Using a Ballistic Pendulum To Demonstrate the Law Of Conservation of Momentum.

November 3, 2003

Using a Ballistic Pendulum

To Demonstrate the Law Of

Conservation of Momentum

Abstract

The concepts of conservation of momentum and conservation of mechanical energy were investigated and used to demonstrate that the law of conservation of momentum is valid. A ballistic pendulum was used to show that the initial velocity of a projectile could be determined using these concepts. To verify the results of the ballistic pendulum experiment, the initial velocity was also calculated by firing it horizontally and calculating the time and distance traveled. The percent difference between the two values for initial velocity was 13.0%.

Introduction

One of the most useful laws of physics is the law of conservation of momentum. For instance, in an inelastic collision, the change in momentum of the combined objects can be used to figure the momentum of the objects before the collision. In this experiment, a ballistic pendulum was used to demonstrate that the law of conservation of momentum is valid.

A ballistic pendulum is a pendulum that begins in a vertical resting position, while a projectile is fire at the end of the pendulum. The pendulum must be capable of incorporating the projectile so that the collision is inelastic and the projectile and pendulum move as a single unit. Examples would be a bullet into a block of wood, or a ball into a trapping basket, such as was used in this experiment. It should be noted that the kinetic energy of the projectile is not conserved during the collision, because of energy losses from such things as sound, heat, or permanent mechanical deformation of the projectile and/or the pendulum.

To test the law of conservation of momentum, a steel ball was fired into a pendulum that had a trap at the end. The ball would enter the trap, and the ball and pendulum would begin to move together in an upward swing of the pendulum. Once the ball has become one with the pendulum, the law of conservation of energy is applicable, because gravity is the only force acting on the system at this point, and gravitational potential energy is conservative.

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By having a swinging pointer attached to the bottom of the pendulum, reverse motion of the pendulum is prevented. This makes it convenient to measure the maximum height obtained by the pendulum. At this point, the kinetic energy of the combined ball and pendulum has been converted to potential energy. The equation:

½ (M+m) V2 = (M+m)gh

Ei (All KE) Ef (All PE)

shows that the initial energy was all kinetic energy, using the combined masses of the ball and pendulum. The final energy was all potential energy, also with the combined masses.

Rearranging this equation, the initial velocity ...

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½ (M+m) V2 = (M+m)gh

Ei (All KE) Ef (All PE)

shows that the initial energy was all kinetic energy, using the combined masses of the ball and pendulum. The final energy was all potential energy, also with the combined masses.

Rearranging this equation, the initial velocity of the combined ball and pendulum can be calculated:

V = (2gh)

Once this velocity was determined, it was possible to calculate the initial velocity of the projectile before the collision. This is because the law of conservation of momentum was conserved. Therefore, the momentum before the collision, mv, was equal to the momentum after the collision, (M+m)V. Rearranging to solve for v results in:

v = ((M+m)/m)V

To check that momentum was actually conserved, the initial velocity had to be verified in a different way. To do this, the ball was fired horizontally and the falling distance and horizontal travel distance were used to calculate travel time, based on an acceleration of 980.0 cm/s/s due to gravity. Once the travel time and average travel range were tested over ten trials, the results were calculated and percent differences were figured.

Purpose

To demonstrate that the law of conservation of momentum is valid.

Procedures

Part 1

Measure and record the vertical distance from the point of the pointer on the pendulum to the base while the pendulum is in the resting position.
Cock the firing mechanism and place the ball on its holder.
Fire the ball at the pendulum.
Measure the vertical height of the pointer at the final position and record.
Repeat for a total of ten trials.

Part 2

Remove the pendulum from the apparatus.
Do a test-firing of the ball over the side of a table to get an estimate of the trajectory of the ball.
Mark the location of the apparatus with tape to ensure same firing location on each attempt.
Place ball in its holder, and measure the vertical distance from the ball to the floor.
Tape a sheet of paper to the floor in the area where the ball will land.
Fire the ball ten times, and measure the horizontal distance covered by the ball from the firing position to the point the ball strikes the paper. Record data.

Sources

Wozniewski, L. (2000). Physics Laboratory Manual: Coefficient of Static and Kinetic Friction. Retrieved October 19, 2003, from Indiana University Northwest, Department of Chemistry, Physics, and Astronomy Web site: http://www.iun.edu/~cpalw/pweb/balpen/balpen.htm

Cutnell, John and Johnson, Kenneth. Physics Sixth Edition. Hoboken, NJ: Wiley and Sons, 2004.

Tables of Experimental Data

Sample Calculations

Results

Conclusion

The ball left the firing mechanism with velocity v, and struck the end of pendulum, where it was trapped and became a combined unit with the pendulum. Because this was an inelastic collision, some of the kinetic energy of the ball was lost during the collision. However, once the ball and pendulum became a single unit, the velocity of the ball/pendulum unit V could be calculated, using the law of conservation of energy. The initial kinetic energy of the combined unit, ½ (M+m)V2 , was the total energy of the system at the point immediately after impact. This kinetic energy caused the pendulum to swing. As the pendulum began its climb, the kinetic energy was converted to gravitational potential energy, which is conservative. By measuring the gravitational potential energy at the final position of the pendulum, the initial kinetic energy was known, since they were equal:

½ (M+m) V2 = (M+m)gh

Since the mass of the pendulum M and the mass of the ball m were known, the equation was solved for the initial velocity V of the combined unit, as in:

V = (2gh)

The calculations of the initial combined velocities of the ten trials ranged from 126.4cm/s to 131.0cm/s, with an average of 129.9 cm/s.

Once the initial velocity of the combined unit was known, the momentum was also known. The momentum before the collision could then be calculated, since the law of conservation of momentum applied. Therefore, the momentum before the collision equaled the momentum after the collision:

mv = (M+m)V

From this equation, the initial velocity of the projectile v could be solved for:

v = ((M+m)/m)V

The calculated initial velocities of the projectile in this experiment ranged from 625.5cm/s to 648.2cm/s, with an average of 642.9 cm/s.

To validify the calculations, the initial velocity of the projectile was also measured using gravity as the measuring stick. The ball was fired ten times from the apparatus, and the travel distance was measured from the firing point of the apparatus to the point the ball hit the ground. The time of travel was calculated from the fact that the ball was 85.0 cm from the ground initially. Since vertical acceleration due to gravity is independent of the horizontal motion of the ball, the time for the ball to reach the ground was (2s/g) where s was the vertical distance traveled, 85.0 cm, and g was the acceleration due to gravity, which is 980 cm/s2.

The vertical travel time was for the same time period as the horizontal travel time. Therefore, the average horizontal velocity of the ball could be calculated by s/t or also by:

Vx = (sx2g/2sy)

which resulted in the same answer, 564.4 cm/s.

The percent difference of the two values of v was calculated to be 13.0%. The difference is most likely due to the fact that the second part of the experiment involved shooting the ball away from the apparatus, which resulted in a certain amount of recoil. While attempts where made to stabilize the apparatus, it still moved backwards on each firing trial. This effect was less noticeable in the pendulum portion of the experiment, because the ball never left the system. This recoil would cause the calculations for the initial velocity of the projectile to be lower than expected, which turned out to be the case. Other possible sources of error could have been human error in the accuracy of the measurements, not returning the apparatus the same point before each trial, and the table and/or ground not being perfectly level. In addition, there was no way to determine if the firing apparatus struck the ball with the same force every time it was fired.