oversimplified to a fault, and can result in student misconceptions. Air particles do not simply move 

directly toward the wing, hit it, and then reflect off. In contrast, air behaves much more smoothly and in “sheets.” (Hence, 

smoke contrails are seen to smoothly go around airfoils, and might not encounter any obstruction). Air particles interact with 

other air particles, and indirectly are affected by the wing even before arriving at it. 

To demonstrate this effect, students or teachers can prepare note cards on sticks or long pencils. Area #1 can be achieved by 

taking two note cards (one above the other), and Area #2 can be achieved by pasting the two cards together, giving the same 

weight, but half the surface area. The airfoils can then be suspended from the ends of the pencils and held in the stream of 

a fan. Students should quickly and easily realize that lift is much greater on an object with greater surface area. For more 

information on lift, see: http://goo.gl/m88nr1
Aeronautics for Introductory Physics: Force Particle 165

Inquiry Lesson/Lab: Measuring Lift with the Wright Airfoils

Richard M. Heavers & Arianne Soleymanloo. The Physics Teacher, 49, 502 (2011).

A large portion of aeronautics research 

is conducted within wind tunnels: 

open or closed systems in which gases 

of varying speeds, pressures, and 

temperatures bombard a full-scale or 

model aircraft or aircraft part, such as a 

wing cross-section. 

From the previous lift demonstrations, 

it should be apparent to students that 

wind speed, surface area, and angle 

of attack all have a significant impact 

on the lift experienced by an airplane 

wing. Although there are a number 

of options to buy commercial wind 

tunnels for the high school laboratory, the following is a smaller-scale, cheaper approach to 

creating a wind tunnel to test miniature airfoils. 

Most scientific probeware companies such as Pasco and Vernier offer rotating rods or plates 

for the study of rotational motion. A protractor can be mounted onto the end in order to help 

students align the airfoil with a given angle of attack. A make-do miniature wind tunnel (made 

with a large PVC pipe and a wind source such as a leaf blower, hair dryer, or focused box fan) 

can be aligned with the rotating rod or plate. 

Objectives 

Determine graphical models 

that relate lift and drag forces 

to wind speed, surface area, 

angle of attack (and stall 

angle), camber, and aspect 

ratio.

Materials 

Rotating platform 

Soda can and scissors 

Tin and tin snips (optional)

PVC pipe 

Wind source 

Pulley system 

Small masses or force meter 

Wind speed sensor 

(mechanical or digital 

anemometer probe, Prandtl 

pressure probe, and / or a 

manometer)

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 2, 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Measuring Lift with 

Wright Airfoils
166 Aeronautics for Introductory Physics: Force Particle

A small airfoil can be constructed out of tin or other 

malleable metal that retains it shape easily but can also be 

bent to change its shape/camber/aspect ratio. (However, 

before introducing these variables, it is best to begin with a 

flat plate airfoil). Aluminum soda cans provide a cheap 

source of tin. Using tin snips, a variety of airfoils can be 

created with varying amounts of surface area. (Caution: 

Metal can have very sharp corners and edges. Always 

caution students to use protective eyewear when cutting 

the tin). The airfoil can also be mounted on the end of the 

rotating rod or plate in such a way as to change its angle of 

attack. A dab of glue from a hot glue gun allows for easy 

mounting and removal of the airfoil.

Once the air source is turned on and the airfoil is exposed 

to the wind, the rotating plate or rod will tend to move 

toward one direction. By simply touching the opposite end 

of the rod, students can literally feel the force of the lift 

produced by the airfoil. To quantify this, the end of the rod 

can be connected via a pulley system, and mass added to 

a small hanger to equal the rotating plate or rod until it is 

“balanced” in the wind.

Wind speed can be measured with the use of a mechanical 

or digital anemometer probe.

Given all of these materials, students can measure lift 

(or drag, if the apparatus is rotated by ninety degrees) as 

they relate to each of the following variables. See Wing 

Geometry Definitions for further information.

• wind speed (relative speed)

• surface area

• angle of attack

• camber

• aspect ratio

Although the following student worksheets do not explicitly 

address each of the stages of the learning cycle for its 

repetitive nature in this kind of in-depth study with multiple 

parts, ensure that students address each step.
Aeronautics for Introductory Physics: Force Particle 167

Teacher Guide/Key: Measuring Lift with Wright Airfoils

Goal: Determine the mathematical and graphical models relating lift and wind speed.

From simple experiments with a fan and a pie pan, students can see that an 

increase in wind speed increases the lift on an airfoil. (Likewise, when driving 

in an open area, students might have placed their hand out a window, tilted it 

up slight, and experienced the upward push of the wind on their hands).

In this and the following experiments, students modify variables associated 

with air flowing around a tin airfoil. Begin by having students work with a flat 

plate airfoil. Doing this helps students to challenge the misconception that lift 

can only be produced on an curved airfoil. Curvature can later be introduced 

during the study of camber.

Perhaps the easiest way to modify wind speed is to use a fan with multiple 

speed settings. An anemometer can be used to measure the wind speed. Given 

that a wind-tunnel effect produced by a standard fan is likely to have only a 

small cross-section of constant air flow, it is preferable to use a very small 

anemometer as opposed to a wide, mechanical one. Alternatively, students can change the wind speed by changing the wind 

source – hair dryer, fan, leaf blower, etc.

In general, students should find that the lift is proportional to speed squared. Hence, a plot of lift versus speed will result in a 

parabolic curve.

Goal: Determine the mathematical and graphical models relating lift and surface area.

Students can cut out various tin airfoils of different surface areas. If the 

concept of aspect ratio has not yet been introduced to students, help them to 

understand that it is also a variable that must also be accounted for. (Aspect 

ratio is the ratio of the long side to the width of the rectangle, or L/W). Not 

only should students be careful to maintain aspect ratio, but they also must be 

careful to mount the airfoil in the same way, so that the leading edge of the 

airfoil is either the length or the width.

This is also an opportunity to introduce to students to the concepts of camber. 

Many students are used to seeing airfoils that are curved – these airfoils have 

a camber. In its simplest form, camber of a curved airfoil can be calculated 

by taking the length of the chord (from the leading edge to the trailing edge) 

divided by the thickness of the airfoil (from the chord to the highest point 

of the curve). While this is beneficial for optimizing the lift/drag ratio at big 

angles of attack, it is not necessary in order to generate lift. For the purposes of 

studying surface area, it is probably best to avoid having any curvature at all.

Lastly, students must also ensure that angle of attack is controlled throughout this experiment. The angle of attack refers 

to the angle from the direction of the wind measured against the chord of the airfoil (which, for a flat airfoil, would simply 

measure to the surface of the airfoil). Students may choose any angle of attack they like, but the angle tends to be small for 

most aircraft (around 3-5 degrees above or below the horizontal, depending upon if the airplane is gaining or losing altitude). 

Angles above 15-30 degrees, depending upon the airfoil, will tend to result in a stall – a dramatic decrease in lift and increase 

in drag. (Caution: This result might not be reproducible for a model wind tunnel using a low wind speed and a self-made 

airfoil, as the wind only blows under the airfoil at some point).

Students should find that the relationship between lift and surface area is linear.
168 Aeronautics for Introductory Physics: Force Particle

Combining the relationship between lift and air speed, and lift and surface area, it can be easy to derive the expression

This relationship offers a wonderful opportunity to perform some dimensional analysis to determine the meaning and source 

of k1. Substituting basic dimensions M (mass), L (length), and T (time), the equation above can be represented as 

The above relationship can be simplified and solved for the dimensions of k1

.

Students should recognize that these units, in SI notation, would be kg/m3, representative of a measure of density! 

By using data points from both graphs for the airfoil in the same situation, students will, however, find that the value of k1 is 

ridiculously high to be the density of the fluid (air) surrounding the airfoil.

The constant, k1, although it does include a measure of fluid density, is dependent upon a number of situational variables. 

Hence:

Students should consider other variables that influence lift, including, most notably, the shape of the airfoil (camber and 

aspect ratio) and angle of attack. Likewise, all of these variables can be evaluated for their effect on lift.

Students can then be asked to perform a number of other experiments to determine relationships. Lift and angle of attack 

are perhaps the easiest of the remaining three variables to study, although its relationship cannot be readily developed into a 

mathematical model. 

Goal: Determine the graphical model relating lift versus angle of attack (and the stall angle).

As the angle of attack increases from zero, the lift increases, initially. The lift then reaches a maximum value until just before 

it hits the stall angle. At this point, lift decreases as angle of attack increases, because most of the momentum transfer from 

the wind now pushes the airfoil backward as opposed to upward.
Aeronautics for Introductory Physics: Force Particle 169

Goal: Determine the graphical model relating lift and camber.

Students can change the camber of an airfoil by curving it various amounts. 

Generally, as camber increases, lift increases, within reasonable limits.

Goal: Determine the graphical model relating lift and aspect ratio.

Students can change the aspect ratio of an airfoil by cutting out rectangular foils 

of the same surface area, but different lengths and widths. Generally, as aspect 

ratio increases, lift increases.

The above three relationships are not linear, and they are not simple. However, 

using simple calculus, it can be demonstrated that k2

 is composed of a coefficient 

of lift (which depends upon the angle of attack, camber, and aspect ratio) and the 

constant, 1⁄2. This relationship can be developed and helps students to see that 

lift is actually generated by the conservation of energy and mass.

For a moving object (such as in fluid flow),

There are plenty of opportunities to engage in data analysis 

with the relationships demonstrated from what is collected. 

The following is a display of some example data provided 

by the article’s authors.

Perhaps the most important aspect in studying the angle of 

attack is identifying the “peak” of lift force, which shows 

the angle at which lift force begins to decrease. This is 

known as the “stall angle,” and should never be exceeded 

unintentionally, as it results in loss of control of the aircraft.

Goal: Develop a visual and descriptive model for why 

long, narrow airfoils produce more lift at small angles 

of attack, as compared to square airfoils with the same 

surface area and camber.

Students can develop visual and descriptive models to 

explain the molecular and fluid behaviors resulting in 

varying lift forces. Because it is a chaotic system, this kind 

of modeling can be hard to do accurately.
170 Aeronautics for Introductory Physics: Force Particle

One thing that can help students is to have them observe fluid flow visually. Flow can be seen using a piece of glass and an 

airfoil section (perhaps made out of Styrofoam, clay, or gelatin to easily modify the airfoil), and then slowly pouring colored 

water around it.

See http://amasci.com/wing/lasrWing.gif for an example of fluid flow around wings.
Aeronautics for Introductory Physics: Force Particle 171

Student Worksheet: Measuring Lift with Wright Airfoils

In the mid to late 1800s, the use of wind tunnels to test the effectiveness of airplane wing shapes became popular. In the early 

1900s, the Wright brothers also began completing their own tests, to determine which kind of airfoil would generate the most 

lift. 

1. What kinds of design aspects do you think influence the lift produced by a wing? 

To measure how much wind impacts an airfoil, create a setup following your teacher’s guidelines. Ensure that the 

device – with the mounted airfoil – is balanced before you turn on the wind generator/fan. (The rotating rod should not 

tend toward one direction or the other). 

2. Turn the wind generator on. The airflow should cause the entire rotating rod to turn toward one direction. Explain why 

this happens by annotating the image of wind currents and an airfoil below:

3. Looking at your setup, explain how you will determine the lift on the airfoil produced by the wind.

Goal: Determine the mathematical and graphical models relating lift and wind speed.

4. If you have ever tried to take a kite out gliding on a calm day, you know that 

wind speeds makes a big difference on lift! Perform an experiment to determine the relationship between lift and wind speed. Collect your data below, 

then draw a sketch of the graph.

5. What is the general mathematical model of the relationship displayed by the graph?
172 Aeronautics for Introductory Physics: Force Particle

Goal: Determine the mathematical and graphical models relating lift and surface area.

6. Perform an experiment to determine the relationship between lift and wind 

speed. Collect your data below, then draw a sketch of the graph.

7. What is the general mathematical model of the relationship displayed by the graph? 

8. Combining the two relationships (from lift and air speed, and lift and surface area), to form a single equation with an 

unknown constant:

9. Using what you know about the units for lift (N), speed (m/s), and surface area (m2), perform dimensional analysis to 

determine the units of the unknown constant.

10. Based upon the unit for the unknown constant, what might be the meaning of this constant? Perform data analysis to 

determine what the unit might be. Ask your teacher for assistance, if necessary.

Aeronautical engineers combine all of these known variables into something called the “Lift Equation,” which, after 

discussing with your classmates, your teacher will help you to derive here:
Aeronautics for Introductory Physics: Force Particle 173

Extension

You will likely find that the unknown constant is actually a combination of constants associated with both the nature of the 

fluid flowing around the airfoil as well as the physical characteristics of the airfoil. We will study three of these variables.

