Apr 27th 2015, Ballistic Pendulum Activity

Purpose:

We want to use our knowledge of conservation of momentum and energy to find initial speed at which the ball is shot into the ballistic pendulum.

Experiment:

- We use the ballistic pendulum shown below. Since the ball is shot from a small distance from the pendulum, we can assume that it's initial velocity right after it leaves the cylindrical cannon is the same as the velocity right before it hits the pendulum.

The pendulum is attached to 4 strings.

- Measurements that we need include mass of ball, mass of ballistic pendulum (the white cubical shape), Length "L" of the 4 strings.
- For the experiment, place the silver ball into the blue cylindrical hoop. Make sure the pendulum stays still on a leveled table, and align in place with the marked lines behind it (on the silver plate). Push down the trigger as this will release the metal rod to push the ball out of the cylindrical loop. 
- After the ball is shot into the pendulum, we would be able to get θ, which is shown in the picture above on the silver plate that is marked in degrees.

Data and Analysis:

M-holder +/- dM = 0.0809 +/- 0.0001 kg

m-ball +/- dm = 0.00763 +/- 0.0001 kg

θ +/- dθ = 17.5° +/- 0.5°

L +/- dL = 0.215 +/- 0.005 m

- Using conservation of momentum: 

- Using conservation of energy: KEo = GPE ; as KEo = kinetic energy right after the ball hits the pendulum. GPE= when the system reaches the highest possible height (h) and velocity is zero.

- Substituting V-hit back into equation of conserved mechanical energy,

- We get Vo = 5.59m/s. The last few equations are equivalent to finding Vo and we're going to use these equations to make partial derivatives in the next part a little easier.

- Calculating propagated uncertainty requires partial derivatives. For all the measured variables we have in equation to find Vo, we will take dv/d"that variable". 

Figure 1.

- To get propagated uncertainty in Vo, we use the partial derivatives found above and multiply them by their appropriate uncertainty. Values of dm, dM, dL, and dθ are chosen depending on how accurate the measurement device is. 

Figure 2.

* dθ needs to be in radians.
- Plugging in numbers for figure 1, then add them up using figure 2., we get,

- We get Vo = 5.59 +/- 0.30 m/sec. Therefore, we are within a 5% propagated uncertainty to say that Vo = 5.59m/sec.

Conclusion:

    Some error and uncertainty include:
- Friction in the cylindrical tube opposing the direction of the ball as it leaves the tube.
- The way that the pendulum is attached makes it more likely that the system tends to lift up rather than moving smoothly along the arc.
- Negligible mass of the the wire pointer that determines θ that might effect the maximum height the system could raise up to.
     Nevertheless, we conclude that we can use conservation of momentum and energy to determine the initial velocity of the ball.