Educational Product

Microgravity: Fall into Math Exhibit Handout

Sir Isaac Newton wrote, “If I have seen far, it is because I have stood

on the shoulders of giants.” By this, he meant that his insights into the

nature of gravity and motion are based on the work Galileo Galilei and

others did before him. Likewise, Albert Einstein’s work built on

Newton’s. Without these three great minds, spacecraft might not be

orbiting Earth today. The following pages provide a historical 

overview of each scientist’s contribution to gravity and microgravity, as well as making mathematical connections to microgravity 

platforms and research. 

Gravity

Gravity can be described as the pull objects exert on each other. Of course,

an object has to be massive for that pull to be felt. How massive does it

need to be? An object needs to be bigger than an elephant, a battleship,

or a skyscraper – say in the neighborhood of a moon, planet, or star.

The massive object closest to us is Earth.

The effects of gravity are subtle and pervasive. Many people are

surprised to find out that the force of gravity affects things like the

shape of a drop of water, toast dropping jelly-side down on the floor, a

thrown baseball arching back to Earth, hot air rising while cool air sinks,

and Italian salad dressing settling in layers. Gravity affects daily life so

much that it is easy not to think about it.

Great Minds Behind Microgravity 

Some of the greatest minds in science and mathematics realized that an underlying

force was at work when it came to falling objects. Galileo, then Newton, and later

Einstein devised experiments to better understand the effects of gravity. Each of these

scientist-mathematicians refined the research of his predecessors. Galileo's work with

free falling objects, Newton's thought experiment for an artificial satellite, and

Einstein's insight into the feeling of weightlessness during free fall, paved the way for

the science of creating environments where some of the effects of gravity are reduced

to a small amount, which is microgravity. Such environments allow scientists to

research a number of phenomena gravity normally affects.

Educational Brief

Microgravity: Fall Into Mathematics 

If I have

seen far, it is

because I

have stood on

the shoulders

of giants.

SIR ISAAC NEWTON

Isaac

Newton

1642  1727

Albert Einstein

1879  1955

National Aeronautics and 

Space Administration

Educators

and Students

Grades 

5-12

EB-1999-03-001-GRC

EB-1999-03-001-GRC

Galileo Galilei

1564  1642
While a professor in Pisa, Galileo was obligated to teach Aristotle's

natural philosophy. However, Galileo's observations differed from

Aristotle's theory that heavier objects fall faster than lighter objects, in

proportion to their weight. Galileo reasoned that objects fall at the same

rate. Legend has it that he dropped a cannon ball and a musket ball from

the Leaning Tower of Pisa, and that the balls hit the ground at the same

time. Unfortunately, this experiment is not documented in Galileo's

writings. 

Since then, scientists have carefully studied free falling objects, taking

measurements with increasing precision. They have indeed found that,

with the absence of air resistance, all bodies, regardless of size, fall with

the same acceleration to Earth's surface. This eventually became known

as the acceleration of gravity. On average at Earth's surface the

acceleration of gravity (1g)

is -9.8 m/s2. 

Galileo also showed the

similarity between something moving at a steady velocity and something

that is not moving at all: no unbalanced force causes the

objects to accelerate. Galileo also discussed an aquarium on

a ship: fish can swim in all directions, regardless of whether

the ship is anchored or sailing. For the fish, the ship's

motion is irrelevant. Similarly, we walk around on Earth,

rarely noticing the Earth's motion as it rotates on its axis

and orbits the Sun at 27,772 m/s. Einstein later used words similar to

Galileo's in the context of falling systems. Galileo’s concepts led to the

creation of microgravity conditions using free fall and to Einstein’s

theory of gravitation.

Facility Distance Time

Glenn Research Center 132 meters 5.2 seconds 

24 meters 2.2 seconds

Marshall Space Flight Center 100 meters 4.5 seconds

Japan Microgravity Center 490 meters 10 seconds

Microgravity: Fall into Mathematics EB-1999-03-001-GRC

Galileo Galilei

Italian Scientist & Mathematician

1564-1642

Objects of Different Mass 

Fall at the Same Rate

A cannon ball and a musket

ball, dropped from the Leaning

Tower of Pisa, would hit the

ground at the same time.