Goal: Determine the graphical model relating lift versus angle of attack (and the stall angle).

11. Perform an experiment to determine the relationship between lift and angle 

of attack. Be certain to collect data from 5 to 50 degrees. Collect your data 

below, then draw a sketch of the graph.

12. Describe, in words, the relationship displayed by the graph.

13. Your graph should display a maximum value known as the “stall angle.” Based upon your graph and your understanding of aeronautical “stall,” explain what happens to a plane at this point, and why.

Goal: Determine the graphical model relating lift and camber.

14. Perform an experiment to determine the relationship between lift and camber. 

Collect your data below, then draw a sketch of the graph.

15. Describe, in words, the relationship displayed by the graph.
174 Aeronautics for Introductory Physics: Force Particle

Goal: Determine the graphical model relating lift and .

16. Perform an experiment to determine the relationship between lift and aspect 

ratio. Be certain to collect data from 5 to 50 degrees. Collect your data below, 

then draw a sketch of the graph.

17. Describe, in words, the relationship displayed by the graph.

18. Look at some data provided by another group of 

students and even the Wright brothers themselves. 

How well does your data compare? If you find that 

you observed other trends, explain why this might be 

the case.

Goal: Develop a visual and descriptive model for why 

long, narrow airfoils produce more lift at small angles 

of attack, as compared to square airfoils with the same 

area and camber.

19. Using the graph to the right, attempt to explain why 

airfoils of different angle of attack, aspect ratio, and 

camber, all result in different amounts of lift. Consider 

component forces, drag, Coander effect, boundary 

layer, turbulence, measuring errors, Bernoulli’s 

principle, etc.
Aeronautics for Introductory Physics: Force Particle 175

Glow With the Flow

http://www.nasa.gov/aero/flow_patterns_image.html#.VHTRmY29-Rg

Analysis Questions:

• Describe the technique observed in the picture. How are scientists visualizing airflow?

• Do a quick Internet search to find out what other information can be collected from wind 

tunnel tests (list at least three, and how they are collected).
176 Aeronautics for Introductory Physics: Force Particle

Problem Set: Lift-Velocity Relationship Problems

Beginner’s Guide to Aerodynamics

BACKGROUND

Lift is created by deflecting a flow of air and drag is generated on 

a body in a wide variety of ways. From Newton’s second law of 

motion, the aerodynamic forces on the body (lift and drag) are 

directly related to the change in momentum of the fluid with time. 

The fluid momentum is equal to the mass times the velocity of the 

fluid. Since the air moves, defining the mass gets a little tricky and 

aerodynamicists usually relate the effect of mass on lift and drag 

to the air density. The mathematical derivation for this conversion 

is given on another slide dealing with momentum effects on lift. 

As a result of this derivation, we find that lift and drag depend on 

the square of the velocity. The velocity used in the lift and drag 

equations is the relative velocity between an object and the flow. 

Since the aerodynamic force depends on the square of the velocity, 

doubling the velocity will quadruple the lift and drag.

PROBLEM SET

Student Worksheet

Teacher Answer Key

You may have been wondering about the photo of the “Spirit of St. Louis” above. We will now use our lift/velocity 

relationship information on an imaginary flight in the Spirit of St. Louis where we again maintain altitude. Data needed to 

continue is found at “Ryan, Spirit of St. Louis.” (http://en.wikipedia.org/wiki/Spirit_of_St._Louis)
Aeronautics for Introductory Physics: Force Particle 177

1. Find “k” as above using the Spirit of St. Louis maximum velocity and the takeoff weight less 20 kg (an arbitrary 

amount of fuel to attain maximum velocity). Once you have determined k, write an equation for velocity as a function 

of lift (as in our example). v= __________.

2. Using a function grapher (such as your graphing calculator), graph this equation (default window settings are 

acceptable). Do you recognize the graph? It is a ______________.

3. Now key in the following RANGE window values: xmin=1000, xmax=2500, xscl=100, ymin=120, ymax=220, yscl=10, and 

graph the equation. Because the domain and range are limited to real Spirit of St. Louis values, the curvature of the 

parabola segment is limited.

4. Return to the equation and solve it for lift. L= ________________. Graph the equation. Then exchange the x and y 

values in the RANGE window; that is, key in: xmin=120, xmax=220, xscl=10, ymin=1000, ymax=2500, yscl=100, and graph. 

The parabola segment now opens upwards, instead of to the side.

(Note: You may wish to try this for other aircraft.)
178 Aeronautics for Introductory Physics: Force Particle

Problem Set: Lift Equation Problems

Beginner’s Guide to Aerodynamics

Lift depends on the density of the air, the square of the 

velocity, the air’s viscosity and compressibility, the surface 

area over which the air flows, the shape of the body, and the 

body’s inclination to the flow. In general, the dependence on 

body shape, inclination, air viscosity, and compressibility is 

very complex.

One way to deal with complex dependencies is to characterize 

the dependence by a single variable. For lift, this variable 

is called the lift coefficient, designated “CL.” This allows us 

to collect all the effects, simple and complex, into a single 

equation. The lift equation states that lift L is equal to the lift 

coefficient CL times the density r times half of the velocity v

squared times the wing area A.

L = CL * A * .5 * ρ * v2 

For given air conditions, shape, and inclination of the object, we have to determine a value for CL to determine the lift. 

For some simple flow conditions and geometries and low inclinations, aerodynamicists can determine the value of CL 

mathematically. But, in general, this parameter is determined experimentally.

In the equation given above, the density is designated by the letter “ ρ.” We do not use “d” for density, since “d” is often 

used to specify distance. In many textbooks on aerodynamics, the density is given by the Greek symbol ρ - “rho” (Greek for 

“r”). The combination of terms “density times the square of the velocity divided by two” is called the dynamic pressure and 

appears in Bernoulli’s pressure equation.

You can view a short movie of Orville and Wilbur Wright discussing the lift force and how it affected the flight of their 

aircraft. The movie file can be saved to your computer and viewed as a podcast.

1. Identify each letter in the lift equation and list acceptable units for each quantity.

2. Use Wikipedia (http://en.wikipedia.org/wiki/Boeing_747), or a search engine, to find information about the Boeing 747. 

What is its wing area?

3. If the Boeing 747 above is cruising at 940 km/h, the density of the air is 1.20 g/L, and the lift coefficient is 1, what is 

the lift?

4. Use Wikipedia (http://en.wikipedia.org/wiki/Boeing_777), or a search engine, to find information about the Boeing 777. 

What is its wing area?
Aeronautics for Introductory Physics: Force Particle 179

5. If the Boeing 777 is cruising at the same velocity under the same conditions as the 747 above, what is its lift?

6. Look up the DC-8 on Wikipedia (http://en.wikipedia.org/wiki/DC-8), or a search engine. What is its wing area?

7. What is the DC-8’s cruising velocity?

8. If the air density is 1.26 g/L and the lift coefficient is 1.5, what is the lift of the DC-8?

9. Find the F/A-18A Hornet on Wikipedia (http://en.wikipedia.org/wiki/F-18), or use a search engine. What is its wing 

area?

10. What is the F/A-18A’s lift while it flies at 700 mi/hr, the air density is 1.31 g/L, and the lift coefficient is 1.8?
180 Aeronautics for Introductory Physics: Force Particle

Model Building: Developing a Model for Lift

Beginner’s Guide to Aerodynamics: Lift Theories 

When two solid objects interact 

in a mechanical process, forces

are transmitted, or applied, at 

the point of contact. But when 

a solid object interacts with a 

fluid, things are more difficult 

to describe because the fluid 

can change its shape. For a solid 

body immersed in a fluid, the 

point of contact is every point 

on the surface of the body. The 

fluid can flow around the body 

and maintain physical contact at 

all points. The transmission, or 

application, of mechanical forces 

between a solid body and a fluid occurs at every point on the surface of the body. And the 

transmission occurs through the fluid pressure.

Variation in Pressure

The magnitude of the force acting over a small section of an object immersed in a fluid equals 

the pressure P times the area A of the section. A quick units check shows that:

As discussed on the fluid pressure slide, pressure is a scalar quantity related to the momentum 

of the molecules of a fluid. Since a force is a vector quantity, having both magnitude and 

direction, we must determine the direction of the pressure force. Pressure acts perpendicular 

(or normal) to the solid surface of an object. So the direction of the force on the small section 

of the object is along the normal to the surface. We denote this direction by the letter n.

Objectives 

Compare and contrast common 

incorrect theories of lift.

Materials 

Incorrect Theory

Readings

NGSS 

CDI HS-PS2 

SEP 2, 6, 7

CCC 4

CCSS 

Reading in Science 

Writing in Science

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Developing a Model 

for Lift
Aeronautics for Introductory Physics: Force Particle 181

The normal direction changes from the front of the airfoil to the rear and from the top to the bottom. We indicate this 

variation on the figure by several small arrows pointing perpendicular to the surface and labeled with an n. To obtain the net 

mechanical force over the entire solid object, we must sum the contributions from all the small sections. Mathematically, the 

summation is indicated by the Greek letter sigma (∑) The net aerodynamic force F is equal to the sum of the product of the 

pressure P times the incremental area delta A in the normal direction n.

In the limit of infinitely small sections, this gives the integral of the pressure times the area around the closed surface. Using 

the symbol ∫ dA for integration, we have:

where the integral is taken all around the body. On the figure, that is why the integral sign has a circle through it.

If the pressure on a closed surface is a constant, there is no net force produced because the summation of the directions of 

the normal adds up to zero. For every small section there is another small section whose normal points in exactly the opposite 

direction.

For a fluid in motion, the velocity has different values at different locations around the body. The local pressure is related to 

the local velocity, so the pressure also varies around the closed surface and a net force is produced. On the figure at the lower 

right, we show the variation of the pressure around the airfoil as obtained by a solution of the Euler equations. The blue line 

shows the variation from front to back on the lower surface, while the red line shows the variation from front to back on the 

upper surface, The black line gives the reference free stream pressure. Summing the pressure perpendicular to the surface 

times the area around the body produces a net force.

Definitions of Lift and Drag

Since the fluid is in motion, we can define a flow direction along the motion. The component of the net force perpendicular 

(or normal) to the flow direction is called the lift; the component of the net force along the flow direction is called the drag.

These are definitions. In reality, there is a single, net, integrated force caused by the pressure variations along a body. This 

aerodynamic force acts through the average location of the pressure variation which is called the center of pressure.

Velocity Distribution

For an ideal fluid with no boundary layers, the surface of an object is a streamline. If the velocity is low, and no energy is 

added to the flow, we can use Bernoulli’s equation along a streamline to determine the pressure distribution for a known 

velocity distribution. If boundary layers are present, things are a little more confusing, since the external flow responds to the 

edge of the boundary layer and the pressure on the surface is imposed from the edge of the boundary layer. If the boundary 

layer separates from the surface, it gets even more confusing. How do we determine the velocity distribution around a body? 

Specifying the velocity is the source of error in two of the more popular incorrect theories of lift. To correctly determine 

the velocity distribution, we have to solve equations expressing a conservation of mass, momentum, and energy for the fluid 

passing the object. In some cases, we can solve simplified versions of the equations to determine the velocity and pressure.

Summary

To summarize, for any object immersed in a fluid, the mechanical forces are transmitted at every point on the surface of the 

body. The forces are transmitted through the pressure, which acts perpendicular to the surface. The net force can be found by 

integrating (or summing) the pressure times the area around the entire surface. For a moving flow, the pressure will vary from 

point to point because the velocity varies from point to point. For some simple flow problems, we can determine the pressure 

distribution (and the net force) if we know the velocity distribution by using Bernoulli’s equation.
182 Aeronautics for Introductory Physics: Force Particle

Teacher Worksheet: Theories of Lift

Read the three explanations on lift and the popular incorrect theories on how lift is produced. Then using the information 

about each of the theories, complete the questions below designed to demonstrate your understanding of the theories.

1. Compare and contrast the incorrect lift theories #1 and #3.

Answers will vary.

2. Contrast the incorrect lift theories #2 and #3.

 Answers will vary.

3. Which theory(ies) cannot explain why airplanes can fly upside down?

 Answers will vary.

4. Which theory(ies) cannot explain why the upper surface of a wing has an effect on lift?

 Answers will vary.

5. Which theory(ies) cannot explain why the air flowing past the lower surface of a wing has an effect on lift?

 Answers will vary.

6. Explain the correct theory of lift in your own words.

Answers will vary.
Aeronautics for Introductory Physics: Force Particle 183

Student Worksheet: Theories of Lift

Read the three explanations on lift and the popular incorrect theories on how lift is produced. Then using the information 

about each of the theories, complete the questions below designed to demonstrate your understanding of the theories.

