Ballistic Pendulum
Introduction
In this experiment, we will use a ballistic pendulum to measure the speed of a projectile. A
ballistic pendulum consists of an object suspended vertically from a support by a rod or
cable. You can fire a projectile horizontally into the pendulum, causing it to swing. The
maximum height attained by the pendulum can then be measured. This height can be used
to calculate the initial speed of the projectile.
There are two parts to this calculation. First the ball hits and sticks to the pendulum. This is
an inelastic collision, by definition. Momentum will be conserved during the collision
because external forces are negligible. The mathematical relationship is as follows:
p = momentum, v = velocity, m = mass, 1 = just before collision, 2 = just after collision, 3 =
pendulum at maximum height, b = ball, p = pendulum
[1] Σp1 = Σp2 (conservation of momentum)
[2] mbvb1 + mpvp1 = mbvb2 + mpvp2 (using the definition of momentum in equation [1])
[3] vp1 = 0 (since the pendulum begins at rest)
[4] vb2 = vp2 = v2 (since the ball and pendulum stick together)
[5] mbvb1 = (mb + mp)v2 (substitution of equations [4] and [3] in equation [2])
[6] vb1 = (1 + mp/mb)v2 (equation [5] solved for the variable of interest)
The masses on the right hand side can be easily measured with a scale, but the velocity of the
group just after the collision is difficult to directly measure. Instead, you can use the final
height of the ball and pendulum to compute the velocity of the group. This is done with an
energy calculation. There are two forces acting on the pendulum and ball system. There is a
variable tension force that does no work (perpendicular force), and a gravitational force by
the earth that can be taken into account by including the earth as part of the system and
using gravitational potential energy as follows:
[7] U2 + K2 = U3 + K3 (conservation of energy)
[8] v3 = 0 (since it reaches maximum height)
[9] K3 = ½*(mb + mp)v32 = 0
[10] (mb + mp)gy2 + ½*(mb + mp)v22 = (mb + mp)gy3
[11] v2 = √[2g(y3 – y2)] (equation [11] solved for v2, the speed of the group)
Conservation of momentum during the collision gives equation [6], while conservation of
energy after the collision gives equation [11]. A substitution of equation [11] into equation
[6] yields a formula for the speed of the projectile in terms of measurable quantities. This is
left as an exercise for the student. Exercise is fun!
Once the initial speed of the projectile is known, the motion of the projectile without the
pendulum in the way may be predicted. In the second part of this experiment, you will
predict and test the motion of the projectile using two-dimensional kinematics.
Experimental Procedures
Ballistic Pendulum
1) Derive a symbolic formula for the initial speed of the ball in terms of measurable
physical quantities.
2) Assemble and test the ballistic pendulum apparatus as directed. You may need to
tighten or loosen the spring to get measurable results, but do not do so in the middle
of the experiment.
3) Measure the mass of the projectile and the mass of the pendulum.
4) Measure the initial height of the pendulum. If you are using a gold ballistic
pendulum, then measure the height of the pin. If you are using another ballistic
pendulum, then measure the height of the center of the pendulum bob.
5) Fire the projectile into the pendulum five times. Record the height attained by the
pendulum in each trial
6) Average the five measurements of height. Note that this is a calculation so it belongs
in that section of your report.
7) Calculate the initial speed of the ball.
Projectile Motion
1) Measure the distance from the floor to the bottom of the ball.
2) Use kinematics for the motion in the y direction to calculate the time it will take for
the ball to hit the ground when it is launched horizontally. Note that the y
component of the initial velocity vector will be zero.
3) Use kinematics for the motion in the x direction and the speed of the ball to
calculate the theoretical horizontal distance the ball will travel.
4) Lock the pendulum up out of the way. Position the apparatus so that the projectile
will clear the bench and strike the floor.
5) Position one member of your team to catch the projectile after it strikes the floor.
6) Place an action-figure at a predicted location.
7) Test your prediction. Note that this will be an observation that belongs in your raw
data section.
Repeat the entire experiment using a different color (gold versus not gold) ballistic
pendulum.