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

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1 11 M36 M36.1 OBJECT THEORY ANALYSIS OF A PERFECTLY INELASTIC COLLISION The object of this experiment is to examine a perfectly inelastic collision between a steel ball and a ballistic pendulum. NOTE: Before coming to the lab, derive the equations for the absolute errors in υi, Vf, pi, and pf in the theory part of your pre-lab. 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 12 M36.2 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, υi, 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 (derived at the tutorial): g υi = r. (1) 2h 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 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 Vf in the same direction as the initial velocity, υi. Applying conservation of mechanical energy to the ball/pendulum system after the collision yields (derived at tutorial): V f = 2 gh (2) where H is the gain in height of the ball/pendulum after the collision. The collision between the steel ball and the ballistic pendulum bob is perfectly inelastic. (The ball and cage move together as a single object.) 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 pi = mυi + M(0) = mυi (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 2 i 2 υ i The magnitude of the total momentum of the system immediately after the collision, pf, is: and the total kinetic energy immediately after the collision is: EXPERIMENT Equipment: WARNINGS: Procedure: pf = (m + M)Vf (5) f ( m M ) V f KE = + (6) electronic balance, tape measure, plumb bob, white paper, carbon paper, ballistic pendulum The spring in the ballistic pendulum gun is very stiff Load the gun from behind, NOT from the front or side, and use both hands. Be careful! (follow instructions given in lab). Before firing the gun, ensure that no students are in the path that the ball will travel. EVERYONE must wear safety glasses (provided in lab) as long as ANYONE is still collecting data. The apparatus has already been clamped to the table and levelled to ensure that it is horizontal. DO NOT UNCLAMP OR MOVE! 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. 2. The vertical height H through which the pendulum rises after impact can be found by firing several shots into the cage. The orange-red 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 Hf, from the to the horizontal base of the apparatus (NOT the table) to the pointer height after the collision. Fire the ball into the cage five times and measure Hf each time. 3. To determine the range, r, swing the pendulum cage up onto the ratchet. Place a piece of plain paper (the target sheet ) into the catchbox, so that the back of the paper is at the back of the box. Tape in place. Put the catchbox on the floor, and, by trial and error, position the box so that the ball lands roughly in the centre of the paper. Place the carbon paper, BLACK SIDE DOWN, on top of the target sheet. Do NOT tape the carbon paper. Place a sheet of Problem Paper on top of the carbon paper. Do NOT tape. Have someone secure the box in position. (eg, stand behind it with your foot against it). Fire 10 (or more) shots into the box. Remove the cover page and the carbon paper. There will be a cluster pattern of black dots on the target sheet. Measure the length of this pattern along the line of the ball s travel. Find the midpoint of this distance and mark on the target sheet. This is the average range of the ball, r. Using a plumb bob and a three metere stick, measure this average range, horizontally, from the centre of the ball on the end of the UNloaded gun to the string of the pumb bob suspended above the centre of the pattern. The error in the average range is the measured distance from the centre of the pattern to either end of the pattern. (Be sure the box has NOT been moved!)

4 14 Checkpoint 1 Before moving the catchbox, ask the TA to review your work for step1. M Measure and record 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.) 4. Take a series of measurements (documented in the box at the bottom of the Data Table) that will be combined to determine the vertical drop of the ball, h, from the bottom of the ball on the end of the Unloaded gun to the surface of the box. (It will take between 2 and 4 separate measurements.) 5. Measure and record the initial height of the cage, Hi, from the horizontal base of the gun to the pointer when the cage is at its lowest point. 6. Calculate the average of your Hf values. The uncertainty in the average Hf value can be obtained from [ 21 (Hfmax Hfmin) + the instrument uncertainty in the measurement of Hf.] NOTE: this is a calculated error, which will be rounded to the usual 2 significant figures, and the average final height will need to be rounded accordingly. Similarly for the calculation of H = H - H, the difference in height of the cage pin before and after the collision. ANALYSIS Checkpoint 2 - Ask the TA to review your work for steps 3 to Calculate the initial speed of the ball, υi (equation (1)), and its absolute uncertainty. 8. Calculate the speed of the ball/pendulum immediately after impact, Vf (equation (2)), and its absolute uncertainty. 9. Calculate the kinetic energy of the ball/pendulum cage system both before and after the collision (equations (4) and (6)). (No error calculations needed. Round to 3 decimal step 9.) Checkpoint 3 ask the TA to review your work for steps 7 through Calculate the momentum of the ball/pendulum cage system (with absolute errors) both before and after the collision (equations (3) and(5)). 11. Compare the momentum of the system immediately before the collision to the momentum immediately after the collision. Use the Algebraic method of comparison. Are you justified in stating that the before and after momentum values are equal within experimental error? 12. Calculate the percentage of initial kinetic energy lost in the collision: loss in kinetic energy KEbefore KEafter % loss in KE = 100% = 100% initial kinetic energy KE Assuming a perfectly inelastic collision and applying conservation of momentum yields: mυi = (M + m)vf 13. As was done in the tutorial, solving the above equation for Vf 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 the theoretical expression that was derived at the tutorial to calculate the expected percentage of initial kinetic energy that should be lost in this collision. before f i

5 15 M Compare the theoretically predicted % loss in kinetic energy to the actually observed experimental % loss in kinetic energy. (Do another algebraic comparison.) Checkpoint 4 ask the TA to review your work for steps 9 through 14. CONCLUSION 15. Discuss whether or not your results indicate that momentum was conserved in the collision. 16. 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? 17. Do your results verify the theoretical expression for the % loss of kinetic energy? KE SOURCES OF ERROR Checkpoint 5 ask the TA to join your discussion of steps 15 through Discuss experimental factors that may affect the values of υi and Vf, (and therefore pi, pf, KEi, and KEf.) Consider all the measured values involved, and whether there are any unaccounted for factors that could change those values (eg outside forces), and therefore change the calculated results of υi, Vf, pi, pf, KEi, and KEf. 19. Does the ball/pendulum cage system satisfy the necessary condition for the law of conservation of linear momentum to apply? Explain. Checkpoint 6 ask the TA to join your discussion of steps 18 and 19..

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 2 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: