Write a lab report on the following lab: 
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Projectile Motion II
After the completion of this laboratory exercise, each student will be familiar with
1. the path of an object in projectile motion,
2. the effect of the initial conditions on the motion of a projectile,
3. the relation between launch angle and the maximum range of a projectile, and
4. solving kinematics equations to predict the flight of a projectile.
Simulation
https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html
Select (double click) the Lab option.
Introduction
The behavior of an object moving within a gravitational field is one of the most recognizable and
fundamental types of motion observed. As a physics student, you must become familiar with the
equations and patterns that underly this motion, categorized as the study of projectile motion.
When an object is in projectile motion, it is solely under the influence of gravity. Although there
must have been some sort of mechanism to launch the object initially, once the object has been
released, the only factor determining its motion is gravity, and the object will move in a predictable
way. The object is only said to be in projectile motion from the instant just after it is launched to
the instant just before it hits an obstacle. When the object, say, reaches the ground, the ground
exerts additional forces on the object, and it is no longer in projectile motion.
Because all projectiles move only due to the influence of gravity, they all exhibit a parabolic
trajectory. However, the initial conditions, such as launch height, launch angle, and launch speed
determine how stretched or compressed this parabola will be. We can use the kinematics equations
Projectile Motion II
to predict the projectile’s maximum height, range, and flight time based on these initial conditions.
This lab will give you practice in performing these calculations, as well as highlight some of the
patterns observed for projectiles.
Part A: Horizontal Launch
The purpose of this portion of the laboratory exercise is to investigate the similarity between
freefall and the y-component of projectile motion.
Step 1. Open the PhET simulation. You will see an image of a cannon that can launch a cannonball
above a road. Begin by setting the launch height to 5 m. To do this, simply click on the cannon
and drag it up or down.
Step 2. Next, set the cannon to launch the cannonball horizontally. To do this, click on the barrel
of the cannon and drag to rotate it until the launch angle reads 0°.
Step 3. Lastly, set the launch speed to 10 m/s using the Initial Speed slider below the cannon.
Review the Initial Values in the box at the top right of the screen to ensure that you have set up
the experiment correctly.
Step 4. Click the launch button (the red button showing the cannon being fired). Observe the path
of the cannonball as it flies. At the top of the screen, there is a toolbox with two tools: a scope
and a measuring tape. The scope can read the flight time, range, and height of the projectile at
points along its path.
Click on the scope and drag it to the location of the cannonball as it hit the ground.
Projectile Motion II
Record the total flight time of the cannonball.
Answer:
Also, record the range of the cannonball.
Answer:
Step 5. Next, while leaving the height and launch angle constant, you will be varying the launch
speed and noticing the effect on the cannonball’s flight. Before running any trials, predict the
relationships between the launch speed and the flight time and between the launch speed and the
range of the object fired horizontally.
As the launch speed is increased, will the flight time increase, decrease, or remain the same?
Answer:
As the launch speed is increased, will the range increase, decrease, or remain the same?
Answer:
Step 6. Now, run the simulation at the launch speeds indicated and complete the table below.
Remember to keep the launch height at 5 m and the launch angle at 0°. (If the projectile lands too
close to the right side of the screen to be able to use the scope, zoom out using the magnifying
glass buttons at the top left of the screen.)
Table 1. Effect of Launch Speed on a Projectile Launched Horizontally.
Launch Speed (m/s) Total Flight Time (s) Range (m)
10
15
20
25
30
Do your results match your predictions?
Answer:
Projectile Motion II
Step 7. To help understand your results, consider the case when the launch speed is zero. This
would correspond to one-dimensional freefall for an object dropped from rest. Use a kinematics
equation to calculate the time it takes an object dropped from rest to fall a distance of 5 m.
Answer:
Now, write a statement to explain the pattern you found in the flight times recorded in this
experiment.
Answer:
Part B: Object Launched at an Angle
The purpose of this portion of the laboratory exercise is to use the PhET simulation to visualize a
typical projectile motion problem.
Step 1. Reset the PhET simulation by clicking on the yellow eraser button to the left of the red
launch button. Now, set the cannon to a height of 0 m and a launch angle of 35°. Don’t worry
about the launch speed just yet. Click the button showing a magnifying glass with a plus sign to
zoom in.
Step 2. Use the tape measure tool to measure the horizontal distance between the cannonball’s
launch point and Michelangelo’s David statue. To use the tape measure tool, click and drag it to
the base of the cannon. Match the crosshairs on the tape measure to the crosshairs on the cannon.
Then, click and drag the right side of the tool to expand the tape measure to reach from the cannon
to the bottom of David.
Projectile Motion II
Record the horizontal distance from the launch point to the statue.
Answer:
Step 3. Use the tape measure to measure the vertical height of the statue of David.
Answer:
Step 4. Use the kinematics equations, and what you know about projectile motion to calculate the
answer to this projectile motion problem:
At what speed must the cannonball be launched so that it just passes over David’s head? What is
the flight time of the cannonball to this point? (Hints: The cannonball may not pass over David’s
head at the exact peak of its trajectory. You will have to solve a system of equations to find the
solution.)
Answer:
Step 5. Now, in the PhET simulation, adjust the launch angle to approximately the angle you
calculated. Does the cannonball pass directly over David’s head?
Answer:
If not, go back to your calculations and find your mistake.