1. Compare and contrast the incorrect lift theories #1 and #3.

2. Contrast the incorrect lift theories #2 and #3.

3. Which theory(ies) cannot explain why airplanes can fly upside down?

4. Which theory(ies) cannot explain why the upper surface of a wing has an effect on lift?

5. Which theory(ies) cannot explain why the air flowing past the lower surface of a wing has an effect on lift?

6. Explain the correct theory of lift in your own words
184 Aeronautics for Introductory Physics: Force Particle

The theory described here is one of the most widely circulated, incorrect explanations. The theory can be labeled the “longer 

path” theory, or the “equal transit time” theory. The theory states that airfoils are shaped with the upper surface longer than 

the bottom. The air molecules (the little colored balls on the figure) have farther to travel over the top of the airfoil than along 

the bottom. In order to meet up at the trailing edge, the molecules going over the top of the wing must travel faster than the 

molecules moving under the wing. Because the upper flow is faster, then, from Bernoulli’s equation, the pressure is lower. 

The difference in pressure across the airfoil produces the lift.

Let’s use the information we’ve just learned to evaluate the various parts of the “equal transit time” theory.

• Lifting airfoils are designed to have the upper surface longer than the bottom. 

- This is not always correct. The symmetric airfoil in our experiment generates plenty of lift and its upper surface 

is the same length as the lower surface. Think of a paper airplane. Its airfoil is a flat plate --> top and bottom 

exactly the same length and shape and yet they fly just fine. This part of the theory probably got started because 

early airfoils were curved and shaped with a longer distance along the top. Such airfoils do produce a lot of 

lift and flow turning, but it is the turning that’s important, not the distance. There are modern, low-drag airfoils 

which produce lift on which the bottom surface is actually longer than the top. This theory also does not explain how airplanes can fly upside down which happens often at air shows and in air-to-air combat. The longer 

surface is then on the bottom!

• Air molecules travel faster over the top to meet molecules moving underneath at the trailing edge.

- Experiments show that the flow over the top of a lifting airfoil does travel faster than the flow beneath the 

airfoil. But the flow is much faster than the speed required to have the molecules meet up at the trailing edge. 

Two molecules near each other at the leading edge will not end up next to each other at the trailing edge. This 
Aeronautics for Introductory Physics: Force Particle 185

part of the theory attempts to provide us with a value for the velocity over the top of the airfoil based on the 

non-physical assumption that the molecules meet at the aft end. We can calculate a velocity based on this 

assumption, and use Bernoulli’s equation to compute the pressure, and perform the pressure-area calculation 

and the answer we get does not agree with the lift that we measure for a given airfoil. The lift predicted by the 

“equal transit time” theory is much less than the observed lift, because the velocity is too low. The actual velocity over the top of an airfoil is much faster than that predicted by the “longer path” theory and particles moving 

over the top arrive at the trailing edge before particles moving under the airfoil.

• The upper flow is faster and from Bernoulli’s equation the pressure is lower. The difference in pressure across the 

airfoil produces the lift.

- As we have seen in Experiment #1, this part of the theory is correct. In fact, this theory is very appealing 

because many parts of it are correct. In our discussions on pressure-area integration to determine the force on a 

body immersed in a fluid, we mentioned that if we know the velocity, we can obtain the pressure and determine 

the force. The problem with the “equal transit” theory is that it attempts to provide us with the velocity based 

on a non-physical assumption as discussed above.
186 Aeronautics for Introductory Physics: Force Particle

The theory described here is often seen on websites and in popular literature. The theory is based on the idea that lift is the 

reaction force to air molecules striking the bottom of the airfoil as it moves through the air. Because this is similar to the way 

in which a flat rock thrown at a shallow angle skips across a body of water, it is called the “skipping stone” theory of lift. It is 

sometimes called a Newtonian theory of lift, since it involves Newton’s third law, but to avoid confusion with the correct

Newtonian theory of flow turning, we shall call it the “skipping stone” theory.

Let’s use the information we’ve just learned to evaluate the “Skipping Stone” Theory.

• This theory is concerned with only the interaction of the lower surface of the moving object and the air. It assumes 

that all of the flow turning (and therefore all the lift) is produced by the lower surface. But as we have seen in our 

experiment, the upper surface also turns the flow. In fact, when one considers the downwash produced by a lifting 

airfoil, the upper surface contributes more flow turning than the lower surface. This theory does not predict or explain this effect.

• Because this theory neglects the action <--> reaction of molecules striking the upper surface, it does not predict the 

negative lift present in our experiment when the angle of attack is negative. On the top of the airfoil, no vacuum 

exists. Molecules are still in constant random motion on the upper surface (as well as the lower surface), and these 

molecules strike the surface and impart momentum to the airfoil as well.

• The upper airfoil surface doesn’t enter into the theory at all. So using this theory, we would expect two airfoils with 

the same lower surface but very different upper surfaces to give the same lift. We know this doesn’t occur in reality. 

In fact, there are devices on many airliners called spoilers which are small plates on the upper surface between the 

leading and trailing edges. They are used to change the lift of the wing to maneuver the aircraft by disrupting the 

flow over the upper surface. This theory does not predict or explain this effect.

• If we make lift predictions based on this theory, using a knowledge of air density and the number of molecules in a 

given volume of air, the predictions are totally inaccurate when compared to actual measurements. The chief problem with the theory is that it neglects the physical properties of the fluid. Lift is created by turning a moving fluid, 

and all parts of the solid object can deflect the fluid.
Aeronautics for Introductory Physics: Force Particle 187

The theory described here is often seen on websites and in popular literature. The theory is based on the idea that the airfoil 

upper surface is shaped to act as a nozzle which accelerates the flow. Such a nozzle configuration is called a Venturi nozzle

and it can be analyzed classically. Considering the conservation of mass, the mass flowing past any point in the nozzle is a 

constant; the mass flow rate of a Venturi nozzle is a constant. The mass flow rate m dot is equal to the density ρ times the 

velocity v times the flow area A:

For a constant density, decreasing the area increases the velocity.

Turning to the incorrect airfoil theory, the top of the airfoil is curved, which constricts the flow. Since the area is decreased, 

the velocity over the top of the foil is increased. Then from Bernoulli’s equation, higher velocity produces a lower pressure 

on the upper surface. The low pressure over the upper surface of the airfoil produces the lift.

Let’s use the information we’ve just learned to evaluate the various parts of the “Venturi” theory.

• The theory is based on an analysis of a Venturi nozzle. But an airfoil is not a Venturi nozzle. There is no phantom 

surface to produce the other half of the nozzle. In our experiments we’ve noted that the velocity gradually decreases 

as you move away from the airfoil eventually approaching the free stream velocity. This is not the velocity found 

along the centerline of a nozzle which is typically higher than the velocity along the wall.

• The Venturi analysis cannot predict the lift generated by a flat plate. The leading edge of a flat plate presents no 

constriction to the flow so there is really no “nozzle” formed. One could argue that a “nozzle” occurs when the angle 

of the flat plate is negative. But as we have seen in Experiment #2, this produces a negative lift. The velocity actually 

slows down on the upper surface at a negative angle of attack; it does not speed up as expected from the nozzle model.

• This theory deals with only the pressure and velocity along the upper surface of the airfoil. It neglects the shape of 

the lower surface. If this theory were correct, we could have any shape we want for the lower surface, and the lift 

would be the same. This obviously is not the way it works - the lower surface does contribute to the lift generated by 

an airfoil. (In fact, one of the other incorrect theories proposed that only the lower surface produces lift!)
188 Aeronautics for Introductory Physics: Force Particle

• The part of the theory about Bernoulli’s equation and a difference in pressure existing across the airfoil is correct. 

In fact, this theory is very appealing because there are parts that are correct. In our discussions on pressure-area integration to determine the force on a body immersed in a fluid, we mentioned that if we knew the velocity, we could 

obtain the pressure and determine the force. The problem with the “Venturi” theory is that it attempts to provide us 

with the velocity based on an incorrect assumption (the constriction of the flow produces the velocity field). We can 

calculate a velocity based on this assumption, use Bernoulli’s equation to compute the pressure, perform the pressure-area calculation, and yet the answer we get does not agree with the lift that we measure for a given airfoil.
Aeronautics for Introductory Physics: Force Particle 189

Student Worksheet: Theories of Lift

Beginner’s Guide to Aerodynamics is a web-based textbook of information prepared at NASA Glenn Research Center to 

help you better understand aerodynamics. Click Beginner’s Guide Index to access the list of slides. Open the slides called 

Aerodynamic Forces, Incorrect Theory #1, Incorrect Theory #2, and Incorrect Theory #3 (with text) and read the explanations 

on lift and the popular incorrect theories on how lift is produced. Then using the information found in these slides, complete 

the questions below designed to demonstrate your understanding of the theories.

1. Compare and contrast the incorrect lift theories #1 and #3.

2. Contrast the incorrect lift theories #2 and #3.

3. Which theory(ies) cannot explain why airplanes can fly upside down?

4. Which theory(ies) cannot explain why the upper surface of a wing has an effect on lift?

5. Which theory(ies) cannot explain why the air flowing past the lower surface of a wing has an effect on lift?

6. Explain the correct theory of lift in your own words.
190 Aeronautics for Introductory Physics: Force Particle

Inquiry Lesson/Lab: Rubber-Band-Driven Airplane Contest

Douglas Oliver & Terry Ng. The Physics Teacher. 37, 108 (1999).

In this activity, students are provided 

with a balsa and rubber band airplane, 

and are asked to make their airplane 

travel as far as possible while supporting 

a load. This goal can be achieved by 

considering how to maximize the amount 

of lift (while also decreasing the amount 

of drag). Students should consider the 

lift equation, which should have been 

partially derived from the wind tunnel 

experiment:

Although the density of the surrounding fluid (air) cannot be modified, students can make 

changes to the coefficient of lift, the surface area, and the velocity of the airplane through the 

fluid. Each of these will be examined separately.

Although the following student 

worksheets do not explicitly address each 

of the stages of the learning cycle for its 

repetitive nature in this kind of in-depth 

study with multiple parts, ensure that 

students address each step.

Objectives 

Design the furthest-traveling 

airplane.

Materials 

Rubber band airplanes 

Pennies / small masses 

Meter stick

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 1, 2, 4, 6, 7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Rubber-Band-Driven 

Airplane Contest
Aeronautics for Introductory Physics: Force Particle 191

Teacher Guide and Key: Rubber-Band-Driven Airplane Contest

Goal: Design the furthest-travelling airplane.

In order to design the furthest-travelling airplane, all part of 

the lift equation must be accounted for and maximized, as 

follows:

Coefficient of lift

Although there are a number of variables that influence 

the actual value of the coefficient of lift, one of them – the 

angle of attack – can be modified by the placement of a 

penny at a point along the long axis of the plane from tip to 

tail. The relationship between angle of attack and coefficient 

of lift is not linear, however, as demonstrated in the graph 

to the right. Ideally, students will want to increase the angle 

of attack just enough, but before it reaches the stall angle, at 

which the coefficient of lift drops precipitously.

Students can identify the stall angle by watching for a “roller coaster” effect of the airplane as it flies. Once this effect has 

been achieved, students will want to slightly move the penny forward so that the airplane moves at an angle of attack just 

below the stall angle.

Students might also attempt to change the wing camber or aspect ratio by replacing the wooden wings provided in the kit 

with other materials.
192 Aeronautics for Introductory Physics: Force Particle

Surface Area

The surface area of the wings can also be modified, however, 

increasing surface area can also increase drag and have a net 

negative effect. Further, an increase in the size of the wings 

could also result in an increase in the mass of the total airplane. 

Students might consider cutting away bits of the balsa wings and 

replace them with paper, or replace them entirely with lighter 

new materials such as thin plastic.

Velocity

The speed of the wings relative to the air can be modified by 

maximizing the acceleration – either by changing propeller 

torque by winding the rubber band as far as possible or by 

decreasing the mass of the airplane. Students might also want to 

modify the airplane propeller to maximize thrust.

Rolling

Another aspect that must be considered, specifically for rubber 

band airplanes, is the rolling that results from the counteracting 

torque. Because most rubber band airplanes have a single 

propeller that moves in one direction, the airplane body itself will 

tend to roll in the direction opposite to the propeller’s spin. This 

can be resolved by appropriately locating the placement for a 

counterweight (such as a penny) to equalize the force of the roll 

opposite the direction of the propeller.
Aeronautics for Introductory Physics: Force Particle 193

Student Worksheet: Rubber-Band-Driven Airplane Contest

Goal: Design the furthest-travelling 

airplane. 

In this activity, you will design a 

rubber-band airplane that can travel as 

far as possible while carrying a load. 

You will be first provided with a 

balsa wood airplane kit. Assemble it 

according to the instructions, and try to 

fly it (release it from ground level).

1. Record your observations. Especially include any observations 

if you notice that the airplane 

doesn’t fly straight.

