The Ballistic Pendulum

Purpose:  To use the ballistic pendulum to determine the initial velocity of a projectile using conservation of momentum and conservation of energy.
Equipment: Ballistic pendulum, carbon paper, meter stick, clamp, box, triple beam balance, plumb. 
Introduction: In this experiment a steel ball will be shot into the bob of a pendulum and the height, h, to which the pendulum bob moves, as shown in Figure 1, will determine the initial velocity, V, of the bob after it receives the moving ball.  If we equate the kinetic energy of the bob and ball at the bottom to the potential energy of the bob and ball at the height, h, that they are raised to, we get:
      

( K.E ) bottom = ( P.E)top

      ½ ( M + m ) V² = ( M + m ) g.h

 where M is the mass of the pendulum and m is the mass of the ball. Solving for V we get:

        V  = √ 2gh            …………….                                                                                                   

(1) Using conservation of momentum we know the momentum before impact (collision) should be the same as the momentum after impact. Therefore:

Pi= Pf

      or

           mv0 = ( M + m) V 

(2) where v0 is the initial velocity of the ball before impact. By using equations (1) and (2) we can therefore find the initial velocity, v0, of the ball. We can also determine the initial velocity of the ball by shooting the ball as above but this time allowing the ball to miss the pendulum bob and travel horizontally under the influence of gravity. In this case we simply have a projectile problem
where we can measure the distance traveled horizontally and vertically (see Figure 2) and then determine the initial velocity, v0, of the ball.

M + m Starting with equations:
    ∆x = voxt + ½axt² 
    (3) ∆y = voyt + ½ayt

 ( 4 )
You should be able to derive the initial velocity of the ball in the horizontal direction (assuming that
∆x and ∆y are known). Include this derivation in your lab report.

We set up our ballistic pendulum loade the triger with the ball and shoot it two times.  From this we got a average hight (h).  Then we toke the pendulum and the ball and got their masses.  Using equations 1 and 2  we were able to determine our initial velocity. 

 

Next we went on to determen initial velocity from range and fall. We placed the pendulum up on the rack so that it wouldnt interfere with balls motion. We clamped the frame to the table so that it wouldnt move and placed a piece of carbon paper on the floor. We shoot the ball a few times and took our average positon to be delta X.  From delta X and delta Y(the hight) we used equation 3 to solve for initial velocity. Having the two velocities we toke a percent diffrence and agreed that the free fall experiment was the more accurate one since there is no interaction with any objects and we toke the average distance of the fall which were almos identical.  We trust the conservation of momentum as well but fell that there could be more room for error since we got diffrent heights almost every time we fired the ball into the pendulum.

Conclusion:     I learned that through the conservation of Energy we can find a initial velocity as long as we are able to get a hight H Since 1/2 ( M + m) V^2 = ( M + m) g *h .  It was also nice to be able to test our regular kinematics to solve for a free fall initial velocity since it was the basis we started with.  Sources of error could be found in the measurments of either the height or the distance for the free fall.  As for the conservation of energy we found that everytime we shoot the ball into the ballistic we would get a diffent reading.  This could be a big source of error since it lacked consistency.  This is why we agreed that the free fall was probably the more accurate testing out of the two.  After all it did land almost in the same exact location every time we shot it.