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

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1 11 M36 M36.1 ANALYSIS OF A COMPLETELY INELASTIC COLLISION OBJECT The object of this experiment is to examine a completely inelastic collision between a steel ball and a ballistic pendulum. NOTE: Before coming to the lab, do the derivation discussed at the end of the Analysis section. THEORY Reference: Sections 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 completely 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). The collision between the steel ball and the ballistic pendulum bob is completely inelastic. To determine the momenta of the steel ball and the pendulum the values of their masses and velocities must be known. Prior to the collision, only the steel ball is moving so the magnitude of the initial momentum of the ball/pendulum system is p i = mυ i + M(0) = mυ i (1) 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 () 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 (3) and the total kinetic energy immediately after the collision is: KE f (4) 1 ( m M ) V f The velocity of the ball before impact, υ i, is obtained by measuring the horizontal range, r, of the ball when it is fired horizontally from a height h above the catchbox. Application of the kinematic equations for vertical acceleration of constant magnitude g yields g υi = r. (5) h

3 13 M36.3 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 (6) where H is the height to which the ball/pendulum rise after the collision. EXPERIMENT Equipment: WARNINGS: electronic balance, tape measure, plumb bob, white paper, carbon paper, ballistic pendulum 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: Before performing the experiment: a) Clamp the base of the apparatus to the table and ensure that it is horizontal. b) 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. Measure the mass of the ball, m, on the electronic balance. Record the mass of the pendulum cage, M. 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 (6 or more) 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 maximum and minimum ranges obtained for your trials. Record the necessary vertical measurements to determine h, the vertical distance from the bottom of the ball to the surface of the box. 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.

4 ANALYSIS Calculate h from the vertical measurements that were recorded. Calculate the average range, r : ( r max rmin ) r and the error in the average range: ( r max rmin ) r Calculate the initial speed of the ball, υ i. {Calculate the absolute error, υ i.} 14 M36.4 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 error in the average H f value can be obtained from 1 (H fmax H fmin ) + the instrument error in the measurement of H f. Calculate H} Calculate the speed of the ball/pendulum immediately after impact, V f. {Calculate V f.} Calculate the momentum and kinetic energy of the ball/pendulum cage system both before and after the collision. {Calculate the absolute errors in the before and after values of the momentum and kinetic energy of the ball/pendulum cage system.} Compare the momentum of the system immediately before the collision to the momentum immediately after the collision. 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 before i 100% Assuming a completely inelastic collision and applying conservation of momentum yields: mυ i = (M + m)v f By solving this equation for V f and substituting into the % loss in KE equation, derive a theoretical expression for the percentage of kinetic energy lost in the collision in terms of m and M only. The derivation is to be done before coming to the lab and is to be written on a separate sheet of paper and submitted at the beginning of the lab period. 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.

5 CONCLUSION 15 M36.5 Discuss whether or not your results indicate that momentum was conserved in the collision. 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? Do your results verify the theoretical expression for the % loss of kinetic energy? SOURCES OF ERROR Does the ball/pendulum cage system satisfy the necessary condition for the law of conservation of linear momentum to apply? Explain. Discuss experimental factors that may affect the values of υ i and V f.

6 16 M36 ANALYSIS OF A COMPLETELY 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) Maximum range (r max ) Minimum range (r min ) = ( ) m = ( ) m Average range of ball, r = ( ) m = ( ) m = ( ) m = ( ) m = ( ) m Change in vertical displacement of ball, h = ( ) m 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)