Lab 5 - Projectile Motion
In this lab, you are going to explore the motion of a ball set into projectile motion after falling off
a ramp and bouncing on the floor. Classic projectile motion carries one simplifying assumption,
which is that air resistance is negligible. Under that condition, the motion of the projectile can
easily be analyzed by separating velocities and positions into their horizontal and vertical
The two quantities of interest that can be used to then predict the entire trajectory are the initial
velocity, vo, and the angle that the initial velocity makes with the x-axis, θ. If these two quantities
are known, then other quantities of interest, such as the total time of flight, maximum height, and
total horizontal distance, or the range, can be predicted. Conversely, if we can physically
measure the total time, maximum height, and range, we can calculate the initial velocity, the
initial angle, and the acceleration of the system. Remember that the only acceleration in classic
projectile motion is that of gravity, downward, in the vertical (the y) direction. There is no
acceleration in the horizontal (the x) direction.
The equations governing the motion are as follows:
𝑥 = 𝑣𝑜 cos 𝜃 𝑡
𝑦 = 𝑣𝑜 sin 𝜃 𝑡 − 𝑔𝑡 2
𝑣𝑦 = 𝑣𝑜 sin 𝜃 − 𝑔𝑡
horizontal position as a function of time
vertical position as a function of time, it is
assumed up is positive and down is negative
vertical velocity as a function of time
One way to confirm projectile motion is to measure the vertical positions at various times. Once
the data is plotted (time on the x-axis and position on the y), if the best trend that fits the data is a
2nd-order polynomial, then projectile motion is confirmed since the vertical position equation is
quadratic. If air resistance acted, the flight times up and down would not be symmetric and
therefore not parabolic. There is an added bonus that, because the acceleration appears in the
coefficient of the quadratic term, we can use the quadratic fit to attempt to measure the
One of the special points in the trajectory of a projectile is at the highest point, or 𝑦𝑚𝑎𝑥 . First, the
path of the projectile is symmetric on the way up, and on the way down, about this point. This
means it takes just as long to rise as it does to fall, and the actual path it travels on the way up is
mirrored on the way back down. Secondly, the vertical velocity, 𝑣𝑦 , is zero at this point. We can
exploit this fact to find the total time of flight. If we set 𝑣𝑦 equal to zero and solve for t, the result
𝑣𝑜 sin 𝜃
To find the total time, we double this, since it takes the projectile as long to fall as it does to rise.
Therefore, the total time of flight is given by:
2𝑣𝑜 sin 𝜃
Next, the maximum height is found by taking the time it takes to reach the peak and plugging it
in for t in the equation for the vertical position. After simplification, the result is
𝑣𝑜2 sin2 𝜃
In the first part of the experiment as outlined below, you will watch a slow motion video of a
bouncing ball set into projectile motion and record times/positions. Plotting the data and
confirming that a second order polynomial is the best simple fit to the data will confirm the ball
obeys the vertical position equation from the previous page.
Secondly, you will watch the video, timing the entire path from launch to landing, as well as
noting the maximum height. You will externally measure these with a stopwatch and the ruler in
the video. We don’t know the initial velocity or the angle of launch, but we can eliminate them
if we solve the total time equation for 𝑣𝑜 sin 𝜃, and square the result, and set it equal to the rearrangement of the 𝑦𝑚𝑎𝑥 equation after solving for 𝑣𝑜2 sin2 𝜃. The result is
= 2𝑔𝑦𝑚𝑎𝑥 .
If we solve this equation for g, we obtain
We can obtain an estimate for the gravitational acceleration if we can measure the maximum
height and the total time.
YouTube video found here: https://www.youtube.com/watch?v=lC_s-MsFf3c&list=PLD3LFNpL8oRSJ5jwNI-JJYyEDQn5qyxi&index=3
Once the link is opened, you will see a video of a ball falling off a ramp, bouncing off the floor,
and being set into projectile motion. The video was shot in 2000 frames per second and encoded
at 24 frames per second. This gives a correction factor of 83.333, meaning that for any time
measurement made with a stopwatch will need to be divided by 83.333 (actually, as a matter of
fact, the factor is 83. 3̅, so you may take as many decimal places in converting your times, but
your final result after the conversion should contain the same number of significant figures as
your stopwatch allows.