Try This!

Calculate your own value for g using data from four

microgravity drop facilities (three at NASA and one in

Japan). Use the equation above, and the data in the table

to solve for g. Keep in mind, d is distance and t is time.

Consider vertical motion to be positive upwards.

The real data will not produce the precise magnitude for g

of 9.8 m/s2. Why not? Keep in mind that making

measurements and calculating results involve the concepts

of accuracy and precision, significant figures, and orders of

magnitude.

d = (1/2)gt2

2
EB-1999-03-001-GRC Microgravity: Fall into Mathematics

As a student at Cambridge, Newton retreated to the country when

the university closed for 18 months because of the plague. While

there, Newton drew upon Galileo's work as he discovered laws of

motion and gravitation; he also invented the mathematical basis

for calculus. His most famous discoveries became known as the

Universal Law of Gravitation and Newton's Three Laws of Motion.

Together they describe the force of gravity and the motion of objects

throughout the universe.

Newton reasoned that the shapes of planetary orbits around

the Sun were due to gravity's presence. He expanded

this conclusion to hypothesize how an artificial

satellite could be made to orbit Earth. In his

thought experiment Newton imagined a tall

mountain extending above Earth's

atmosphere, so that air friction would not

be a factor. He then imagined a cannon at

the top of that mountain firing cannon

balls parallel to the ground. The cannon

ball would travel in a parabolic path,

arching to Earth's surface. (Without air

drag, the cannon ball would travel in a

straight line; gravity causes it to fall, arching

to Earth's surface.)

With more and more black powder, the cannon ball path

would lengthen. If the cannon could fire with enough force, the cannon

ball would fall entirely around Earth, orbiting it. The Space Shuttle

Orbiters and the International Space Station are spacecraft that do, in fact,

fall at fast enough rates to orbit Earth. 

Try This!

Given the two equations to the right, calculate how fast a Space Shuttle

Orbiter and the International Space Station travel in order to orbit Earth.

Solve for velocity (v), using the data below.

G = Universal Constant of Gravitation = 6.67 x 10-11 m3

/kg s2

mEarth = Earth's mass = 5.98 x 1024 kg

r = radius of Earth + spacecraft's altitude

rEarth = 6.37 x 106 m Orbiter altitude = 290 km

International Space Station altitude = 350 km

Sir Isaac Newton

British Scientist & Mathematician

1642-1727

The First Artificial Satellite

Imagined

Newton thought how a

cannon ball could orbit

Earth: a cannon on top

of a tall mountain would

need to fire a cannon ball

fast enough to fall entirely

around Earth.

3

r

F = GmEarthmsatellite

r2

GmEarth

r v =
EB-1999-03-001-GRC

As a young man, Einstein graduated with a physics degree and

struggled for a year to find a job. He finally found a position

working for the patent office in Bern, Switzerland. There, he

thought about physics and had the insight quoted on the left. 

Einstein realized that in a state of free fall, a person or object

would appear to be weightless. Actual weight is a measure of

Earth’s gravitational pull on a mass. Conditions can be changed,

however, to affect an object’s apparent weight. One example is

Einstein’s free falling person: the person still has the same mass, but

appears weightless. The illustration below shows how apparent

weight changes for a person in an elevator that is stationary,

accelerating upward, accelerating downward, and in free fall. Similar

feelings of becoming heavier and lighter are more obvious on a roller

coaster. Rapid accelerations upward press riders back into the seat.

Downward accelerations make riders feel lighter, like they would

come out of their seat, except for the safety restraints. 

If all safety mechanisms broke on an elevator, everything in the

elevator would fall at the rate of gravitational acceleration. The

person pictured in the elevator would have no apparent weight

because he or she would no longer press down on the scale. Note:

This picture is misleading. The person and the scale would not

spontaneously start floating. The person would need to jump up in

order to float. 

