1M40.41 Ballistic Pendulum
Abstract
The ballistic pendulum demonstrates the relationship between the law of conservation of momentum and the
law of conservation of energy. In this demonstration a spring loaded gun is used to fire a brass ball at a stationary
pendulum. When the ball collides with the end of the pendulum it attaches itself to it. The energy in the brass
ball is transferred to the pendulum causing it to begin to oscillate. A catching mechanism has been added to
the apparatus to catches the pendulum at its maximum height. The strength of the spring loaded gun can be
adjusted in order to show the direct relationship between the velocity of the ball, and the energy transferred to
the pendulum. A quantitative aspect can be added to the demonstration by determining the initial velocity of
the ball bearing. This can be done by removing the catching mechanism and then treating the ball bearing as a
horizontally launched projectile.
Picture
Safety Concerns
Ensure that the pendulum is stationary before firing the ball bearing. Movement in the pendulum can cause
the bearing to be deflected and not enter the pendulum, possibly causing injury to bystanders. Do not attempt
to conduct the measurement of the ball bearing’s initial velocity without first removing the catching mechanism,
as the ball bearing will collide with it.
NOTE: Make sure that the apparatus is not stored with the gun cocked as this could result in the
accidental firing of the gun while being transported. Instead leave the gun such that the spring is relaxed.
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Equipment
• Ballistic pendulum apparatus (Central Scientific Co 75425 000)
• carbon paper
• tape
• measuring tape
Procedure
Put the ball bearing on the end of the gun and push the plunger of the gun back until it clicks into place.
Release the pendulum from the catching mechanism so that it can hang freely in front of the gun and ball bearing.
Measure the height of the point of the arrow on the pendulum with respect to the base of the apparatus. Once the
pendulum is stationary fire the ball into the pendulum. Measure the height of the arrow on the pendulum, with
respect to the base, to determine its change in vertical position. Note that the scale on the catching mechanism
is not calibrated and should only be used to make reference points and not for making measurements. Use this
change in height and Equation 6 to calculate the ball’s velocity. Turn the knob at the rear end of the gun to
adjust the tension in the spring, and therefore the velocity of the ball.
The apparatus must be modified in order to measure the velocity of the ball bearing using the projectile motion
method. This can be done by removing the wing nuts that hold the catching mechanism in place. To measure
the horizontal displacement of the ball fire a test shop and determine the approximate spot where the ball hits
the ground. Put a piece of carbon paper on top of a sheet of paper and tape them both to the floor at the spot
where the test shot fell. Fire a second shot so that the ball bearing hits the carbon paper causing a black mark
to be made on the sheet of paper. Measure the horizontal and vertical positions of this dot relative to the launch
point of the ball bearing. Use these measurements and Equation 10 to calculate the velocity of the ball.
Theory
A spring loaded gun accelerates the brass ball to an unknown velocity. The brass ball then collides with the
pendulum causing it rise to a measurable height above its original position. By using the laws of conservation of
energy and momentum it is possible to calculated the initial velocity of the brass ball. This velocity of the ball
can also be measured by treating the ball like a projectile and determining the horizontal distance that the ball
travels in the amount of time it takes the ball to fall to the ground.
From the law of conservation of energy, the kinetic energy of the pendulum at its lowest point is equal to its
potential energy at its maximum height. This relationship is described by the expression,
(mp + mb ) vs2
= (mp + mb ) gh,
2
(1)
where mp is the mass of the pendulum, mb is the mass of the ball, vs is the center of mass velocity of the joined
system, g is the acceleration due to gravity and h is the height reached by the system’s CM (center of mass)
above it’s original position. The velocity of the system’s CM is therefore,
vs2 = 2gh.
(2)
Similarly an expression for the initial velocity of the system after the collision can be obtained using the law
of conservation of momentum which states that,
vb mb + vp mp = vbf mb + vpf mp ,
(3)
where vb and vbf are the initial and final velocities of the ball bearing respectively, mb is the mass of the ball
bearing (mb = 67.2g), vp and vpf are the initial and final velocities of the pendulum respectively and mp is the
mass of the pendulum (mp = 332.3g). This expression can be simplified by observing that since the pendulum
in initially at rest then vp must be zero, and that since the bearing and pendulum stick together that the final
velocity of the ball and the pendulum are the same. In other words,
vbf = vpf = vs .
2
(4)
After these simplifications have been applied Equation 3 becomes,
vb mb = vs (mb + mp ) .
(5)
by substituting Equation 5 into Equation 2 and by rearranging, an expression for the initial velocity of the ball is
obtained which states that,
mb + mp p
vb =
2gh.
(6)
mb
Finally, since the mass of the pendulum and the mass of the ball bearing are known, it is possible to indirectly
measure the initial velocity of the ball bearing by directly measuring the height to which the fired ball bearing
causes the pendulum to swing.
The velocity of the ball bearing can be confirmed independently by treating the ball bearing like a projectile.
By removing the catching mechanism ball bearing can be fired horizontally from a elevated surface and allowed
to fall to the ground. The time that the ball traveled can be determined by plugging the bearing’s displacement
in the vertical direction into the kinematics equation,
y=
f t2
2
where y is the ball’s vertical displacement. The expressionfor time as a function of y is,
r
2y
.
t=
g
(7)
(8)
Now that the projectile’s flight time is known it is possible to substitute the kinematics equation,
vb =
x
t
(9)
where x is the horizontal displacement of the ball, into Equation 8 in order to obtain the final equation which is,
r
g
vb = x
.
(10)
2y
What remains is an expression for the velocity of the ball bearing as a function of its horizontal and vertical
displacements, which can be determined experimentally.
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References
[1] G. D. Freier and F. J. Anderson. A Demonstration Handbook for Physics, ”Mi-3. Ballistic Pendulum”.
American Association of Physics Teachers One Physics Ellipse, College Park MD, 1996. pg M-26.
[2] P.D. Gupta. ”Blackwood Pendulum Experiment and the Conservation of Linear Momentum”. American
Journal of Physics, 53, March 1985. pg 267-269.
[3] C.N. Walls. ”Apparatus Review: The Beck Ballistic Pendulum”. American Journal of Physics, 36, December
1968. pg 1161.
[4] Richard Manliffe Sutton. Demonstration Experiments in Physics, ”M-124. Velocity of Bullet by Ballistic
Pendulum”. Mcgraw-Hill Book Company Inc, New York and London, 1938, pg 55-56.
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