PASCO ME-6831 Ballistic Pendulum_Projectile Launcher - Manual (Page 10)

B a l l i s t i c P e n d u l u m / P r o j e c t i l e L a u n c h e r B a l l i s t i c P e n d u l u m T h e o r y
®
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Ballistic Pendulum Theory
Overview
The ballistic pendulum is a classic method of determining the velocity of a projectile. It is also a good demonstra-
tion of many of the basic principles of physics.
The ball is fired into the ballistic pendulum, which then swings up a measured amount. From the height reached by
the pendulum, you can calculate its gravitational potential energy. The gravitational potential energy is equal to the
kinetic energy of the pendulum at the bottom of the swing, just after the collision with the ball.
You cannot equate the kinetic energy of the pendulum after the collision with the kinetic energy of the ball before
the swing since the collision between ball and pendulum is inelastic, and kinetic energy is not conversed in inelas-
tic collisions. Momentum is conserved in all forms of collisions, so you know that the momentum of the ball
before the collision is equal to the momentum of the pendulum after the collision. Once you know the momentum
of the ball and the ball’s mass, you can determine the initial velocity.
There are two ways of calculating the velocity of the ball. The first method (called the “approximate method”)
assumes that the pendulum and the ball together act as a point mass located at their combined center of mass. This
method does not take rotational inertia into account. It is somewhat quicker and easier than the second method
(called the “exact method”), but not as accurate.
The second method (exact method) uses the actual rotational inertia of the pendulum in the calculations. The equa-
tions are slightly more complicated, and it is necessary to take more data in order to find the moment of inertia of
the pendulum, but the results are generally better.
Please note that the subscript “cm” used in the following equations stands for “center of mass”.
Approximate Method
Begin with the potential energy of the pendulum at the top of its swing after the collision with the ball:
where M is the combined mass of the pendulum and ball, g is the acceleration due to gravity, and
h is the change
in height. Substitute for the change in height:
where R
cm
is the distance from the pivot point to the center of mass of the pendulum/ball system. This potential
energy is equal to the kinetic energy immediately after the collision:
where v
p
is the speed of the speed of the pendulum just after collision. The momentum of the pendulum just after
the equation is:
which you can substitute into the previous equation to give:
Solving this equation for the pendulum momentum gives:
PE
Mg
h
cm
=
h
R 1

cos –
 
=
PE
MgR
cm
1

cos –
 
=
KE
1
2
---Mv
p
2
=
P
p
Mv
p
=
KE
P
p
2
2M
-------- =
P
p
2M KE
 
=