In the advent of the space age, Einstein’s insight played a

significant role in the development of microgravity

research platforms. Platforms, like drop towers, are

extensions of Einstein’s thought experiment. Like Galileo

and his fish, Einstein also realized that frames of reference

and relative motion are vital concepts. In fact, Einstein’s

principle of relativity simply states that the laws of physics

are the same in every inertial frame of reference. The far

reaching implications of this statement are still being

studied today.

Albert Einstein

American Scientist & Mathematician

1879-1955

"I was

sitting in a chair in

the patent office at Bern when all of

a sudden a thought occurred to me:

'If a person falls freely, he will not

feel his own weight.' I was startled.

This simple thought made a deep

impression on me. It impelled me

toward a theory of gravitation."

 ALBERT EINSTEIN

Normal

weight

Heavier than

normal

Lighter than

normal

No apparent

weight

Try This!

Would your apparent weight in an elevator

going up be the same going down? Use the

equations and data provided to calculate a

persons apparent weight (P) in an accelerating

elevator. Note: Vertical accelerations are

considered to be positive upwards.

Calculate weight for a 60 kg person in an accelerating

elevator, if the elevator accelerates at:

1. 1 m/s2

2. 0.5 m/s23. free fall

F = Total force m = Persons mass

P = Apparent weight g = Earths gravity

aelevator = Acceleration of the elevator

W = Actual weight (mass x Earths gravity)

4

Microgravity: Fall into Math Exhibit Handout
The environment where astronauts float in the International Space Station

is an example of microgravity. Microgravity is a condition where some of

the effects of gravity are reduced compared to those experienced on

Earth. As mentioned previously, astronauts and the Space Station are

actually in a state of free fall around Earth.

Creating Microgravity

There are various ways to create microgravity. As mentioned before, one

way to create microgravity is to put an object in a state of free fall.

Researchers have devised facilities that use free fall to create

microgravity conditions for varying lengths of time. These include drop

facilities, parabolic aircraft, sounding rockets, and orbiting spacecraft. 

Another way to achieve microgravity is by sending a spacecraft far into

space to escape Earth’s gravitational pull. To reach a point where Earth’s

gravity is reduced to one-millionth of that on Earth’s surface, the

spacecraft would have to travel almost 17 times farther away than the

Moon (6.37 million km)! Such travel is not a practical alternative. 

Microgravity Research Environment

Microgravity is a useful research environment because it reduces the

effect gravity has on convection and sedimentation. These processes

affect fluids and the way materials solidify. Microgravity provides a new

research environment so scientists can study a wide range of phenomena.

On Earth, buoyancy-driven convection occurs when less dense fluids rise

and denser fluids sink. Boiling water on a stove is a common example of

such convection. As water at the bottom of a pot heats, small bubbles of

gas form that are less dense, and rise to the top. At the same time, cooler,

denser water replaces the bubbles at the bottom, causing a flow pattern.

Similarly, during combustion hot, less dense products of a flame rise, and

cooler, denser surrounding air sinks to the flame’s base, creating similar

flow patterns. In microgravity, buoyancy-driven convection is reduced,

causing a candle flame to appear rounder and dimmer.

Sedimentation occurs when fluids and particles of different densities form

layers, with denser layers at the bottom. Italian salad dressing is a

common example of sedimentation. However, in microgravity, fluids with

different densities do not settle out. For example, water traveling through

a pipe could contain pockets of air, instead of water being on the bottom

and air staying on top. 

Microgravity: Fall into Mathematics

Microgravity

Microgravity is an

environment or condition

where some of the effects of

gravity are reduced

compared to those

experienced on Earth.

Hot air rises.

Cool air sinks.

