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NORTHERN ILLINOIS UNIVERSITY
PHYSICS DEPARTMENT
Physics 253 – Fundamental Physics Mechanics
Lab #8 The Ballistic Pendulum
Meet in FR 233.
Sections A, B, C, G: Tuesday Oct. 18; Sections D, E, T: Thursday Oct. 20.
Read Giancoli: Chapter 9
Lab Write-up due Tue. Oct. 25 (Sections A, B, C, G); Thur. Oct. 27 (Sections D, T)
Apparatus
A ballistic pendulum is a device consisting of three parts: a spring gun, a ball that
can be launched from the gun, and a cup at the end of a pendulum to catch the ball. The
spring gun is designed to fire a ball of mass mb with an initial velocity vi. The pendulum
and cup can be moved out of the way. This permits the ball to be fired as a projectile and
the initial velocity measured.
When the pendulum is in its lower position, the cup with mass mc is ready to catch
the ball when fired. When the ball is caught in the cup, the energy of the combined cup
and ball is used to swing the pendulum up by a height h. A ratchet catches the cup and
allows you to read a position measurement by means of a pointer that catches in a groove
of the ratchet.
The measurement on the ratchet is marked in units that go from 0 to 40. There are
marks at the halfway points (5, 15, 25, 35), and individual ratchet positions represent
steps of 1. The ratchet measure can be converted to height by means of the following
table:
Table 1: Conversion of ratchet measurement to height
Ratchet
Measurement
0
10
20
30
40
Height
(mm)
65
73
80
88
96
Determining the conversion factor for other ratchet measurements requires
interpolation. Interpolation means estimating the value between two known values. We
do this by assuming that the conversion function is a straight line between the two known
points. The two points are used to create the equation of a straight line: y  mx  b.

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For our specific table, we treat the ratchet measurement x as the independent
variable, and the height of the ball h as the dependent variable. To interpolate, look at
your measurement of x. If it’s not in the table, find the measurement, x1, in the table that
is immediately below your measurement and its corresponding height h1. Find the
measurement, x2, in the table that is immediately above your measurement and its
corresponding height h2. The interpolated height h for your measurement is
h  h1 
x  x1
h2  h1
x 2 x1
(1)
Theory

In an earlier laboratory
we studied projectile motion. A ball moving horizontally
with initial velocity vi was allowed to fall freely to the ground a distance d below the
launch point. The range R beyond the launch point was derived from the kinematics of an
object subject only to gravitational acceleration g. The relation between the velocity,
range, and vertical distance was:
g
(2)
vi  R
2d
Kinetic energy is the energy of motion. It depends on the mass and velocity of an
object. When the ball is initially launched it has an initial kinetic energy Ki.

1
K i  mbv i2
2
(3)
When the ball is caught by the cup both the mass and velocity will have changed. The
new mass will be the sum of the mass of the ball and the cup, mb  mc . The final velocity

vfi immediately after the ball is in the cup is given by
v fi 
mb
v
i
mc  mb
(4)
(Here we have used the principle of conservation of linear momentum which we will
develop in Chapter 9). Using the new mass and the velocity in Eq. (4), the kinetic energy

after the capture Kf is
Kf 
mb  mc
 mb 
2

 mb  mc 
2
vi2 
1 mb2
vi2
2 mb  mc
(5)
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Gravitational potential energy is related to the height of an object. Objects that
are elevated can gain kinetic energy as they descend. This gravitational potential energy
U can be represented as
U= (mb+mc)gh
(6)
where we have used the total mass of the ball and cup, acceleration due to gravity, and the
height that the pendulum ended at after catching the ball. Conservation of energy is a
fundamental principle of physics. In mechanical systems like the ballistic pendulum,
energy conservation predicts that the kinetic energy can be fully converted into potential
energy.
Data Collection
You must show all your calculations in Parts A&B below to your TA before you leave the
lab. This includes standard deviation and propagation error calculations.
Part A: Muzzle velocity
(1) Make sure that the apparatus is clamped to the table and that a stop wall is available,
and swing the pendulum arm out of the way.
(2) Measure the vertical drop d from the end of the gun to the floor and record this value
and its uncertainty.
(3) Fire the gun and note where the ball lands. Fasten sheets of carbon paper at that
location.
(4) Fire the gun 10 times so that the impacts are recorded on the carbon paper.
(5) Measure the horizontal distance from the spot on the floor directly beneath the tip of
the gun, to the edge of the paper.
(6) Measure and record the distance of each mark on the paper from the edge of the
paper, and find the mean distance, the standard deviation of the mean, and the
standard error of the mean.
(7) Add the average distance on the paper to the distance of the paper from the gun along
the floor to get the measured range R and determine its uncertainty.
(8) Calculate the initial velocity vi using Eq. (2), and find its uncertainty through the use
of propagation of errors; your lab TA will give you the equations to use.
Part B: Conservation of energy
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(9) Weigh and record the mass of the ball mb and its uncertainty, and record the mass of
cup mc written on the apparatus (and estimate its uncertainty).
(10) Fire the gun with the ball into the cup 10 times and record the ratchet measurement,
x, for each trial.
(11) Calculate the mean ratchet measurement, the standard deviation of the mean, and
the standard error of the mean. Use the interpolation formula in Eq. (1) and the
data in Table 1 to find the height h, and determine its uncertainty.
(12) Use the initial velocity, vi, and Equation 3 to find the initial kinetic energy Ki and
determine its uncertainty.
(13) Use the initial velocity, vi, and Equation 5 to find the final kinetic energy Kf and
determine its uncertainty.
(14) Use the height, h, and Eq. 6 to find the potential energy U and determine its
uncertainty.
(15) Compute a percent difference between Kf and U where
% Difference = 100 [ | meas. #1 – meas. #2 | / ½ (meas. #1 + meas. #2) ]
Note that we must use the percent difference relationship to compare the two values
(rather than the percent error described in an earlier lab) because neither
measurement is a commonly accepted value of the energy.
Analysis
From your results, how well was energy conserved? Do Kf and U agree to within their
uncertainties? Does doubling the uncertainty in vi and h result in energy being conserved
to within their uncertainties? How much larger would your uncertainties in vi and h have
to be to result in order for energy to be conserved? What factors may account for energy
not being completely conserved? Try to take into account all processes that may lead to
this discrepancy.