2. Unless you inserted a counterweight on one of the wings, you might have noticed that the airplane tended toward one 

direction. This is a result of “torque.” To understand torque, suspend an airplane from the tail, nose down, from a piece 

of string. Then, wind the propeller, and release, allowing the propeller to unwind. What happens to the body of the 

plane? 

3. Before re-designing your airplane, develop a baseline distance of the plane without modification, to see if your changes 

improve or worsen the airplane’s performance. Modification to your plane can include changes in shape, material, or 

size, so long as your plane is powered by a single rubber band and has only one propeller. Record your baseline values 

here:

Then, consider the lift equation partially derived from the wind tunnel lab. By increasing lift, your plane is more likely 

to travel further.

4. Look at all parts of the lift equation. Which terms are constants over which you have no control? Explain why you have 

no control over these constants.

5. Which terms are variables that you can modify? Explain how you plan to do this.
194 Aeronautics for Introductory Physics: Force Particle

Use the following tips to assist you in your re-design of the rubber band plane:

Tip #1: Angle of Attack

Recall the graph of lift vs. angle of attack from your wind tunnel 

activity. Note that the peak value is the stall angle. For this example 

from the graph, what angle would be ideal for the airplane?

You can also change your own plane’s angle of attack by inserting a 

penny along the length of the fuselage. Once in flight, the airplane 

will adjust its angle so that the center of lift and center of gravity are 

directly over one another. 

In order to identify the stall angle on your plane, continue to adjust the location of the penny until you notice that the path of 

the plan becomes “bumpy.” These bumps are locations at which the plane is stalling.

Tip #2: Get Creative

Feel free to make modifications to your airplane. Remove parts of the aircraft and replace it with your own materials. 

Whatever you choose to do, be certain to provide a rationale based upon the lift equation or other reasonable principles.

Tip #3: Test and Revise

You will not know the effect of your modifications (good or bad) until you test them multiple times. Keep a recording of your 

tests. Work to increase both the total distance travelled as well as the mass supported.

Recalling a Record: X-43A Scramjet Sets New Hypersonic Record a Decade Ago

http://www.nasa.gov/centers/armstrong/Features/X-43A_recalling_a_record.html#.VHTOoo29-Rg

Analysis Questions:

• What is “Mach”?

• The flight of the X-43A was described in the article as “the first demonstration of an integrated 

scramjet in atmospheric flight.” What is a scramjet? How does it work?

• What are the challenges of achieving hypersonic flight in the Earth’s atmosphere?

• What are the challenges of achieving hypersonic speeds outside of the Earth’s atmosphere?
Aeronautics for Introductory Physics: Force Particle 195

Real-World Applications: Helicopter Toy and Lift Estimation

Said Shakerin. The Physics Teacher, 51 310 (2013).

Rotocraft use similar principles as static wings on airplanes in order to generate lift. The 

popular toy, Wacky Whirler, can be used to estimate lift. This is a good activity to help 

students come up with reasonable estimates by making some basic assumptions. This activity 

could be extended to see how small groups compare in their estimates.

Objectives 

Estimate the maximum lift 

generated by a Wacky Whirler.

Materials 

Wacky Whirler

Meter stick / ruler 

Video recorder 

Video analysis tool 

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 3, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Force Particle-Inertia

ACTIVITY

Helicopter Toy and 

Lift Estimation
196 Aeronautics for Introductory Physics: Force Particle

Teacher Guide and Key: Helicopter Toy and Lift Estimation

Goal: Estimate the maximum lift generated by a Wacky Whirler.

Students should be provided with a Wacky Whirler, meter stick or ruler, and the ability to record a video of the Wacky 

Whirler in motion, along with video analysis software. (Caution: Always advise students to use eye protection when releasing 

toy helicopters, as they can spin with high frequency and behave erratically).

Because the Wacky Whirler will have its greatest spin frequency immediately after being released from the hands, that is the 

point at which the toy will also have its greatest lift. (Note: Some students might intuitively come up with the correct idea 

that the highest frequency is reached some time after releasing the propeller, due to the propellers moment of inertia. After 

praising this precise observation they may feel positively challenged by devising more accurate ways of measuring.) Using 

the lift equation for an airplane, it is safe to assume that lift can be calculated with

Starting with the last variable, the velocity, v, can be estimated by using a video camera in one of many ways. Perhaps the 

simplest approach is to record the spin frequency at take-off using video analysis software. From this, determine the period of 

one rotation of the wings. Measure the diameter of the full set of helicopter wings, and determine the distance traveled by the 

edge of the wings in the given period for one rotation. Although velocity for the edge of the wings can be calculated by 

taking the circumference they travel and dividing it by the period of rotation, this is a tangential velocity that is most accurate 

only for the edge of the propellers. A more accurate average velocity would be the edge velocity divided by two. (Note: 

Actually, the average velocity is about 79.37 % of the wing tip’s track velocity, since the square of the velocity contributes to 

the lift.) An alternate approach to resolving this problem can be found in the reference article in The Physics Teacher.

The second to last variable, the surface 

area, A, can be easily estimated using 

a meter stick and treating the wings as 

approximate rectangles.

The density of air, ρ, is typically around 

1 kg/m3

, but can be better estimated 

given the known room temperature and 

pressure. 

The coefficient of lift, CL, is perhaps the 

most difficult value to estimate for this 

activity, because it is typically a value 

that is determined experimentally when 

the magnitude of lift is measured and 

known. However, some simple research 

into typical CL values should result in 

students seeing that values are typically 

around CL = 1 for a 10 degree angle of 

attack (see the previous lab activity), and 

that flat plates different very little from 

modern wings as this angle of attack. 

Therefore, it is not unreasonable to 

choose a value of 1. 

After the estimation of lift, students should compare their estimates between groups and defend their estimation values. The 

values should also be considered in comparison to the weight. The maximum lift value must exceed the weight if any upward 

acceleration was observed. 
Aeronautics for Introductory Physics: Force Particle 197

Student Worksheet: Helicopter Toy and Lift Estimation

Objective: Estimate the maximum lift generated by a Wacky Whirler.

Just like some airplanes, helicopters use propellers. However, while airplanes tend to use propellers for 

propulsion, helicopters use them for lift. Because helicopter “wings” follow similar rules as airplane wings, 

we can use the lift equation:

(Note: For very high frequencies, rotor lift actually will decrease, which is different than the behavior of 

airplane wings).

Play with a helicopter toy, and get it to fly. (Caution: Always use eye protection, especially when the 

helicopters are released near eye level). Watch the path of the toy after it is released from your hands.

1. At which point of its flight path is the maximum lift generated? How do you know?

In order to estimate the lift at this point in the path, you need to know the value of each variable in the 

equation. With your small group, discuss how you might be able to measure or calculate the value of each 

variable in the lift equation.

Speed

The only thing that really makes the helicopter toy’s wings different from an airplane is that their motion is 

in a circle – not in a straight line. Decide how you can measure the speed of the propellers.

2. How can you determine the speed of something moving in a circle?

3. One of the challenges of dealing with propellers is that each part of the propeller has a different linear 

speed. Where is the speed the fastest? Slowest? Average? Annotate the image, and explain.

4. Using a ruler, determine the radius of the propeller at the point of average speed, and label the image 

above.
198 Aeronautics for Introductory Physics: Force Particle

5. Using a video recorder and a video analysis tool, determine the period of one rotation of the propeller. Hint: Paint one end of the propellers with a bright color, so that you can observe a single side 

as it rotates completely. Consider recording from the below the release, above it, to the side, or even 

changing lighting and background contrast in order to clearly measure a single rotation.

6. Using the distance and time measurements, determine the average speed of the propellers as they 

move through the air.

Surface Area

7. Using a meter stick, estimate the total surface area spanned by the wings. Record measurements and 

total surface area below on the image.

Surface Area

8. The density of air, ρ, is typically around 1.2 kg/m3 at STP, but can be better estimated given the 

known room temperature and pressure. Look online to determine the air density at your current location (consider altitude, temperature, and humidity). Record that value here.

Coefficient of Lift

The coefficient of lift, CL

, is perhaps the most difficult value to estimate for this activity, because it is 

typically a value that is determined experimentally when the magnitude of lift is measured and known. 

However, the graph to the right gives some typical values of the coefficients of lifts. 

9. From this graph, what would you consider to be the average coefficient?
Aeronautics for Introductory Physics: Force Particle 199

Lift Calculation

10. Using the lift equation, calculate 

the total estimated maximum lift.

11. How does this value for maximum lift compare to weight? Is this value reasonable? Why or why not? 

12. How should this value for lift compare when the helicopter toy is accelerating up versus accelerating 

down? Explain.
200 Aeronautics for Introductory Physics: Constant Net Force

Constant Net Force

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
Aeronautics for Introductory Physics: Constant Net Force 201

Real-World Applications: Parachute Lab

Although students familiar with wind tunnels might 

have a sense for how lift and drag can be measured, 

there are alternative ways to measure forces 

without the use of a force meter.

The following activity helps students to use the 

concepts of force diagrams and Newton’s second 

law to infer the amount of drag force acting on an 

accelerating parachuter.

(Note: Although students are asked to design a 

parachute – intended to help the object achieve 

terminal velocity – in this activity, students should 

design an object that actually does accelerate 

constantly, albeit at a much lower value than 

acceleration due to gravity. If students are unable 

to design a parachute with such requirements, 

consider asking students to instead design an object 

to fall attached to a paper or Styrofoam ball.)

Objectives 

Calculate the forces acting on a 

parachute as it falls.

Materials 

Parachuter object 

Parachute material 

String 

Tape 

Electronic Balance

Meter stick 

NGSS 

CDI HS-PS-1

SEP 1-7

CCC 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Net Force

ACTIVITY

Parachute Lab
202 Aeronautics for Introductory Physics: Constant Net Force

Teacher Guide and Key: Parachute Lab

Objective: Calculate the forces acting on the parachute as it falls. 

For this activity, students must build a parachute system that allows an object to accelerate at a value less 

than 9.8 m/s/s (so that it can be easily timed), but so that it is not falling at terminal velocity. At terminal 

velocity, the frictional drag force simply is equal to the weight of the falling object, which overly simplifies 

this learning experience.

To ensure that the parachute does not reach terminal velocity, students might be encouraged to use 

more massive objects, such as golf balls, instead of plastic parachute men, that have a very low mass. 

Alternatively, use Styrofoam balls or crumpled paper to slow down a falling object without it reaching 

terminal velocity. It can be difficult to determine if an object has reached terminal velocity or not. Students 

can check for this by holding a digital motion sensor above the parachute after it has been released and has 

begun dropping, or by recording the drop with a video recorder for video analysis. (Alternatively, teachers 

can just ensure that students are getting average acceleration, even if terminal velocity is reached during a 

portion of the fall).

In order to measure the acceleration of their parachuters, they will need to know both the distance of the 

fall as well as the time. Teachers can help speed up the process by measuring the height of the fall in 

advance. To make it possible for students to time the fall, parachuters should be dropped the equivalent of 

approximately one story or the height of a three-meter stairwell with an opening in the middle. Balconies 

and windows are also good places from which to release the parachute. (Caution: Ensure that there are 

spotters on the ground to observe parachutes as they fall. Ensure that no one is standing under the falling 

parachutes. If using a balcony or window, ensure that students always keep the center of gravity of their 

bodies below the ledge).

1. Look back at the “Drag Stations Lab.” How might you design a parachute to decrease velocity of a 

parachuter as it falls, but ensure that it is still accelerating, and does not attain terminal velocity?

Mass should be moderate, so that a small parachute will not cause it to reach terminal velocity. 

Parachutes can slow down falling objects by increasing size and having a shape that “captures” the 

air as it falls.

2. Consider your available materials and decide on a design to try.

3. Build your parachute. When it falls, ensure that it has a constant acceleration towards the ground. 

How will you determine the acceleration of the parachute? (Hint: You may use a meter stick and a 

timer to help you. What equation will you use? What measurements will you take?)

Measure the distance fallen with the meter stick (or use a string and bob hanger to measure long 

distances). Measure the time it takes to fall with the timer. Acceleration can be calculated from 

4. Calculate the acceleration of your parachute. 
Aeronautics for Introductory Physics: Constant Net Force 203

5. Draw a force diagram for your parachute 

system as it accelerates downward. Make sure 

relative force vector lengths are correct. Label 

the net force on the system.

The drag force points upward, and gravitational force points downward (and is longer 

than the drag force). The net force is the excess 

force, part of the gravitational force.

6. Measure the mass of your parachute system:

 kg

7. What is the value of the net force acting to pull 

your parachute down?

Net force can be calculated using Newton’s 

second law. (Fnet = ma).

8. What is the weight of your parachute system?

This value comes from Fg = (9.8 N/kg) m

9. What is the value of the frictional drag force acting on your parachute system? Does this value seem 

reasonable?

Because the magnitude of weight minus the magnitude of the net force equals drag force, these two 

numbers simply need to be subtracted from one another.