Candle in 1 g

Candle in microgravity

No convection

5

EB-1999-03-001-GRC
Microgravity: Fall into Mathematics EB-1999-03-001-GRC

Researchers use four types of platforms to create microgravity

environments: drop facilities, parabolic aircraft, sounding rockets, and

orbiting spacecraft. http://microgravity.nasa.gov/NASA_Carrier_User_Guide.pdf

Drop Facilities

Drop facilities include drop towers and drop tubes. NASA Glenn

Research Center has two drop towers. One is 24 meters high, providing

2.2 seconds of microgravity. The other is 132 meters deep for 5.2 seconds

of microgravity. The towers drop self-contained experiments with

equipment, computers, and cameras on board. NASA Marshall Space

Flight Center has a drop tube 105 meters high that allows molten metal

droplets to be studied while they fall for 4.6 seconds.

Parabolic Aircraft

NASA’s KC-135 is an aircraft that flies on a parabola-shaped path. The

plane is referred to as the Weightless Wonder or the Vomit Comet for the

nausea it causes. The aircraft has padded walls, foot restraints, hand holds,

and devices for securing the experiments during flight. During research

campaigns, the KC-135 can fly 40 - 60 parabolas. First, the plane climbs

at a 45-degree angle to the horizon. Then, the pilot cuts back the engines

to slow the aircraft and pitches the nose down to complete the parabola.

The experiment and the researchers experience 15 - 25 seconds of

microgravity for each parabola. As the aircraft pulls out of the dive, the

nose angles up, pulling 2 g, as it begins the next parabola. 

Sounding Rockets

Like the KC-135, sounding rockets follow a parabolic arc. Unlike the

plane, rockets like the Black Brant travel above the atmosphere. There the

experiment compartment detaches to experience up to six minutes of free

falling microgravity, before reentering Earth’s atmosphere and

parachuting to the ground. Sounding rockets produce higher-quality

microgravity conditions for longer periods of time; however, researchers

cannot fly with the experiment. 

Orbiting Spacecraft

Spacecraft like Space Shuttle Orbiters

and the International Space Station carry special

laboratories into orbit for the longest periods of

microgravity of all the platforms. The Orbiters can

carry experiments for up to 17 days. The

International Space Station will have the capability to fly experiments for

several months or even years. 

Microgravity Platforms 6

http://microgravity.nasa.gov/mf.html

http://microgravity.grc.nasa.gov/kjenks/

kc-135.htm

http://www.wff.nasa.gov/~web/

sndroc_pics.html

http://station.nasa.gov

http://www.ksc.nasa.gov/ksc.html
EB-1999-03-001-GRC Microgravity: Fall into Mathematics

Scientists conduct microgravity research to better understand fundamental

processes and structures that gravity affects. Microgravity research is

conducted in the fields of biotechnology, combustion science, fluid

physics, fundamental physics, and materials science.

Biotechnology 

Biotechnology applies what scientists know about biology to solve

problems. For example, scientists use microgravity to grow higher quality

protein crystals and viruses, and larger three-dimensional samples of cells

and tissues than can be grown on Earth. Such research leads to advances

in crop growth, medicine development, and artificial organs.

Combustion Science

Combustion, or burning, is a chemical reaction that releases heat and

soot. Microgravity scientists research fuels, the burning process, flame

spread, and sooting. This research applies to controlling pollution, forest

fires, and general fire safety. Since 85% of the energy used on Earth is

combustion based, the applications for industry are enormous. 

Fluid Physics

Fluids are liquids, gases, and particles that flow in response to a force.

The relevance of fluid research ranges from blood circulation to weather

patterns, from engines to cosmetics. Microgravity fluid research examines

the way fluids circulate, move with other fluids, respond to different

forces, and wet surfaces, among other phenomena. Research results

impact spacecraft cooling and life support, fuel tank design, aerosol

sprays, gels, smoke, paints, and foams (like shaving cream).

Fundamental Physics

Fundamental Physics tests the laws that govern the physical world. In

microgravity this science researches laws affecting critical transition

points between solids, liquids, and gases. Other areas of research examine

the behavior of individual atoms (which affects atomic clocks) and

Einstein’s theory of gravitation. 