Part C: Maximum Range of an Object in Projectile Motion
The purpose of this portion of the laboratory exercise is to discern both experimentally and
algebraically the launch angle that corresponds to the maximum range of a projectile.
Step 1. Reset the simulation and zoom back out to the standard view. Set the cannonball’s height
to zero, launch speed to 15 m/s, and launch angle to 20°. Fire the cannon, and use either the tape
measure or the scope tool to determine the range of the projectile. Then, while leaving the height
and launch speed constant, adjust the angle to each of the values indicated and complete the table
below.
Projectile Motion II
Table 2. Dependence of Projectile Range on the Angle of Projection
Angle of Projection (°) Range (m)
20
30
40
45
50
60
70
Step 2. Using Excel or Google Sheets, create a graph of range (y-axis) versus angle of projection
(x-axis) based on the data in the table above. Include a title for your graph as well as axis labels
with units. Fit a trendline to your data.
What type of trendline best fits your data, and what is the equation of this trendline?
Answer:
Step 3. Using your trendline, find the launch angle that corresponds to the maximum range.
What is this launch angle?
Answer:
Step 4. Let us next show that this launch angle for maximum range will be the same for all
projectiles, regardless of launch speed (as long as air resistance is ignored). First, we know that
the launch velocity, v0, can be resolved into its x and y components.
𝑣0𝑥 = 𝑣𝑜𝑐𝑜𝑠𝜃
𝑣0𝑦 = 𝑣𝑜𝑠𝑖𝑛𝜃
Consider a projectile that is launched from ground level and returns back to the ground. What is
the vertical displacement, 𝑦 − 𝑦𝑜, of the projectile in this case?
Answer:
Use the kinematics equation for the vertical component of an object in projectile motion
𝑦 − 𝑦0 = 𝑣0𝑦 − 1
2 𝑔𝑡2
to find an equation for the flight time of the projectile from the instant it is launched to the instant
it hits the ground. Leave your equation in variable form. Do not plug in any numbers.
Answer:
Next, combine this equation with the kinematics equation for the horizontal component of an
object in projectile motion
𝑥 − 𝑥0 = 𝑣0𝑥𝑡.
Projectile Motion II
Replace 𝑥 − 𝑥0 with 𝑅 (for range) and be sure to plug in the equation above for 𝑣0𝑥. Your final
answer should correlate the range to the launch velocity, 𝑣𝑜, launch angle, 𝜃, and gravitational
acceleration constant, 𝑔.
Answer:
Use the trig identity
2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 = 𝑠𝑖𝑛(2𝜃)
to simplify your equation.
Answer:
Now, you should see that if the launch velocity and the gravitational acceleration are held constant,
your range will be maximized when 𝑠𝑖𝑛(2𝜃) is maximized. For what value of 𝜃 is 𝑠𝑖𝑛(2𝜃)
maximized? (Recall that a sine function varies between -1 and 1; What must 𝜃 be so that 𝑠𝑖𝑛(2𝜃) =
1?)
Answer:
Does this value for the angle corresponding to maximum range agree with your experimentally
determined angle above?
Answer:
Part D: Symmetry of Launch Angles and Ranges
The purpose of this portion of the laboratory exercise is to determine why two launch angles may
give the same range.
Step 1. Re-examine Table 2 above and your corresponding graph. Were there any pairs of launch
angles that resulted in the same range? If so, what are the pairs?
Answer:
Based on your observations, write a rule that describes when two launch angles will result in the
same range.
Answer:
Step 2. Next, you will want to show that this rule holds for any launch speed. Begin with the last
equation you found in Part C. Recopy that equation here.
Answer:
Projectile Motion II
Revisit the rule you wrote in step 1. You should have found that when the launch angle is
symmetric about 45°, the ranges are the same. Another way to state this is that when 𝜃 = 45° +
𝛥𝜃, the range is the same as when 𝜃 = 45° − 𝛥𝜃, where 𝛥𝜃 is any angle between 0 and 45°.
Plug 𝜃 = 45° + 𝛥𝜃 into the equation above. Simplify the equation as much as you can, while still
leaving it in variable form (i.e. don’t plug any numbers in for 𝑣𝑜 or 𝑔). (You may want to use the
trig identity
𝑠𝑖𝑛(90° + 𝜙) = 𝑐𝑜𝑠𝜙
which stems from the fact that sine and cosine functions are the same, but shifted 90°.)
Answer:
Also, plug 𝜃 = 45° − 𝛥𝜃 into your range equation. Again, simplify as much as you can while still
leaving the equation in variable form. (This time, you may want to use the trig identity above,
plus the additional identity that
𝑐𝑜𝑠(−𝜙) = 𝑐𝑜𝑠𝜙
which reflects the fact that the cosine function is symmetric about the y axis.)
Answer:
Compare these two equations. What conclusion can you draw?
Answer:
Questions
1. How does the vertical portion of projectile motion compare to one-dimensional free fall?
2. How does adding air resistance affect the motion of a projectile? What differences in the
projectile’s path, height, and range do you observe? Experiment with the PhET simulation to draw
specific conclusions. Systematically compare cannonball flights with the same initial conditions
but with and without air resistance.
3. With all other initial conditions held constant, how does increasing the launch height affect
range?
4. With all other initial conditions held constant, how does increasing launch speed affect
maximum height?
5. With all other initial conditions held constant, how does increasing the launch angle affect
range?
6. With all other initial conditions held constant, how does increasing the launch angle affect flight
time?