The region of interest in the video is right when the ball bounces the first time, rises, and then
falls back and hits the floor. This is approximately from the 37 second mark to the 1:35 mark on
the time ribbon along the bottom of the video in YouTube.
Procedure 1: - Tracking the Trajectory
1.1 Position vs. Time
While carefully watching the video, start your stopwatch right as the ball leaves the surface, and
measure how long it takes to reach the following positions on both the way up and down. If
using the lap button on your smartphone, you will need to add each previous lap to obtain the
total for that measurement. The first position should start at 0.0 s and 0.0 m. Note that the
measuring stick is in cm, so these need to be converted to meters. The positions you should
measure the times for are symmetric; from 0.00 m to 0.55 m is the ball rising, and then from 0.55
m to 0.00 m is the ball falling.
Once your final times have been obtained, plot your positions vs. time in Excel. Fit your data
with a 2nd order polynomial and display the equation on your chart. Be sure to include labels on
your axes and include your plot in your report.
From the equation for vertical position as a function of time, the leading coefficient on the
quadratic term represents 2 𝑔. Therefore, multiply the quadratic coefficient by 2, and this
provides an estimate for 𝑔.
Estimate for 𝑔:____________
Compare your estimate for 𝑔 with the accepted value of 9.8 m/s2 using a percent error
1.2 - Velocity vs. Time
Using the same procedure to calculate velocities from position/time data as the previous lab,
calculate an estimate for the instantaneous velocities, and plot the velocity vs. the time for these
values. The result should be a linear trend with a negative slope, according to the equation for
vertical velocity as a function of time, and the slope should be the acceleration due to gravity.
Note: there may some outliers, where do they occur, and are they outlier enough to discard?
Fit the data with a linear trendline and record the slope. Compare this with the accepted value of
gravity, 9.8 m/s2 using a percent error.
Next, find the uncertainty on the slope of the trend using a LINEST function.
Procedure 2 - Measuring 𝑔 from Time of Flight and Maximum Height
For this procedure, rewind the video to just before the ball bounces the first time. As the ball
bounces, start your stopwatch, timing the entire flight, stopping the stopwatch as the ball lands.
Now, we want to assign a value for the uncertainty on the total time. Typical reaction time for a
human is around 0.1 - 0.2 seconds; it takes about that amount of time for your finger to press the
stopwatch button once your brain realizes it needs to. This is complicated further by the fact that
there isn’t a specific instant at which you ‘know’ when the button is to be pressed, at either end.
The ball is in contact with the floor for some length of time in the slow motion regime and
knowing exactly when the button is to be pressed is not trivial. As a result, it is probably not
unfair to assert that there is about 0.3 seconds of reaction time at both ends of the flight. As a
result, the total uncertainty on your time measurement may be somewhere around 0.6 seconds. If
you are extremely confident that the start and end times were performed very precisely, this may
be around 0.4 or so. It is up to you to estimate your uncertainty, and there is not a standard
procedure, such as for using a ruler with the 20% rule as used previously. It is entirely your
estimate as to what uncertainty to assign, within reason. Afterall, uncertainty is your estimate for
how well you know your measurement.
Once you have your measurement and determined your uncertainty (remember the measurement
and uncertainty must have the same number of decimal places), divide both by the 83.333 factor
to obtain the actual total time of flight and associated uncertainty (yes, its valid to divide both by
the conversion). Denote the total time by t.
Time (s): __________ + _________ seconds
t = actual time (s): __________ + _________ seconds
Next to obtain a measurement for the maximum height, watch the video again, carefully
attempting to note the location of the maximum height. Fortunately, this occurs very close to
where the ball crosses the meter stick. However, we cannot be 100% precise about this. So,
watching the ball carefully, and attempting to note its precise vertical position relative to the
meter stick, obtain an estimate for the maximum height and record this in meters. Estimate the
uncertainty (note, this should be larger than the 20% of the least count as explained in the power
point and used previously, due to the very small size of the least count relative to the perception
of the ball as it crosses the meter stick, and the fact that we cannot be 100% certain where the
exact maximum height occurs.) Record your estimate for the maximum height and your estimate
for its uncertainty below:
𝑦𝑚𝑎𝑥 : ________ + ________ m
From your values for the actual time and the maximum height, calculate an estimate for the
acceleration due to gravity from the formula
𝑔: _________ m/s2
Now for the uncertainty. It can be shown that the uncertainty on the gravitational acceleration is
𝜎𝑦 𝑚 2
) + 4( ) ,
𝜎𝑔 = 𝑔√(
where σym is the uncertainty on the maximum height and σt is the uncertainty on the time.