Materials Science

Materials Science studies the properties of materials we produce to make

things. A material’s structure is based on the arrangements of its atoms.

Research seeks to improve the properties of materials like metals and

alloys, electronic devices, glasses and ceramics, and polymers. As a

result, the quality of soda cans, airplanes, computers, communications

systems, and bioceramic artificial bones improves.

Microgravity Research

For more information about microgravity

research, visit these websites:

Microgravity Research Program Office

http://microgravity.nasa.gov/

Microgravity Research Division

Headquarters

http://microgravity.hq.nasa.gov/

Microgravity Science Division at Glenn

Research Center

http://microgravity.grc.nasa.gov/

The National Center for Microgravity

Research on Fluids and Combustion

http://www.ncmr.org/

For more information about NASAs

Education Programs, visit:

NASA Spacelink

http://spacelink.nasa.gov/

Insulin crystals grown on Earth.

Insulin crystals grown in microgravity.

7
Microgravity: Fall into Mathematics EB-1999-03-001-GRC

Galileo Try This!

Calculate your own value for g using data from four

microgravity drop facilities. Consider vertical motion to

be positive upwards. 

Use this equation: d = (1/

2)gt

2

.

Rewrite to solve for g. g = 2d/t2

Solve the equation using the data in the table.

1. g = 2 (-132 m) / (5.2 s)2 = -9.8 m/s2

2. g = 2 ( -24 m) / (2.2 s)2 = -9.9 m/s2

3. g = 2 (-100 m) / (4.5 s)2 = -9.9 m/s2

4. g = 2 (-490 m) / (10 s)2 = -9.8 m/s2

Newton Try This!

Calculate how fast a Space Shuttle Orbiter and the

International Space Station travel in order to orbit Earth.

Solve for velocity (v), using the data below.

G = Universal Constant of Gravitation = 6.67 x 10-11m3/kg s2

mEarth = Earth's mass = 5.98 x 1024 kg

r = radius of Earth + spacecraft's altitude

rEarth = 6.37 x 106

m Orbiter altitude = 290 km

International Space Station altitude = 350 km

Orbiter: v = (6.67 x 10-11

m3/kg s2) (5.98 x 1024 kg)

6.66 x 106

m

= 7.74 x 103

m/s

Station: v = (6.67 x 10-11

m3/kg s2) (5.98 x 1024 kg)

6.72 x 106

m

= 7.70 x 103m/s

d t

1. 132 m 5.2 s

2. 24 m 2.2 s

3. 100 m 4.5 s

4. 490 m 10.0 s

GmEarth

r

v =

Einstein Try This!

Would your apparent weight in an elevator going up be the

same going down? Calculate a persons apparent weight

(P) in an accelerating elevator. Note: Vertical accelerations

are considered to be positive upwards.

1. Elevator's acceleration is 1 m/s2.

a

elevator

= -1 m/s2 (elevator down) aelevator= +1 m/s2 (elevator up)

P = (60kg)(-1m/s2 - (-9.8 m/s2

)) P = (60 kg) (+1 m/s2 - (-9.8 m/s2))

= 528 kg m/s2 = 648 kg m/s2

2. Elevator's acceleration is 0.5 m/s2.

aelevator = -0.5 m/s2

(elevator down) a

elevator

= +0.5 m/s2(elevator up)

P = (60kg)(-0.5 m/s2 - (-9.8 m/s2

)) P = (60 kg) (+0.5 m/s2- (-9.8 m/s2))

= 558 kg m/s2 = 618 kg m/s2

3. Elevator is in free fall. 

a

elevator

= -9.8 m/s2 (free fall)

P = (60 kg) (-9.8 m/s2 - (-9.8 m/s2

))

= 0 kg m/s2

F = Total force

W = Actual weight (mass x Earths gravity)

m = Persons mass = 60 kg

aelevator = Acceleration of the elevator

g = Earths gravity

P = Apparent weight

P = m(aelevator - g)

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Answers to Try This! 8