Ballistic Pendulum

 

When a projectile is fired at the pendulum and captured by it, the combination of the pendulum and projectile will swing upwards by an amount which is indirectly related to the original muzzle velocity of the bullet.

According to the law of conservation of energy, the increase in height, Dh, of the pendulum block and projectile is such that the increase in potential energy is equal to the kinetic energy just after the collision.:

(m + M) g Dh = 1/2 (m + M) V2

where m is the mass of the projectile, M is the mass of the pendulum, and V is the velocity of the bob and projectile just after the projectile has been captured. This equation can be simplified:

g Dh = 1/2 V2

Figure 2. Schematic Drawing of Motion

From Figure 2 we can write:

Dh = L ( 1 - cos a)

Combining these equations and solving for V we get:

The momentum of the projectile before it is captured is equal to the momentum the combination of the bob and pendulum has after the capture:

m v = (m + M) V

or

One way to check this out is to find the initial velocity another way. We do this by going backwards through the equations of motion. If we measure how far the projectile travels we can then solve for the initial velocity and compare results. In order to confirm that the laws of conservation of energy and momentum have given the correct muzzle velocity, we will determine the muzzle velocity. Position the "gun" so that the projectile is launched out over the floor. (You may wish to go out into the hall and do this so the projectile hits the soft carpet.) Be sure to wear safety glasses when you shoot the projectile. Use caution to avoid the possibility of striking another person. (Points off if you hit someone else. Hah!) The horizontal distance the projectile will travel before striking the floor is related to the muzzle velocity by the following:

y = 1/2 g t 1/2

x = v t

If these equations are rearranged to eliminate t we find the initial velocity to be simply::

Compare these two values and compute the percentage difference. Discuss with the reader which you think is more reliable a measure of the initial velocity.

Procedure:

Caution: Do not aim the "gun at anyone at any time. The impact of the projectile can cause serious bodily injury.

Caution: Do not put hands or fingers near the "gun" or breech bolt. The impact of the compressed spring can hurt. Ouch!!

1. Place the instrument on a solid lab table about a metre above the floor. Leave a space of about 3 metres clear in front of the gun. Level the instrument (you can tell this when the block is aligned with the projectile.) with the two leveling screws on the base.

2. You'll need to measure the mass of the pendulum block alone. This is difficult if you have to take it off the strings. Try to measure the mass carefully by placing the mass on a scale moved adjacent to the instrument. Set up the instrument again.

3. Determine the initial height of the pendulum bob.

4. Pull the breech bolt back until it is latched. there are three settings. It is strongly advised to use one of the first two actches. Avoid the third catch as this gives a fairly high muzzle velocity.

5. Fire the projectile and see how closely the ball lodges into the block. If you carefully set up the pendulum when leveling, this should be no problem.

6. Repeat this about 5 times using the same breech bolt setting.

7. Calculate the value of the muzzle velocity.

Mass of Projectile

m

Mass of Pendulum Bob

M

Length of Pendulum

L

 Position of Pendulum Bob
       
       
       
       

8. Calculate the Muzzle Velocity, v

9. Move the pendulum block up to the highest point so as to clear the path to fire the projectile to determine its range and velocity.

10. Measure the height of the gun off the floor. Get real! Use appropriate significant figures. Don't try to snow your mother by suggesting you can measure things any better than you really can.

11. Load the gun and set the breeh bolt to the same setting as before.

12. Fire the gun and record wher the projectile hits the floor. Repeat thsi four or five times to get an average distance. After all, you will not be able to measure this down to a tenth of a millimetre.

13. Calculate the initial velocity v and compare with the value determined earlier in step 8.

Starting Height of projectile
Horizontal Distance y
   
   
   
   
   

 

Limitations and Discussion, Data Analysis, Summary and Conclusions