11 M36 M36.1 ANALYSIS OF A PERFECTLY INELASTIC COLLISION OBJECT The object of this experiment is to examine a perfectly inelastic collision between a

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1 11 M36 M36.1 ANALYSIS OF A PERFECTLY INELASTIC COLLISION OBJECT The object of this experiment is to examine a perfectly inelastic collision between a steel ball and a ballistic pendulum. THEORY Reference: Section 6.3, College Physics, Serway and Vuille The law of conservation of momentum is a universal law that applies to all interactions between two or more bodies (provided there are no external forces acting on the two bodies). It is experimentally verified by observations of astronomical bodies, everyday objects, atomic and subatomic particles. One type of interaction between two bodies that is governed by the law of conservation of momentum is a collision. In a perfectly inelastic collision the two bodies remain together after colliding and consequently have the same final velocity. Consider the following apparatus:

2 1 M36. Figure 1. Ballistic Pendulum Apparatus A spring-loaded gun is used to fire a steel ball. The steel ball can be fired across the room into a wooden box or caught in flight by a small cage which forms the bob of a pendulum (the ballistic pendulum). By firing the steel ball across the room into a wooden box, the speed of the ball, υ i, when it came off the gun, can be calculated from the measured value of the vertical distance, h, that the ball fell as it moved a measured horizontal distance, r. Application of the kinematic equations for vertical acceleration of constant magnitude g yields g υi = r. (1) h The collision to be studied is the perfectly inelastic collision resulting when the steel ball of mass m is caught in flight by the pendulum bob of mass M. After the collision the ball and pendulum bob swing together, reaching a maximum height, H, above the initial height of the bob. Immediately after impact, both the pendulum bob and the steel ball move off together with a velocity V f in the same direction as the initial velocity, υ i. Applying conservation of mechanical energy to the ball/pendulum system after the collision yields V f = gh () where H is the measured height to which the ball/pendulum rise after the collision. The collision between the steel ball and the ballistic pendulum bob is perfectly inelastic. Prior to the collision, only the steel ball is moving, so the magnitude of the initial momentum of the ball/pendulum system is

3 13 M36.3 p i = mυ i + M(0) = mυ i (3) where m is the mass of the ball and M is the mass of the pendulum bob. The kinetic energy prior to the collision is KE m (4) 1 i υ i The magnitude of the total momentum of the system immediately after the collision, p f, is: p f = (m + M)V f (5) and the total kinetic energy immediately after the collision is: KE f (6) 1 ( m M ) V f EXPERIMENT Equipment: electronic balance, tape measure, plumb bob, white paper, carbon paper, ballistic pendulum WARNINGS: The spring in the ballistic pendulum gun is very stiff use two hands and be careful when loading the ball on the gun (follow instructions). Before firing the gun, ensure that no students are in the path that the ball will travel. Procedure: The apparatus has already been clamped to the table and levelled to ensure that it is horizontal. 1. Fire the ball into the cage and check that the cage holds the ball and that the system (pendulum cage and ball) gets caught in the ratchet after the impact.. The vertical height H through which the pendulum rises after impact can be found by firing several shots into the cage. The pointer on the side of the pendulum cage is a point in the horizontal plane containing the centre of mass of the loaded pendulum. Measure the pointer height, H i, relative to the base of the apparatus before the collision and then measure the pointer height, H f, relative to the same reference, after the collision. For better results, fire the ball into the cage five times and measure H f each time. Checkpoint 1 ask the TA to review your work for step. 3. To determine the range, r, swing the pendulum cage up onto the ratchet. Fire the ball and note the point at which it strikes the floor. Place the catchbox so that the ball lands in the centre of the box. Tape a piece of white paper into the box and cover it with a piece of carbon paper. Now fire several shots onto the paper in the box. Carefully remove the carbon paper, but not the white paper. Noting that the range is the horizontal distance from the centre of the ball on the unloaded gun to the centre of each impact point, measure the average range and estimate a reasonable uncertainity in this value. Measure h, the vertical distance from the bottom of the ball to the surface of the box. Document your method of measuring h on the data sheet. 4. Measure the mass of the ball, m, on the electronic balance. Record the mass of the pendulum cage, M. (The value of M in grams is stamped into the side of the cage.) Checkpoint before moving the catchbox, ask the TA to review your work for steps 3 and 4.

4 14 M36.4 ANALYSIS 5. Calculate the average range, r : ( r max rmin ) r and the uncertainty in the average range: ( r max rmin ) r 6. Calculate the initial speed of the ball, υ i (equation (1)), and its percentage uncertainty. 7. Calculate the average of your H f values and determine H = H f - H i, the difference in height of the cage pin before and after the collision. A reasonable estimate of the uncertainty in the average H f value can be obtained from 1 (H fmax H fmin ) + the instrument uncertainty in the measurement of H f. 8. Calculate the speed of the ball/pendulum immediately after impact, V f (equation ()), and its percentage uncertainty. Checkpoint 3 ask the TA to review your work for steps 5 through Calculate the momentum and kinetic energy of the ball/pendulum cage system both before and after the collision (equations (3), (4), (5), and (6)). 10. Compare the momentum of the system immediately before the collision to the momentum immediately after the collision. Estimate the experimental uncertainty in the before and after values of momentum. Are you justified in stating that the before and after momentum values are experimentally equal? 11. Calculate the percentage of initial kinetic energy lost in the collision: KE KE 1 1 loss in kinetic energy before after mυi ( M m) V f % loss in KE = 100% 100% 1 initial kinetic energy KE mυ Assuming a perfectly inelastic collision and applying conservation of momentum yields: mυ i = (M + m)v f 1. As was done in the tutorial, solving the above equation for V f and substituting into the % loss in KE equation results in a theoretical expression for the percentage of kinetic energy lost in the collision in terms of m and M only. Use this theoretical expression to calculate the expected percentage of initial kinetic energy that should be lost in this collision and compare it to the corresponding actual experimental value of % kinetic energy loss that was observed. before Checkpoint 4 ask the TA to review your work for steps 9 through 1. i 100%

5 CONCLUSION 15 M Discuss whether or not your results indicate that momentum was conserved in the collision. 14. Discuss whether or not your results indicate that kinetic energy was conserved in the collision. Do you expect conservation of kinetic energy for this collision? Discuss why or why not. If kinetic energy is lost, where did it go? 15. Do your results verify the theoretical expression for the % loss of kinetic energy? SOURCES OF ERROR 16. Does the ball/pendulum cage system satisfy the necessary condition for the law of conservation of linear momentum to apply? Explain. 17. Discuss experimental factors that may affect the values of υ i and V f. Checkpoint 5 ask the TA to join your discussion of steps 13 through 17.

6 16 M36 ANALYSIS OF A PERFECTLY INELASTIC COLLISION DATA & RESULTS mass of ball, Mass of pendulum cage, m = ( ) kg M = ( ) kg Determination of initial velocity, v i : (speed of ball immediately before collision) Average range of ball, r = ( ) m Vertical displacement of ball, h = ( ) m (see box at bottom of page) Determination of final velocity, V f (speed of ball/cage immediately after collision) Initial height of cage, H i = ( ) m Final height of cage: Final Height, H f (m m) Trial 1 Trial Trial 3 Trial 4 Trial 5 Average Final Height H f (m) Change in height of cage, H = ( ) m (H= H f H i ) Before Collision After Collision Speed (m/s) Momentum (kg m/s) Kinetic Energy (J) In the space below, describe the method that you used to measure h: