Part A
Consider a small mass m with initial velocity v_{o} which collides and sticks to a pendulum of mass M and length L, as shown. After the collision, the m/M combination rises to a height h.
1. What can be said about the momentum and kinetic energy of the ball/pendulum system just before and just after the collision?
2. What is the velocity of the combination just after the collision in terms of variables given above?
3. Assuming that energy is conserved after the collision, determine the height that the combination rises to after the collision. (Kinetic energy K = 1/2 m_{total} v^{2} and gravitational potential energy U_{g} = m_{total} gh.)
Part B
A ball is fired horizontally from a height H with speed v_{o} and hits the ground a distance D from the launch position. Determine the intitial velocity in terms of the variables H, g, and D.
Part C
1. Load the ball into your spring gun to the maximum compression. (Make sure that the ball stays back in contact with the spring and does not roll forward.) Fire the ball horizontally and determine the initial speed of the ball, using your result from Part B. Fire and measure the distance three (3) times to get the average distance traveled and the error in the range.
Trial 1 

Trial 2 

Trial 3 

Average distance ± error: 

Also record the height from which the ball was fired:
Calculate the initial speed of the ball and the error in that value. (The error is given by the fractional errors in the terms that make up v: Dv_{o} = v_{o} [ DD/D + 1/2 DH/H ].)
2. Given that the mass of the ball is 65.9 ± 0.5 g and that the pendulum mass is 240.5 ± 0.5 g, predict the height the pendulum should rise when the ball is fired into it. Also calculate the uncertainty in that value using the fractional error rule: Dh = h [ 2Dv_{o}/v_{o} + 2 Dm/m + 2 (Dm + DM)/(m + M) ].
predicted height ± error: 

3. Carry out three trials with your ballistic pendulum to find the average height its center of mass rises. Record the angle in each of the three trials: these will be used to find the height change. Find the length of the pendulum to its center of mass (R) or use the value of 28.7 cm and use this to calculate the height change in each trial using h' = R (1  cosq ).
Angle 1: 
Height 1: 
Angle 2: 
Height 2: 
Angle 3: 
Height 3: 
Average height ± error: 
4. What is the percentage difference between the predicted and measured values? Do they fall within each others error ranges?
5. Give possible explanations for any difference in the two values.