10. How does the value of the frictional drag force on your parachute compare to others’ parachutes? Do 

you notice any relationship between frictional drag and heavier/lighter or bigger/smaller parachutes?
204 Aeronautics for Introductory Physics: Constant Net Force

Student Worksheet: Parachute Lab

Objective: Calculate the forces acting on the parachute as it 

falls.

1. Look back at the “Drag Stations Lab.” How might you 

design a parachute to decrease velocity of a parachuter as 

it falls, but ensure that it is still accelerating, and does not 

attain terminal velocity?

2. Consider your available materials and decide on a design 

to try.

3. Build your parachute. When it falls, assume that it has a constant acceleration towards the ground. 

Look back at your “Constant Acceleration” concept map. How will you determine the acceleration of 

the parachute? (Hint: You may use a meter stick and a timer to help you. What equation will you use? 

What measurements will you take?)

4. Calculate the acceleration of your parachute. 

5. Draw a force diagram for your parachute 

system as it accelerates downward. Make sure 

relative force vector lengths are correct. Label 

the net force on the system.
Aeronautics for Introductory Physics: Constant Net Force 205

6. Measure the mass of your parachute system: kg

7. What is the value of the net force acting to pull your parachute down?

8. What is the weight of your parachute system?

9. What is the value of the frictional drag force acting on your parachute system?

10. How does the value of the frictional drag force on your parachute compare to others’ parachutes? Do 

you notice any relationship between frictional drag and heavier/lighter or bigger/smaller parachutes?
206 Aeronautics for Introductory Physics: Constant Net Force

Inquiry Lesson/Lab: Falling and Air Resistance (Quantitative)

The following is a higher-level mathematical approach to determining air drag’s dependency 

upon the velocity of an object through a medium (as opposed to the Inquiry Lab: Falling and 

Air Resistance (Quantitative).) Although simple investigations in mass or weight dependency 

on terminal velocity likely have helped students realize that drag is approximately 

proportional to the square of the velocity, the reality is that this is not a perfect model. Help 

students critically analyze this relationship using a general expression for drag dependence 

(simplified as mass) upon velocity.

(Note: To acquire more data points, consider placing half or portions of coffee filters inside of 

the bottom filter!)

Objectives 

Derive an expression for 

mass and terminal velocity for 

an object falling with 

air resistance.

Materials 

Coffee filters

Electronic balance 

Motion detector 

Data analysis software

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 1, 2, 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Force

ACTIVITY

Falling and Air 

Resistance 

(Quantitative)
Aeronautics for Introductory Physics: Constant Net Force 207

Teacher Guide and Key: Falling and Air Resistance (Quantitative)

Objective: Derive an expression for mass and terminal velocity for an object falling with air 

resistance.

Before performing an experiment, it is important to have a good conceptual understanding of the 

variables that can influence drag. Work through the following prompts to prepare your for your laboratory 

investigation.

1. Envision an object falling through a friction-free atmosphere near the surface of the Earth. According 

to Newton’s second law, Fnet = ma, which in this case is Fnet = mg, where g = 9.8 N/kg. Draw a force 

diagram for this scenario.

The point mass should have a single, downward force vector (weight).

2. Now, envision that the object is falling and accelerating downward, but that there is some air drag 

present. Draw a force diagram for this scenario.

The point mass should have the downward force of weight, as well as a smaller upward force of air 

drag.

3. Write a mathematical expression to describe the net force acting on the object.

4. Now, assume that the coffee filter has achieved terminal velocity. Draw a force diagram for this 

situation.

The point mass should have weight as well as an equal-length drag vector.

5. Re-write the expression above where Fnet = 0.

6. From prior experience, we know that Fdrag has a dependence upon speed, but that relationship is 

not exact. While, in general, it is a parabolic relationship, this is not always the case for objects 

with irregular surfaces or at very high or very low speeds. As a result, we can describe that Fdrag is 

proportional to vn, or Fdrag = b vn, where b is a constant that factors all other variables that influence 

drag (i.e. shape, fluid viscosity, etc.). Substitute the new expression for Fdrag into the equation above.

7. Because we want to find out the exponent of the velocity, solve for v.

8. Re-arrange the equation so that it can be plotted linearly on a logarithmic graph (i.e. “ln y = (m) x + b”).
208 Aeronautics for Introductory Physics: Constant Net Force

9. Going back to the goal statement, remember that we are looking at the relationship between terminal 

velocity and mass. If we were to plot data on a logarithmic graph, what would be the significance of 

each piece of the graph, from the equation above?

a. y-axis variable: natural log of terminal velocity

b. slope: 1/n

c. x-axis variable: natural log of mass

d. y-intercept: (1/n)( natural log of g – natural log of b)

10. Using stacks of coffee filters, decide how to collect a wide range of multiple-trial data for terminal 

velocity and mass. (Feel free to simply use average velocity for terminal velocity – however, if you 

do this, provide a justification for why this is acceptable). 

11. Plot the data on a log-log graph (terminal velocity on the vertical axis, and mass of filter sets on the 

horizontal axis). 

12. What is the value of slope, from the log-log graph? How does this value compare to the typically 

estimated value of 2?

13. What is the value of the y-intercept, from the log-log graph? What is the value of “b” in this 

experiment?

14. Now plot the data by inverting the axes. What is the value of the slope? Explain! (Careful – this is not 

a simple “inverse.” Remember that you are using a log-log graph).

The value of the slope should be equal to the negative inverse of the slope from the first graph.

15. If you assumed that terminal velocity was approximately equal to the average velocity of the falling 

coffee filter, how did this influence your value for n and for b?

Average velocities are lower than terminal velocity (because time is spent speed up to terminal 

velocity), so the effect would be an increase in the value of n.

16. Write your final equation, based on your data, for the dependence of drag on velocity for your 

experiment. Compare this value to the rest of the lab groups in your class.
Aeronautics for Introductory Physics: Constant Net Force 209

Student Worksheet: Falling and Air Resistance (Quantitative)

In previous lessons, you have learned about fluid friction’s dependence upon speed. In some cases (for 

small, aerodynamic objects), as the speed through the fluid is doubled, the frictional drag is doubled. In 

other causes (for larger, less aerodynamic objects), as the speed through the fluid is doubled, the frictional 

drag is quadrupled. In the “Falling and Air Resistance Lab (Qualitative),” you likely saw that coffee filters 

tend to behave more like large, less aerodynamic objects. However, sometimes objects don’t display linear 

or parabolic (square) relationships. Sometimes they display relationships right through the middle, or 

something totally different. This is the same case with airplanes and flying objects. Every object’s drag 

experience is different, and that dependency can change based upon speed, air composition and density, 

and airplane structure. 

In the following experiment, you will not just look to see if drag is proportional to v or v2. Rather, you will 

determine how drag is proportional to vn, and you will solve for n using a logarithmic graph.

Objective: Derive an expression for mass and terminal velocity for an object falling with air 

resistance.

Before performing an experiment, it is important to have a good conceptual understanding of the 

variables that can influence drag. Work through the following prompts to prepare your for your laboratory 

investigation.

1. Envision an object falling through a friction-free atmosphere near the surface of the Earth. According 

to Newton’s second law, Fnet = ma, which in this case is Fnet = mg, where g = 9.8 N/kg. Draw a force 

diagram for this scenario.

2. Now, envision that the object is falling and accelerating downward, but that there is some air drag 

present. Draw a force diagram for this scenario.

3. Write a mathematical expression to describe the net force acting on the object.

4. Now, assume that the coffee filter has achieved terminal velocity. Draw a force diagram for this 

situation.

5. Re-write the expression above where Fnet = 0.
210 Aeronautics for Introductory Physics: Constant Net Force

6. From prior experience, we know that Fdrag has a dependence upon speed, but that relationship is 

not exact. While, in general, it is a parabolic relationship, this is not always the case for objects 

with irregular surfaces or at very high or very low speeds. As a result, we can describe that Fdrag is 

proportional to vn, or Fdrag = b vn, where b is a constant that factors all other variables that influence 

drag (i.e. shape, fluid viscosity, etc.). Substitute the new expression for Fdrag into the equation above.

7. Because we want to find out the exponent of the velocity, solve for v.

8. Re-arrange the equation so that it can be plotted linearly on a logarithmic graph (i.e. “ln y = (m) x + b”).

9. Going back to the goal statement, remember that we are looking at the relationship between terminal 

velocity and mass. If we were to plot data on a logarithmic graph, what would be the significance of 

each piece of the graph, from the equation above?

a. y-axis variable: 

b. slope: 

c. x-axis variable: 

d. y-intercept: 

10. Using stacks of coffee filters, decide how to collect a wide range of multiple-trial data for terminal 

velocity and mass. (Feel free to simply use average velocity for terminal velocity – however, if you 

do this, provide a justification for why this is acceptable). 

11. Plot the data on a log-log graph (terminal velocity on the vertical axis, and mass of filter sets on the 

horizontal axis). 
Aeronautics for Introductory Physics: Constant Net Force 211

12. What is the value of slope, from the log-log graph? How does this value compare to the typically 

estimated value of 2?

13. What is the value of the y-intercept, from the log-log graph? What is the value of “b” in this 

experiment?

14. Now plot the data by inverting the axes. What is the value of the slope? Explain! (Careful – this is not 

a simple “inverse.” Remember that you are using a log-log graph).

15. If you assumed that terminal velocity was approximately equal to the average velocity of the falling 

coffee filter, how did this influence your value for n and for b?

16. Write your final equation, based on your data, for the dependence of drag on velocity for your 

experiment. Compare this value to the rest of the lab groups in your class.
212 Aeronautics for Introductory Physics: Constant Net Force

Hitchhiking Sensors Capture Curiosity’s Entry

http://www.nasa.gov/topics/aeronautics/features/medli.html#.VHjBzo29-Rg

Analysis Questions:

• Scientists from NASA collected data on both temperature and pressure. What effect could 

extremes of both of these variables have on a flying aircraft?

• How might scientists expect Curiosity to behave differently (in terms of motion) and 

experience different effects if it were to fall through the Earth’s atmosphere, as opposed to 

through Mars’?
Aeronautics for Introductory Physics: Constant Net Force 213

Problem Set: Simplified Aircraft Motion 

Beginner’s Guide to Aeronautics

BACKGROUND

This image below shows some rules for the simplified 

motion of an aircraft. By simplified motion we mean 

that some of the four forces acting on the aircraft are 

balanced by other forces and that we are looking at 

only one force and one direction at a time. In reality, 

this simplified motion doesn’t occur because all of the 

forces are interrelated to the aircraft’s speed, altitude, 

orientation, etc. But looking at the forces ideally and 

individually does give us some insight and is much 

easier to understand.

In an ideal situation, an airplane could sustain a 

constant speed and level flight in which the weight 

would be balanced by the lift, and the drag would 

be balanced by the thrust. The closest example of 

this condition is a cruising airliner. While the weight 

decreases due to fuel burned, the change is very small 

relative to the total aircraft weight. In this situation, the aircraft will maintain a constant cruise velocity as described by 

Newton’s first law of motion.

If the forces become unbalanced, the aircraft will move in the direction of the greater force. We can compute the acceleration 

which the aircraft will experience from Newton’s second law of motion

Where a is the acceleration, m is the mass of the aircraft, and Fnet is the net force acting on the aircraft. The net force is the 

difference between the opposing forces; lift minus weight, or thrust minus drag. With this information, we can solve for the 

resulting motion of the aircraft.

If the weight is decreased while the lift is held constant, the airplane will rise:

Lift > Weight - Aircraft accelerates upward

If the lift is decreased while the weight is constant, the plane will fall:

Weight > Lift - Aircraft accelerates downward

Similarly, increasing the thrust while the drag is constant will cause the plane to accelerate:

Thrust > Drag - Aircraft accelerates forward (speeds up)

And increasing the drag at a constant thrust will cause the plane to slow down:

Drag > Thrust - Aircraft accelerates backward (slows down)
214 Aeronautics for Introductory Physics: Constant Net Force

Problem Set from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Answer Key

1. What is the smallest angle in degrees between the lift and drag forces?

a. 0° 

b. 0°

c. 180°

d. 270°

e. 360°

2. What is the smallest angle in degrees between the weight and lift forces?

a. 0° 

b. 0°

c. 180°

d. 270°

e. 360°

3. What is the smallest angle in degrees between the thrust and drag forces?

a. 0° 

b. 0°

c. 180°

d. 270°

e. 360°

4. What is the smallest angle in degrees between the weight and thrust forces?

a. 0° 

b. 0°

c. 180°

d. 270°

e. 360°

5. When an airplane (or any object) is moving at a constant speed, the forces acting on the airplane must be

a. unbalanced, which means that the total force is not zero

b. balanced, which means that the total force is zero

A fighter jet is flying at 37,000 feet with a constant speed of 1,310 km/h. The jet has a weight of 110,000 N 

and its engines provide a thrust of 106,752 N (using afterburners).

