Energy and Momentum Conservation: The Ballistic Pendulum
Introduction: With energy and momentum conservation, you can calculate the initial velocity of a metal
ball fired by a spring launcher. Given this initial velocity, the projectile motion equations predict the
firing distance of a Ballistic Pendulum. You will confirm your prediction by shooting a metal ball at a
target. This will require precise measuring skills, a sharp eye, serious calculations, and strong arms.
Required Equipment: Beck ballistic pendulum assembly w/
spring gun, ball, pendulum, meter sticks, rulers, plumb bob
Study I: Measure Parameters
1. [DT1] Center of mass of ball/pendulum system: Place the
ball in the pendulum cup. Remove the pendulum from
the support arm and place it on a “knife edge” and locate a
position where it balances. Mark the pendulum with a
felt-tip pen “dot” to locate the center-of-mass. Use this
dot as a reference for all y measurements. Reattach the
pendulum securely to the support arm.
2. [DT1] Record the mass of the pendulum and ball. Ignore errors in these mass values.
3. [DT1] Measure and record the height, yo, of your dot (the center of mass of the ball-pendulum
system) above the pendulum base. You can neglect the uncertainty in yo. Measuring in midair is
difficult, so you’ve been provided with a wooden ruler with metal edges/triangles and a plastic ruler.
Make a caliper with the wooden ruler and use the metal edges to accurately estimate the height. The
plastic ruler is not necessary, but some groups find it helpful to measure between the metal edges.
Study I: Firing the gun
4. Load the spring-gun: put the ball in place and push back until the spring is latched. This is difficult!
5. Lower the pendulum and make sure that it can swing freely and the opening in the bob is lined up
with the gun. Each time you fire the spring gun, make sure that the pendulum is securely fastened to
the support arm assembly at the top.
6. [DT2] Starting with the pendulum at rest, shoot the ball into the cup. The pendulum (and ball) will
swing up and lock into some particular tooth in the rack. Measure the elevation (ymax) of the center
of mass above the pendulum base. Ultimately you care about the change in height of the pendulum,
but it is easier to line the bottom of the ruler with the base than yo.
7. [DT2] Take five trials. If the position for a trial is much different from the others, something may
have happened during the firing and you should discard this measurement and repeat it.
8. [DT3] Use Graphical Analysis to calculate the average and standard deviation of your data. Use an
“index” variable, create a plot, and hit the STAT button.
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Study I: Collision of Ballistic Pendulum
Data Table 1: Measured Parameters – Ballistic Pendulum
yo
mball
M pendulum
Table 2: Height of pendulum swing
1
Trial
2
3
4
5
ymax
Table 3: Statistical Data for ymax (don't print graph for STATS!)
Average
Standard Deviation
ymax
 ymax 
Uncertainty
 ymax = /  N
Steps 1-3 will help you to calculate the initial velocity OF THE PENDULUM using the change in
energy AFTER the collision. Remember that MKS (meters, kilograms, seconds) make your life easier.
1. Calculate the change of the height of the (center of mass of the) pendulum:
2. The pendulum undergoes a change in potential energy due to the change in height. Determine the
mass of the pendulum while it swings and calculate:  U=mass∗g∗ y pend
3. The pendulum loses kinetic energy as it swings. What is its final kinetic energy? Why?
4. You want the initial velocity of the pendulum, which appears in the formula for kinetic energy.
In order to calculate vo of the pendulum, assume that energy is conserved  E=0 and
combine the kinetic and potential energy.
2
2
 K=1/2∗mass∗ v f −v o− pend 
5. Calculate the initial velocity OF THE BALL using conservation of momentum DURING the
collision. The formula is simple pi= p f . Initially only the ball has momentum, so
pi=mball∗v o−ball . After the collision, both masses move together with a common velocity
v o− pend . Calculate the momentum AFTER the collision and solve for v o−ball :
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Study II: Projectile Motion (DON'T Fire without a TA Witness!).
Remove the pendulum from the support arm and set it aside. You're going to check your calculation from
Study I with a projectile launch (no collision). If you have room at your station to set up a firing range,
use your table for the following measurements. If you are at the center of the classroom, set up a target
range at one of the tables or benches at the perimeter of the classroom.
1. Measure Y, the height of the ball above ground and use the equations from projectile motion to
predict the horizontal range of the ball. You will need your result from Study I for the initial
velocity of the ball.
y t=−1/2gt 2v oy t  y o and x t=v ox tx o
2. You need to calculate an error in the range value, so you can make a paper target. This is simple
if you've done a symbolic calculation for v ox−ball in terms of the change of height  y pend ,
but quite difficult otherwise.
a) It turns out v ox is proportional to
roots:
  y pend
and you can use the following rule for square
 v ox   y pend
=
v ox 2 y pend
b) Once you've found the relative error in v ox , the range will have the same relative error
(ignore uncertainty in the height measurement). Calculate the absolute error in the range.
3. Once you’ve completed the calculations, you are ready to make a paper target and move to your
firing range (remember DON'T FIRE!). The idea is that you can use the uncertainty calculated
above to make real life error bars. That way you're not trying to hit a single place on your target.
Target
Range
+/- uncertainty
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4. Bonus point opportunity (5 possible points): Carefully measure where you expect your ball to
land and place your paper target. Get it aligned as best as you can. Have a TA look at your work
and set up. Be sure you have a TA witness before firing your shot. You can earn 5 bonus
points for hitting the target on the first try.
Alignment will be tricky: use the long meter sticks, the plumb bob, the floor tiles, and your eyes
so you can find to get the pendulum lined up with the catch box. A plumb bob is designed to
make a vertical line so that you can more accurately measure from heights; get a lesson from the
TA if you don’t know how to use one.
5. When ready get a piece of carbon paper and your TA. After the TA has had a look at your
calculations and target, place the carbon paper carbon-side down on the target (don’t use tape).
FIRE!
Hit Target!
Missed Target!
TA Witness Initials:
6. After the TA has witnessed the first shot, replace the carbon paper and shoot 5 more times. Use
masking to tape to check for recoil of the entire assembly. You may have to move the pendulum
between firings so that it always has the same initial starting position.
Trial
1
2
3
4
5
Range
7. After completing the trials, estimate the uncertainty in the range based on the spread of points on
the paper.
8. Calculate the initial velocity (given the average range value). Calculate the uncertainty in vox
based on your estimated uncertainty in the range.
Discussion Questions.
1. Is the mechanical energy conserved in the collision between the ball and the pendulum? Does this
matter in your calculations (i.e. did you use conservation of energy to describe this part of the
experiment?).
2. Why is it a reasonable assumption that  E=0 during the pendulum swing? What are some of the
problems with this assumption? Identify sources of systematic and random uncertainty in this lab.
3. How do your results support the concepts of conservation of momentum and energy? Explain with
your data.
4. What explains the range of values you measured for ∆y and X, the range of the ball? Could these be
attributed to “human error” or is something else going on?
5. Compare your values of vox from Studies I&II.
Do they agree within error?
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