Arranging the formula into this form simplifies the calculation, as well as puts the expression in
the form of the relative uncertainty, or the ratio of an uncertainty to its measurement. This allows
us to see which measurements are more precise. For the ones that are less precise, it can also
serve as a starting point to re-evaluate our methodology and determine if a better measuring
procedure exists for that quantity.
Calculate the uncertainty for 𝑔, and record below:
𝜎𝑔 : ________ m/s2.
1. What are the measured values of gravity from fitting your data with the parabola and linear
line from Part 1? What are the % errors of each with the accepted value?
2. Discuss how you might estimate the uncertainty for the positions of the ball as for some of
your measurements the ball is not close to the meter stick? Discuss how the uncertainties in the
position measurements may affect your data for Procedure 1.1.
3. From Procedure 1.2, after performing the LINEST function, does the accepted value of gravity
fall within the uncertainty of the slope?
4. What is your value of gravity from Procedure 2? Include your uncertainty result in reporting
this value. Does the accepted value for gravity fall within the bounds of the result/uncertainty?
5. Discuss your reasoning for assigning your uncertainties that you did in part 2. How accurate
do you feel you were when pressing the stopwatch button relative to the ball launching/landing
on the floor? How confident are you that you were able to determine and accurately measure
where the maximum height occurred?
Lab 6 - The Atwood Machine
The Atwood machine is a device where two masses hang over a pulley (or system of pulley) and
connected by a light string. Assume one mass is resting on the ground and the other is suspended
in the air. If the mass on the ground is equal to or greater than the suspended mass, the system
will not move unless disturbed by an external stimulus. However, if the suspended mass is larger,
once the system is released from rest, the larger mass will fall, and the smaller mass will rise.
Because they are connected by a string, the system will move with one magnitude of
acceleration; one will be down and the other will be up. If we call the initially suspended mass
m1 and the resting mass m2, using Newton’s 2nd law, the acceleration of the system can be shown
𝑚1 − 𝑚2
𝑚1 + 𝑚2
It is this equation we will seek to verify.
Navigate to the following website: https://www.thephysicsaviary.com/Physics/singlepage.php?ID=17
Once you have navigated to the webpage, click anywhere inside the black rectangle. This will
take you to a second screen with a ‘Begin’ button. Click on Begin. This will open the simulation,
and you should see two masses connected by a string suspended over two horizontally-aligned
pulleys. Mass 1 is suspended, while Mass 2 is hanging just over the motion sensor which will be
used to collect the data.
Adjust the masses to the following values:
Mass 1: 130 g (or 0.130 kg)
Mass 2: 125 g (or 0.125 kg)
Use the above equation to predict the value of the acceleration of the system (use 9.8 m/s2 for g).
𝑎𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛 : ___________ m/s2.
Hit the Start button above Mass 1. This will set the system into motion. After doing so, scroll the
page down and you will see two graphs; one a position vs time and the second a velocity vs.
From the velocity vs. time graph, pick and record the time and velocity for 10 data points.
Record your data in Excel and plot. Once plotted, fit with a linear trend line, and record the
slope. The slope should also represent the acceleration of the system.
Run a LINEST function on the data to then also obtain the uncertainty and record.
Acceleration: _____________ + ___________ m/s2
Repeat the above procedure for the following values of the masses:
Mass 1: 170 g
Mass 2: 165 g
Mass 1: 200 g
Mass 2: 195 g.
1. For the three trials, give a general statement about the agreement between your predicted and
2. Perform a % error between the predictions and the measured values. State all three % errors. If
these are more than 5%, it is likely that either there was a calculator error when using the
formula to make the predictions, or there were mistakes in recording the data from the plot (be
sure to pay attention to the units and axis labels.)
3. From the results of the LINEST functions, do the predicted accelerations fall within the range
of the measured accelerations and their uncertainties?
4. What are the major sources of error in this procedure? Even though this is a simulation, there
are a few possible sources that could affect the overall result negatively.
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