6. What is the jet’s height (altitude) in meters?

a. 1.13 x 108 m

b. 61,248 m

c. 8,679 m

d. 356 m

e. 11,277.6 m

7. What is the jet’s mass? [Hint: Use F = mag = W, where ag = 9.8 m/s2.]

a. 520,120 kg

b. 11,224.49 kg

c. 3,527 kg

d. 614,853 kg

e. 215 kg

8. How much lift force is being applied to the jet?

a. 512,128 N

b. 5,634 N

c. 110,000 N

d. 9,935 N

e. 106,752 N
Aeronautics for Introductory Physics: Constant Net Force 215

9. If the jet engines are creating a thrust of 106,752 N, what is the drag force applied to the jet?

a. 512,128 N

b. 5,634 N

c. 110,000 N

d. 9,935 N

e. 106,752 N

Suppose the jet engines are providing a thrust of 100,102 N while the drag force is only 90,167 N.

10. What is the total unbalanced (or net) force on the jet?

a. 512,128 N

b. 5,634 N

c. 110,000 N

d. 9,935 N

e. 106,752 N

11. What will be the jet’s acceleration?

a. 0.89 m/s2

b. 4.35 m/s2

c. 0.36 m/s2

d. 4,253 m/s2

e. 65.3 m/s2

12. If the jet is cruising at 223.47 m/s and accelerates at 0.25 m/s2 for 1,530 seconds, what is the jet’s final speed?

a. 401.28 m/s

b. 605.97 m/s

c. 731.67 m/s

d. 634,521.6 m/s

e. 3,452 m/s

13. Using the data from the previous problem, calculate how far the jet traveled while it accelerated during the 1,530 

seconds.

a. 401.28 m

b. 605.97 m

c. 731.67 m

d. 634,521.6 m

e. 3,452 m

14. If the jet cruised at a speed of 1,310 km/h for 2.53 hours, how far did it travel?

a. 517.79 km

b. 4,639.00 km

c. 218.42 km

d. 6,547.23 km 

e. 3,314.30 km
216 Aeronautics for Introductory Physics: Constant Net Force

Problem Set: Forces in a Climb

Beginner’s Guide to Aeronautics

BACKGROUND

(Note: This background is most appropriate after

students have been introduced to these concepts 

through constructivist and/or inquiry activities.)

There are four forces that act on an aircraft in flight: 

lift, weight, thrust, and drag. The motion of the aircraft 

through the air depends on the relative size of the 

various forces and the orientation of the aircraft. For 

an aircraft in cruise, the four forces are balanced, and 

the aircraft moves at a constant velocity and altitude. 

On this slide, we consider the relations of the forces 

during a gradual climb. We have drawn a vertical and 

horizontal axis on our aircraft through the center of 

gravity. The flight path is shown as a red line inclined 

to the horizontal at angle c. The lift and drag are 

aerodynamic forces that are defined relative to the flight 

path. The lift is perpendicular to the flight path and the 

drag is along the flight path. The thrust of the aircraft is 

also usually aligned with the flight path. Some modern fighter aircraft can change the angle of the thrust, but we are going to 

assume that the thrust is along the flight path direction. The weight of an airplane is always directed towards the center of the 

Earth and is, therefore, along the vertical axis.

Forces are vector quantities. We can write two component equations for the motion of the aircraft based on Newton’s 

second law of motion and the rules of vector algebra. One equation gives the vertical acceleration av

, and the other gives the 

horizontal acceleration ah in terms of the components of the forces and the mass m of the aircraft. If we denote the thrust by 

the symbol F, the lift by L, the drag by D, and the weight by W, the vertical component equation is:

where sin and cos are the trigonometric sine and cosine functions. Similarly, the horizontal component equation is:

We can simplify the equations a little by using the definition of excess thrust Fex:

The resulting equations of motion are:

For small climb angles, the cos(c) is nearly 1.0 and the sin(c) is nearly zero. The equations then reduce to:

The resulting simplified motion is described on another slide. The horizontal equation is integrated on another slide to give 

the velocity and location as functions of time.

For more moderate angles, high excess thrust can provide an important contribution to the vertical acceleration. The next 

time you visit an airport, notice the high climb angles used by modern airliners. This flight path is possible because modern 

turbine engines develop high excess thrust at takeoff. The pilot climbs sharply to get the aircraft as high as possible within 

the confines of the airport which produces the least noise for nearby homes.
Aeronautics for Introductory Physics: Constant Net Force 217

Problem Set

Student Worksheet

Teacher Answer Key

A Boeing 737-600 takes off from a major airport at a speed of 160 knots. At 15,946 feet from the liftoff point this plane must 

be 2812 feet (about 1/2 mile) high.

1. Assuming the angle of ascent is a constant, find the angle the plane must use to achieve this height at this distance - to 

the nearest degree. Make a diagram and explain how you arrived at your answer.

(Open the Beginner’s Guide to Propulsion and open the slide Forces in a Climb under Aircraft Motion.)

(Note: Don’t forget to change radians to degrees.)

2. We are going to use this information to find vertical and horizontal net forces.

a. Find the thrust for a Boeing 737-600 airplane.

To find the thrust of the airplane go to Boeing 737-600

http://www.boeing.com/commercial/737family/pf/pf_600tech.html.

Thrust =____________

b. What is the take off weight of the plane?

Change this weight to newtons

Weight in lbs. = ___________ Weight in newtons =___________

c. Use the first equation on the Forces In A Climb page to find vertical net force.

Use 700,000 N as the lift and 35,000 N as the drag.

Show the original equation and then show the substitutions with your answer. Label answer with correct units.
218 Aeronautics for Introductory Physics: Constant Net Force

d. Use the second equation to find the horizontal net force, use lift and drag from part c above.

Show the original equation and then show the substitutions with your answer. Label answer with correct units.

3. Divide your net force from 2c and 2d by the weight of the airplane to get the vertical and horizontal acceleration in 

“g’s” (ratioed to gravitational acceleration).

a. Vertical acceleration = _________________g’s

b. Horizontal acceleration = _______________g’s
Aeronautics for Introductory Physics: Constant Net Force 219

Interactive Demonstration: Demonstration of Newton’s Third Law 

Using a Balloon Helicopter

Yee-kong Ng, Se-yuen Mak, and Choi-man Chung. The Physics Teacher, 40, 181 (2002).

Cheap balloon helicopter whistles can be 

purchased from toy stores online.

As students observe the balloon take off and 

then remain in the air for up to a half-minute, 

students can draw force diagrams and explain 

how the balloon remains in the air due to the 

“exhaust” through the air channel. Students 

should consider both horizontal forces causing 

rotation, as well as vertical lift forces from 

the blades. (Students can construct their own 

nozzles and wings).

Objectives 

Draw force diagrams to explain 

how the balloon helicopter 

whistle works.

Materials 

Balloon helicopter whistle

Balloon

NGSS 

CDI HS-PS2 

SEP 2, 7

CCC 2, 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Net Force

ACTIVITY

Newton’s 3rd Law 

using a Balloon 

Helicopter
220 Aeronautics for Introductory Physics: Constant Net Force

Student Worksheet: Demonstration of Newton’s 3rd Law Using a Balloon Helicopter

Observe the balloon helicopter as it launches. 

1. Explain why the balloon helicopter rotates. Include separate force 

diagrams for points on the helicopter blades as well as for parcels of air 

just outside of the air nozzles. Be certain to identify any force pairs that 

interact with the helicopter system.

2. Explain why the balloon helicopter lifts vertically. Include separate 

force diagrams for points on the helicopter blades as well as for parcels 

of air surrounding the blades. Be certain to identify any force pairs that 

interact with the helicopter system. 

3. Get your balloon helicopter to hover for a few moments, and be certain to record a video of the blades spinning at the 

required frequency for it to hover. During the time of the hover:

a. Estimate the lift produced by the helicopter. Explain how you measured/calculated this amount.

b. Estimate the total surface area of the blades (single side, for all three). Show work.

c. Using a coefficient of lift of 1 (a reasonable value for toys such as these), determine the average required velocity 

of the blades through the air.

d. Determine the average radius of the blades.

a. Work backwards to estimate the time for one rotation of the set of blades. 

b. Use a video analysis tool to estimate the actual time for one rotation of the set of blades. Is your estimated value 

very different? Why might this be?
Aeronautics for Introductory Physics: 2-D Combination 221

2-D Combination

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
222 Aeronautics for Introductory Physics: 2-D Combination

Real-World Applications: Measuring the Flight Speeds of Fire 

Bombers from Photos: An In-Class Exercise in Introductory 

Kinematics

Greg W. Lowe & Eric Ayars. The Physics Teacher, 48, 106 (2010).

Image and video analysis software 

can be used to determine the speed 

of a fire bomber, as the plane 

leaves a nice “track” with the 

leading edge of the falling flame 

retardant.

In this activity, students are asked 

to evaluation another hypothetical 

student’s claim to be able to 

determine the speed of the aircraft 

based upon the photograph of the 

fire bomber an its trail. Students 

should identify potential:

Assumptions:

• Wind is still

• Airplane does not propel the flame retardant (it is simply dropped)

• Airplane is flying level

Mistakes:

• Give the above assumptions, it would be a mistake to assume that the lowest leading edge of the flame retardant was released first – the back end would have been 

dropped first!

Objectives 

Estimate flight speed from a 

firebomber.

Materials 

Firebomber photos 

Photo analysis software 

Firebomber aircraft statistics

NGSS 

CDI HS-PS2 

SEP 2, 4, 5

CCC 4, 5

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

2-D Combination

ACTIVITY

Measuring Flight 

Speeds from Photos
Aeronautics for Introductory Physics: 2-D Combination 223

Student Worksheet: Measuring the Flight Speed of Fire-Bombers

Fire bombers are aircraft that eject flame retardant on large fires, 

and are regularly used to put out massive forest fires. Frequently, 

the retardant has a visible color. 

The following is an example of another student’s work and 

calculations, in which s/he is attempting to determine the speed of 

the airplane. It is your job to identify any assumptions and/or 

potential mistakes (if any!) made by the student.

A single known value (the plane length) can be found online. In this 

particular case, the S-2A is 13.3 meters long. This value can be 

used to estimate the vertical and horizontal distances of the flame 

retardant.

When the retardant is ejected, it creates a parabolic shape at the 

trailing (bottom) edge.

Because the flame retardant appears to have fallen a distance of about 1.1 plane lengths (15 m), the total time since the 

release of the retardant can be estimated – assuming there is no air resistance.

Using the kinematics equation:

t is equal to about 1.7 seconds.

Given that the length of the horizontal trail appears to be 6.7 airplane lengths, or 89 m, the constant horizontal velocity 

should be about 52 m/s, which seems to be a reasonable value!

What assumptions and potential mistakes did the student make about this image?
224 Aeronautics for Introductory Physics: 2-D Combination

NASA Tests Radio for Unmanned Aircraft Operations

http://www.nasa.gov/topics/aeronautics/features/uas_prototype_radio.html#.VHjAeI29-Rh

Analysis Questions:

• There has been some use of autonomous vehicles to fight wildfires. Why might this be beneficial, compared to the use of standard manned aircraft?

• There has been some concern over maintaining the quality of signals when communicating 

with autonomous vehicles. What factors – especially those in the vicinity of a wildfire – might 

be cause for concern?
Aeronautics for Introductory Physics: 2-D Combination 225

Problem Set: Aircraft Motion Activity

from Beginner’s Guide to Aeronautics

Student Worksheet

Teacher Answer Key

The pilot of an airplane sent to drop supplies to victims of an accident that are stranded on a small island. The plane’s 

altitude is always 500 meters and your speed is 89.61 m/s.

1. Once released, how much time will elapse before the supply package reaches the island’s surface?

(Hint: Use the distance equation in the y-direction.) 

time = ____________

2. Will the descent time of the supply package change if the airplane’s speed changes?

a. Yes

b. No

3. At what (horizontal) distance in front of the island should the package be released in order land safely on the island?

dx = ______________

4. What is the package’s horizontal speed when it makes landfall on the island? 

vx = _____________

5. What is the package’s vertical speed when it makes landfall?

vy = ____________

6. What will be the package’s flight angle with respect to the level of the island as it descends?

q = ______________
226 Aeronautics for Introductory Physics: 2-D Combination

Suppose the victims on the island can retrieve supply packages that land within 30 meters of the island. The length of 

the island is 50 meters along the direction you are approaching.

7. How far from the island would the airplane have to release the supply package for it to land 30 meters offshore from 

the island’s nearest point?

dx (front) = _____________

8. How far from the island would the airplane have to release the supply package for it to land 30 meters offshore from 

the island’s farthest point?

dx (back) = _____________

9. Time1 will be the time on your watch when you release the package and it lands 30 meters in front of the island. Time2 

will be the time on your watch when you release the package and it lands 30 meters in back of the island. Calculate the 

amount of time you have to successfully drop the package, namely, time1- time2.

time1- time2 = ________________

10. What could be done to decrease the package’s speed when it reaches the ground?
Aeronautics for Introductory Physics: Energy 227

Energy

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
228 Aeronautics for Introductory Physics: Energy

Literary/Data Analysis: A Hard Day’s Flight

Henk Tennekes. The Simple Science of Flight. The MIT Press: 

Cambridge, Massachusetts. Ch. 2

Have students read Chapter 2 of The Simple Science of Flight. 

This chapter provide a great example of the connections 

between physics, biology, and chemistry, as it discusses 

metabolic energy, combustion, and flight mechanics.

Have students use the questions on the following page for 

literary and data analysis.

*Cover image and graphics from The Simple Science of Flight 

are reproduced here with permissions from The MIT Press.

Objectives 

Estimate flight speed from a 

firebomber.

Materials 

Firebomber photos 

Photo analysis software 

Firebomber aircraft statistics

NGSS 

CDI HS-PS2 

SEP 2, 4, 5

CCC 4, 5

CCSS 

Reading in Science 

Writing in Science

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Energy

ACTIVITY

A Hard Day’s Flight
Aeronautics for Introductory Physics: Energy 229

Teacher Answer Key: A Hard Day’s Flight

1. How does the basal metabolic rate of birds compare to humans? Why do you think this is the case?

A birds’ basal metabolic rate is about 10 times faster than humans’. This is because birds need to access energy from 

their food much more quickly: flight demands a lot more energy than anything that humans typically do throughout the 

day.

2. Only about 25% of metabolic energy gets converted to mechanical energy. Where does the remainder go? How does 

this value compare to the chemical to mechanical energy conversion of airplanes? (You might need to do some internet 

research to figure this out!)

Heat! For most fuel types, only about 15% of the energy is extracted from fuel.

3. Looking at Figure 6, how does speed required for maximum efficiency compare between a 5 degree climb, horizontal 

flight, and 5 degree descent? Why?

The speed for maximum efficiency is about the same for a climb or horizontal flight (8 m/s). It is about 9 m/s for a 5 

degree descent. It makes sense that the value is higher for descent, because the bird is falling and picking up speed 

without much effort – attempting to slow down would take energy.

4. Looking at Figure 6, why is it so difficult to fly slowly or quickly, compared to at a medium speed?

Speed has to be relatively high in all cases, because flying slowing means that less lift is achieved. Going too fast, 

however, increases total drag, and results in it being harder to fly.

Energy, Work, and Power

5. Relating Figure 6 to Figure 7, explain why the speed at which drag is the smallest is not the same as the speed at which 

the required power is a minimum.

Figure 7 shows how much energy is needed per unit distance, while figure 6 shows how much energy is needed per unit 

time. Figure 6 is looking at energy rate, not total energy for a given distance of travel.

6. Look at Figure 8 displaying power and velocity for a variety of vehicles. What would this graph look like on a 

Cartesian, non-logarithmic graph? Explain why the data demonstrates a curved, non-linear relationship.

The graph would curve to the upper right and then flatten out to an asymptote. Doubling speed requires power to 

increase logarithmically, partly because friction is more difficult to overcome as one moves more quickly.

Nutrition and Combustion

7. What is the difference between fuel consumption and energy consumption?

Fuel consumption refers to the food or chemical energy taken in. Energy consumption refers to the number of calories 

that can actually be used or processed from the intake.

8. Looking at Table 3, which foodstuff and which fuel provide the greatest energy density?

Vegetable oil and natural gas
230 Aeronautics for Introductory Physics: Energy

Student Worksheet: A Hard Day’s Flight

1. How does the basal metabolic rate of birds compare to humans? Why do you think this is the case?

2. Only about 25% of metabolic energy gets converted to mechanical energy. Where does the remainder go? How does 

this value compare to the chemical to mechanical energy conversion of airplanes? (You might need to do some Internet 

research to figure this out!)

3. Looking at Figure 6, how does speed required for maximum efficiency compare between a 5 degree climb, horizontal 

flight, and 5 degree descent? Why?

4. Looking at Figure 6, why is it so difficult to fly slowly or quickly, compared to at a medium speed?

Energy, Work, and Power

5. Relating Figure 6 to Figure 7, explain why the speed at which drag is the smallest is not the same as the speed at which 

the required power is a minimum.

6. Look at Figure 8 displaying power and velocity for a variety of vehicles. What would this graph look like on a Cartesian, non-logarithmic graph? Explain why the data demonstrates a curved, non-linear relationship.

Nutrition and Combustion

7. What is the difference between fuel consumption and energy consumption?

8. Looking at Table 3, which foodstuff and which fuel provide the greatest energy density?
Aeronautics for Introductory Physics: Energy 231

ACCESS II Confirms JetBiofuel Burns Cleaner

http://www.nasa.gov/aero/access-ii-confirms-jet-biofuel-burns-cleaner/#.VHS9BY29-Rg

Analysis Questions:

• What evidence does NASA have that blended fuel is better than typical jet fuel?

• What is the new biofuel made of? Why is it considered “renewable”?

• Compare the benefits and disadvantages to using biofuels that were not mentioned in this 

article. (What kind of impact would the mass usage of biofuel have on the economy, pollution, 

agriculture, food availability, deforestation, etc.?)

• One of the goals of ACCESS is to study contrail formation. What is a contrail, and why are 

they generally perceived as negative for the environment?
232 Aeronautics for Introductory Physics: Energy

Literary/Data Analysis: In Wind and Weather

Henk Tennekes. The Simple Science of Flight. The MIT Press: 

Cambridge, Massachusetts. Ch. 3

Have students read Chapter 3 of The Simple Science of Flight.

This chapter continues to provides additional examples of 

the connections between physics, biology, and chemistry. It 

discusses the effect of wind on the achievement of lift, the 

ability to soar, the effect of thermals and slopes, long-distance 

migration, and the mechanics of take-off and landing. 

Have students use the questions on the following page for 

literary and data analysis.

*Cover image and graphics from The Simple Science of Flight

are reproduced here with permissions from The MIT Press.

Objectives 

Explain how birds and planes 

store and use energy during 

flight.

Materials 

The Simple Science of Flight, 

Ch. 3

NGSS 

CDI HS-PS3

SEP 4, 6

CCC 1, 4, 5

CCSS 

Reading in Science 

Writing in Science

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Energy

ACTIVITY

In Wind and Weather
Aeronautics for Introductory Physics: Energy 233

Teacher Guide and Key: In Wind and Weather

1. How does wind direction influence flight times, takeoff, and landing? Explain.

Wind vectors can cause an airplane to arrive sooner (as with a tailwind) or arrive later (as with a headwind). Airplanes benefit from having a headwind during takeoff, however, because it generates lift. In contrast, a tailwind necessitates that an airplane achieve a greater groundspeed before takeoff. For an ultralight plane (with a takeoff speed of 

approximately 40 miles per hour), just a 10 mile per hour tailwind almost doubles the necessary takeoff speed.

2. How does wind influence insects, small birds, large birds, and very large planes? Why?

Wind has a much greater impact on the direction and speed of small animals than large ones. Smaller animals have 

less inertia to keep them on their track.

3. What are the two primary methods for soaring? Explain how they work.

1. Birds “slope soar” in areas where wind moves upward as it pushes against tall structures such as boats or ridges.

2. Birds track down thermals (rising pockets of hot air) that form above hot ground.

4. How do birds prepare to have enough energy for migration? (Likewise, how do airplanes prepare for long flights?) 

What is the downside to being overly prepared, in terms of energy storage, during the first portions of the flight?

Birds gorge on food and gain fat. Airplanes fill up on fuel (although not much more than they know they will need). The 

downside is that the initial part of the trip will require more energy expenditure to sustain such heavy mass.

5. Verify the calculations (that R = 2160 m) for the Boeing 747-400 using kinematics equations. (see pg. 52).

6. Describe the flight procedures for takeoff and landing for birds, and compare this to takeoff and landing procedures for 

airplanes.

It might seem unusual, but birds actually prefer to have a bit of headwind when they take off in order to generate lift at 

low speeds. Preferably, birds will take off from a high point so that they can use the force of gravity to speed themselves 

up and generate lift. When landing, birds with often “backwing,” which is a form of air braking. Winds are very important for lift, especially during takeoff and landing. Both takeoff and landing should occur with a headwind, because 

planes need to have a high speed with respect to the wind in order to gain lift. A tailwind could cause a stall much 

more easily at the slow speeds of takeoff and landing. 
234 Aeronautics for Introductory Physics: Energy

Student Worksheet: In Wind and Weather

1. How does wind direction influence flight times, takeoff, and landing? Explain.

2. How does wind influence insects, small birds, large birds, and very large planes? Why?

3. What are the two primary methods for soaring? Explain how they work.

4. How do birds prepare to have enough energy for migration? (Likewise, how do airplanes prepare for long flights?) 

What is the downside to being overly prepared, in terms of energy storage, during the first portions of the flight?

5. Verify the calculations (that R = 2160 m) for the Boeing 747-400 using kinematics equations. (see pg. 52).

6. Describe the flight procedures for takeoff and landing for birds, and compare this to takoff and landing procedures for 

airplanes.
Aeronautics for Introductory Physics: Energy 235

NASA Helicopter Drop Test a Smashing Success

http://www.nasa.gov/larc/helicopter-drop-test-a-smashing-success/#.VHS7go29-Rg

Analysis Questions:

• Explain how the floor of the helicopter was modified to improve crash-landing safety.

• What was the purpose of the dots on the helicopter? Explain how this works.
236 Aeronautics for Introductory Physics: Rotational Motion

Rotational Motion

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
Aeronautics for Introductory Physics: Rotational Motion 237

Real-World Applications: Circular Motion Studies with a Toy 

Airplane/The Airplane Experiment

Frank Butcher, The Physics Teacher, 25, 572 (1987).

Lee Larson & Roderick Grant, The Physics Teacher, 29, 564 (1991). 

There are a number of mobiles and 

battery-powered airplanes (flyers or 

remote controlled) that can be easily 

attached to a ceiling and caused to 

move in a horizontal circle. These 

toys provide a wonderful opportunity 

for students to predict the tension in 

the string, which can then be checked 

against a measurement taken by an 

analog or digital force meter.

Under normal conditions, in which 

an airplane is traveling at a constant 

velocity, the lift force would equal 

the weight. When flying in a circle, 

however, a net inward force must be 

directed on the plane to cause it to 

accelerate (and change direction). 

Objectives 

Determine the tension in the 

string of an airplane flying in a 

circle.

Materials 

Toy airplane 

String 

Meter stick 

Flashlight (opt.)

Force meter 

Electronic balance (opt.)

Stop watch 

NGSS 

CDI HS-PS2 

SEP 1-7

CCC 4, 6

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Constant Net Force

ACTIVITY

Circular Motion 

Studies with Toy 

Airplane
238 Aeronautics for Introductory Physics: Rotational Motion

The force on the airplane can be easily identified as Fg

 and FT. However, FT is unknown. A force diagram can be drawn in 

which FT can be resolved into both a horizontal, FTx, and vertical FTy. Because the plane does not accelerate vertically, it can 

be assumed that Fg is equal to FTy, and that FTx is the net force 

(centripetal force), for which we have an expression and for which 

we can calculate a value. Using the measured weight of the plane and 

equating it to FTy, the angle can be estimated and used to solve for FT, 

which can be checked against a spring scale or digital force meter 

from which the flying plane is supported.

Additionally, the wings of an airplane can sometimes be removed or 

broken off, and data compared to determine the effect of lift due to 

the wings.
Aeronautics for Introductory Physics: Rotational Motion 239

Teacher Worksheet: Circular Motion Studies with a Toy Airplane

Objective: Determine the tension of string on an 

aircraft flying in a circle.

1. Observe the toy airplane as it flies in a circle. What 

variables must be known about it in order to determine the tension in the string?

(Hint: Feel free to stop the toy at any time, remove 

it from its string, etc. Consider using the “flashlight method” to determine the radius of the circle 

it makes as it flies. To do this, place a flashlight 

directly below the flying airplane. The smallest, 

crispest shadow will be produced when the flashlight is directly below it, point directly up. A small 

mark can be made on the ceiling where the shadow 

is, and a measurement can be made from the pivot 

point to the mark from the location of the shadow.)

Students will need to collect the mass of the airplane, radius, and rotational speed (using the radius 

and average time to complete a cycle).

2. Decide what data you will need to collect, and how 

you will go about doing it.

3. Once you have collected your necessary data, draw 

a force diagram (with components) and label any 

known values. Solve for the tension force.

Students should note that weight (determined from kg 

on a balance or directly read on a force meter) is 

equivalent to the magnitude of the vertical force. 

Using the length of the string and the radius, they can then determine the angle. Using the vertical 

force and the angle, the tension force and the centripetal force (for #4 below) can be easily calculated.

4. With the help of your teacher, attach a force 

meter to flying airplane device. Record the force 

measured by the force meter. Address any discrepancies between your calculated tension force, 

and the measured tension force.

Answers will vary.

5. Determine the centripetal force on the aircraft.

See #3.

6. If possible, remove the wings of the aircraft. How 

do you anticipate this will affect the tension and 

the centripetal forces? Why? Perform an experiment, and check.

Weight will decrease, but so will lift. So, the required vertical component of the tension force will 

decrease because the weight will decrease, but it will also increase because the lift generated by the 

wings will decrease. The net result on the centripetal force will also be dependent upon the mass, 

which is now smaller.
240 Aeronautics for Introductory Physics: Rotational Motion

Student Worksheet: Circular Motion Studies with a Toy Airplane

Objective: Determine the tension of string on an 

aircraft flying in a circle.

1. Observe the toy airplane as it flies in a circle. What 

variables must be known about it in order to determine the tension in the string?

(Hint: Feel free to stop the toy at any time, remove 

it from its string, etc. Consider using the “flashlight method” to determine the radius of the circle 

it makes as it flies. To do this, place a flashlight 

directly below the flying airplane. The smallest, 

crispest shadow will be produced when the flashlight is directly below it, point directly up. A small 

mark can be made on the ceiling where the shadow 

is, and a measurement can be made from the pivot 

point to the mark from the location of the shadow.)

2. Decide what data you will need to collect, and how 

you will go about doing it.

3. Once you have collected your necessary data, draw 

a force diagram (with components) and label any 

known values. Solve for the tension force.

4. With the help of your teacher, attach a force meter 

to flying airplane device. Record the force measured by the force meter. Address any discrepancies 

between your calculated tension force, and the 

measured tension force.

5. Determine the centripetal force on the aircraft.

6. If possible, remove the wings of the aircraft. How do you anticipate this will affect the tension and 

the centripetal forces? Why? Perform an experiment, and check.
Aeronautics for Introductory Physics: Rotational Motion 241

Real-World Applications: Measuring the Moment of Inertia of an 

Airplane

Lyle F. Minkler, The Physics Teacher, 13, 46 (1975).

The general public is typically 

surprised to find that, even in rough 

turbulence, pilots are generally “riding 

it out” as opposed to fiercely gripping 

the aircraft controls white-knuckled, 

trying to maintain control. This is 

because aircraft are made to be stable. 

Airplanes “want” to fly level, so to 

speak, and are designed to be that way.

The stability of an aircraft in flight is 

directly tied to the center of mass and 

moment of inertia of the aircraft. 

Because setups for this laboratory 

experience are likely to be so diverse, 

no student worksheet is provided for 

this activity.

Objectives 

Determine the center of mass 

and moment of inertia of a 

model aircraft.

Materials 

Model aircraft 

Razor blades 

Rubber stoppers 

Small compression spring 

Timer 

Electronic balances 

NGSS 

CDI HS-PS2 

SEP 1, 2, 5, 6

CCC 4, 6, 7

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Rotational Motion

ACTIVITY

Measuring the 

Moment of Inertia 

of an Airplane
242 Aeronautics for Introductory Physics: Rotational Motion

Objective: Determine the center of mass of a model aircraft.

Finding the center of mass of an aircraft can be easily accomplished in a variety of ways. For model aircraft, it is easy enough 

to simply attach a loop of string to the fuselage, and to place it in such a way that the aircraft is balanced when suspended 

from a given point.

However, realistically, this cannot be easily accomplished with a fullsized aircraft. Alternatively, the aircraft can be treated as a “lever” with 

the reference datum point starting at the nose of the aircraft. Students can 

measure the force exerted by the forward and aft gear (placed on two or 

three balances), as well as the distances from the reference datum line to 

the points of contact with the balances. Because the sum of the torques 

produced at each contact point must equal the total torque of the aircraft 

measured from the reference datum line, 

Fforward dforward + Faft daft = Fweight dcenter of mass. 

The following NASA resource has a very good video describing the above 

process:

Video: Flight Testing Newton’s Laws – Weight and Balance

http://www.nasa.gov/mov/194114main_020_weight_and_balance.mov

Objective: Determine the moment of inertia of a model aircraft.

An aircraft with a high moment of inertia about its center of mass is 

very stable, while an aircraft with a low moment of inertia is less stable 

(however, it is easier to maneuver). An impart part of testing aircraft 

before flight is to identify their moment of inertia.

Ideally, moment of inertia should be measured with respect to the 

center of gravity. However, to determine the moment of inertia with 

an oscillating spring (as is used in this approach, and has been used 

historically in the aircraft industry), the center of rotation cannot 

be placed at the center of gravity, otherwise it would not oscillate 

effectively.

The frictionless pivot for the study of moment of inertia should be placed at a location that is not at the center of gravity. The 

aircraft can be balanced at this point with the use of razorblades mounted onto corks, and attached to a solid platform. The 

front end of the aircraft can then rest upon a coiled compression spring of known spring constant, k.

Using an expression for torque and moment of inertia, the period of oscillation can be used to calculate the moment of inertia 

of the aircraft.

Students must then locate the center of mass, and use the transformation formula:

Additionally, an historical document demonstrates how this process has been used throughout aeronautical history:

http://naca.central.cranfield.ac.uk/reports/arc/rm/3620.pdf
Aeronautics for Introductory Physics: Additional Resources 243

Additional Resources

A joint project of 

NASA Aeronautics 

and the 

American Association of 

Physics Teachers
244 Aeronautics for Introductory Physics: Additional Resources

Literary/Data Analysis Activity: The 747 and its Competitors

Henk Tennekes. The Simple Science of Flight. The MIT Press: 

Cambridge, Massachusetts. Ch. 6

Have students read Chapter 6 of The Simple Science of Flight, 

and use the following guiding questions for literary analysis.

*Cover image and graphics from The Simple Science of Flight

are reproduced here with permissions from The MIT Press.

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Additional Resources

ACTIVITY

The 747 and its 

Competitors

Objectives

Compare the 747 and its 

competitors

Materials

The Simple Science of Flight, 

Ch. 6

CCSS

Reading in Science

Writing in Science
Aeronautics for Introductory Physics: Additional Resources 245

Teacher Guide and Key: The 747 and its Competitors

1. Compare the benefits and disadvantages of the Boeing 747 contrasted with the Concorde.

Benefits of 747: Uses less fuel, carries more people and cargo, can go twice the range, higher finesse, cheaper

Disadvantages of 747: Goes more slowly. (It also has much smaller stall angle).

2. List the four requirements for the Boeing 747 design team, and include a brief rationale for each requirement.

Requirement Rationale

High speed and finesse Save time and money

Fly just below speed of sound Greater engine efficiency

Fly in lower stratosphere Cooler temps for engine, avoids weather

Don’t fly any higher! Wings are good design for particular height

3. Look at Table 6. At which listed altitude would an airplane fly to be most efficient? Consider all variables, and how they 

influence engine performance, drag, and lift. Keep your answers based upon qualitative analysis only.

Around 10,000-11,000 m. At this altitude, you are reaching the maximum coldness that makes for efficient jet engines. 

Going higher would also require larger wings to sustain flight, as the density of the air decreases to a sub-optimal 

value.
246 Aeronautics for Introductory Physics: Additional Resources

Student Worksheet: The 747 and its Competitors

1. Compare the benefits and disadvantages of the Boeing 747 contrasted with the Concorde.

2. List the four requirements for the Boeing 747 design team, and include a brief rationale for each requirement.

Requirement Rationale

3. Look at table 7. At which listed altitude would an airplane fly to be most efficient? Consider all variables, and how they 

influence engine performance, drag, and lift. Keep your answers based upon qualitative analysis only.
Aeronautics for Introductory Physics: Additional Resources 247

NASA Resources

NASA Aeronautics

http://www.nasa.gov/topics/aeronautics/index.html

NASA Aeronautics Education

http://www.aeronautics.nasa.gov/k_12.htm

Museum in a Box

http://www.aeronautics.nasa.gov/mib.htm

• Dressing for Altitude

• History of Flight

• Parts of an Airplane

• Principles of Flight

• Structures and Materials

• Propulsion

• Future Flight

• Careers in Aeronautics

• Airspace

Beginner’s Guide to Aeronautics

• Beginner’s Guide to Aerodynamics 

http://www.grc.nasa.gov/WWW/k-12/BGA/BGAindex.html

• Beginner’s Guide to Propulsion

http://www.grc.nasa.gov/WWW/K-12/airplane/bgp.html

• Beginner’s Guide to Wind Tunnels

http://www.grc.nasa.gov/WWW/K-12/airplane/bgt.html

• Beginner’s Guide to Compressible Aerodynamics

http://www.grc.nasa.gov/WWW/K-12/airplane/bgc.html

• Beginner’s Guide to Hypersonics

http://www.grc.nasa.gov/WWW/BGH/bgh.html

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Additional Resources

NASA RESOURCES
248 Aeronautics for Introductory Physics: Additional Resources

AAPT Resources

American Association of Physics Teachers

http://www.aapt.org/

The Physics Teacher magazine

https://www.aapt.org/Publications/

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Additional Resources

ACTIVITY

AAPT Resources
Aeronautics for Introductory Physics: Additional Resources 249

Additional Resources

Literature Resources

Tennekes, Henk. The Simple Science of Flight. MIT Press. (1996.) *Cover image and graphics from The Simple Science of 

Flight are reproduced in this document with permissions from The MIT Press.

Leybold Lessons (and Laboratory Equipment) 

http://www.ld-didactic.de/en.html

• P1.8.5.1 Static Pressure in a reduced cross section – Measuring the pressure with the precision manometer

• P1.8.5.2 Determining the volume flow with a Venturi tube – Measuring the pressure with a precision manometer

• P1.8.5.3 Determining the wind speed with a pressure head sensor – Measuring the pressure with the precision 

manometer

• P1.8.6.1 Measuring the air resistance as a function of the wind speed – Measuring the wind speed with the precision 

manometer.

• P1.8.6.2 Drag coefficient cW: Relationship between air resistance and body shape – Measuring the wind speed with 

a precision manometer.

• P1.8.6.3 Pressure curve on an airfoil profile – Measuring the pressure with the precision manometer.

• P1.8.7.1 Recording an airfoil profile polar in a wind tunnel

• P1.8.7.2 Measuring students’ own airfoils and panels in the wind tunnelP1.8.7.3 Verifying the Bernoulli equation – 

Measuring with the precision manometer

Other Aeronautics Education Resources

FliteTest – Webshows, Podcasts, Articles, and Forums

Civil Air Patrol – Find a local squadron, become an Educator member

WITH YOU WHEN YOU FLY: Aeronautical Physics

UNIT

Additional Resources

ACTIVITY

Additional Resources
Developed by:

Rebecca E. Vieyra, 2014-2015 Albert Einstein Distinguished Educator Fellow placed at NASA ARMD, lead

Tony Springer, NASA ARMD Director of Communications and Education, lead

Lillian Gipson, Maria Werries, Jim Schultz, NASA ARMD Office of Communications and Education, graphic design, editing

Reviewers/Contributors: 

Dr. Beth Cunningham, Executive Officer, AAPT

Dr. Robert Hilborn, Associate Executive Officer, AAPT

Dr. André Bresges, Managing Director, Institute of Physics Education, University of Cologne/Germany, AAPT member

Florian Genz, Doctoral Candidate, Institute of Physics Education, University of Cologne/Germany, AAPT member 

Dr. Colleen Megowan-Romanowicz, Executive Officer, American Modeling Teachers Association, AAPT member

Dr. Philip Blanco, Grossmont College, AAPT member

Dr. Roger Freedman, Physics Education, UC Santa Barbara, AAPT member

Ron Bowerman, St. Wendelin High School, AAPT member

Col. George Zamka (USMC, ret.), NASA Astronaut, Test Pilot

Daniel Scopel, Pilot 

April Lanotte, Science Education, University of Colorado-Boulder, 2011-2013 Albert Einstein Distinguished Educator Fellow

Thomas Benson, NASA Glenn, Beginner’s Guide to Aeronautics

National Aeronautics and Space Administration, Aeronautic Research Mission Directorate

Washington, DC 20546-0001

http://www.aeronautics.nasa.gov/education.htm

http://www.nasa.gov/aeronautics

American Association of Physics Teachers

1 Physics Ellipse Dr.

College Park, MD 20740

http://www.aapt.org

National Aeronautics and Space Administration

Headquarters

300 E. Street, SW

Washington, DC 20546

www.nasa.gov

EG-2015--03-005-HQ