TA name Lab section Date TA Initials (on completion) Name UW Student ID # Lab Partner(s) EXPERIMENT 6: COLLISIONS CONSERVATION OF ENERGY & MOMENTUM IN COLLISIONS 117 Textbook Reference: Walker, Chapter 9-1,3,4,5,6 SYNOPSIS Thiswek slabinvolvesenergyandmomentumconservationlawsandtheconceptofimpulse. The impulse J is the momentum change produced by the action of a force over a time period and is defined as J Fdt F t p mv final mv initial. F is the average force acting over a time interval t. The product F t can be obtained as the area under graphs of force vs. time for the interval t 2 - t 1 = t. The examples we will consider involve motion and forces in one dimension so a single graph will suffice. To investigate these ideas, we will be colliding carts in three distinct ways. One end of each cart contains two magnets. When the magnet ends of the carts are facing each other, the carts will repel each without actually touching. This collision is very nearly an elastic collision. (An elastic collision is one in which no kinetic energy is lost. Few collisions of macroscopic objects are perfectly elastic, but this one is close.) The other end of the carts contains a spring-loaded plunger and a set of Velcro patches, either of which can be chosen to provide the interaction in the collision. When the carts collide hitting the Velcro, they stick together and produce an inelastic collision. In the third kind of collision, the plunger on one of the carts is partially compressedbeforethecolisionsothatit explodes duringthecolision. This is sometimes called a super-elastic collision, in that the cocked spring adds energy to the collision. You will investigate all three types of collision in the course of this lab. When you are through with these activities, you should understand the conditions under which mechanical energy and momentum will be conserved in a collision and the conditions under which the mechanical energy and momentum will change during a collision. For elastic collisions, you will also investigate the relation between impulse and change of momentum. APPARATUS The apparatus utilizes two motion detectors and two carts. Cart #1 is the cart near the motion detector that is plugged into Channels 1 and 2. Motion Detector 1 Cart #1 projectile Cart #2 target Motion Detector 2 Fig. 6.1 Physics 117 Copyright 2007, Department of Physics, University of Washington Winter 2007
Fig. 6.2 PROCEDURE 1. Place a 500g bar in each cart, then determine the mass of each cart plus the added bar and record the results below. You will need to enter the masses into the computer. Use these masses for all the measurements to follow. Cart #1+500g bar = ± kg Cart #2+500g bar = ± kg Before observing collisions, level the track using the leveling screws. By making the track level, you minimize an external horizontal force, in this case the horizontal component of the gravitational force. Collision #1 Place the carts on the track with their magnetic ends facing each other so that the carts can colide withoutactualytouching. 2. Without taking data, observe what happens when Cart 1 collides with Cart 2 that is initially at rest. The initial velocity of Cart #1 should be sufficient so that Cart #2 easily reaches the end of the track after the collision, but not so great that the carts touch. Describe how the final velocities of the two carts compare with the initial velocities. 3. From your observations in 2, predict how the total initial momentum of the system of the two carts compares to the final total momentum. Explain your prediction. Physics 117 6-2 Autumn 2006
4. From your observations in 2, predict how the total kinetic energy of the system of the two carts compares to the final total kinetic energy. Explain your prediction. After answering these questions, use DataStudio with the file Lab 6 found in the Lab 6- Momentum and Collisions folder to take data. When you open the file, four screens will appear. One will be the Calculator screen. Enter the mass of cart 1 plus the 500g bar and click Accept. Select Mass 2 from a pull-down menu and enter the mass of cart plus bar. Click Accept. Close the Calculator window. Click Start and collide the carts as you did for question 2 with their magnet ends facing each other. Be sure the carts do not touch during this collision. 5. On the graph of momentum vs. time, zoom-select the momentum data from a time when the collision just begins, t begin, to a time when the collision just ends, t end. Maximize the graph and print a copy for both partners. Determine t begin and t end and enter the two times and their uncertainties below. The uncertainties should be your estimate of the uncertainty of determining the beginning and end of the collision rather than that of reading time from the graph. t begin = ± s t end = ± s 6. From the momentum vs. time graph, use Smart Tool to determine the momentum p 1 of cart 1 and the momentum p 2 of cart 2 at time t begin. Also determine the momentum p 1 ' of cart 1 and the momentum p 2 ' of cart 2 at time t end. Adopt a consistent sign convention for the momenta. A convenient one is to take momenta directed away from motion detector 1 to be positive. The uncertainties in the momenta should be based on your estimate of how closely you can determine the momenta given the fluctuations in the data. Momenta in the SI system have the units of kg m/s. There is no special name for the unit of momentum. p 1 = ± kg m/s p 2 = ± kg m/s p 1 + p 2 = ± kg m/s p 1 ' = ± kg m/s p 2 ' = ± kg m/s p 1 ' + p 2 ' = ± kg m/s 7. From the momenta in 6, determine the change in the total momentum, p, over the time of the collision. p = (p 1 ' + p 2 ') - (p 1 + p 2 ) = ± kg m/s Another technique can be applied to your data to obtain the change in momentum over the collision that is less subject to uncertainties arising from the fluctuations in the momentum data. Physics 117 6-3 Autumn 2006
8. With a ruler, draw a best-fit line through the total momentum curve on the graph. Determine the slope of that line and estimate its uncertainty. Show your work. From the slope, determine the change in momentum, p, over the time of the collision. slope = ± kg m/s 2 9. From the data in 8, determine the direction and magnitude of the external force acting on the cartsduringthecolision[hint:newton s2 nd law can be stated as F = p/ t]. F = ± N 10. Zoom-select the kinetic energy data for a time interval just before t begin to a time just after t end. Maximize the graph and print a copy for both partners. From the kinetic energy vs. time graph, determine the kinetic energy K 1 of cart 1 and the kinetic energy K 2 of cart 2 at t begin. Also determine the kinetic energies of carts 1 and 2, K 1 ' and K 2 ' at t end. K 1 = ± J K 2 = ± J K 1 + K 2 = ± J K 1 ' = ± J K 2 ' = ± J K 1 ' + K 2 ' = ± J 11. Compare the total kinetic energy of the system before and after the collision. Does this agree with your prediction in 4? If it does not agree, what do you think is responsible for the discrepancy? 12. From the force vs. time graph, determine the impulse on each cart during the collision. As described in the introduction, this requires finding the area under the graph of force vs. time for the duration of the collision. The simplest way to do this for your data is make use of the Trapezoidal Rule. To use this rule, the area under the force curve is divided into trapezoids of width t, where t is the time interval between data samples, and height F(t) and F(t+ t), the forces at either end of the interval. The area of such a trapezoid is F( t) F ( t t) A t 2. The sum of the areas included between N data points is Δt A (F 2 F 2 2 F3 F N F N ) 2 1, 1 Physics 117 6-4 Autumn 2006
which is the Trapezoidal Rule. F 1 is the first data point of the collision and F N is the last. N typically ranges from 8 to 10. Maximize and print the force vs. time graph for both partners. Use Smart Tool measure the forces F 1.F 2,,F N. The time interval t is 1/25 th second for a sample rate of 25/s. Use the value of t given below. Calculate the area under the force curve for both Cart 1 and Cart 2 (Be careful with the signs of the forces and remember that thearea under thecurveistheareabetwenthe curve and the time axis). cart 1 cart 2 F 1 = F 1 = F 2 = F 2 = F 3 = F 3 = F 4 = F 4 = F 5 = F 5 = F 6 = F 6 = F 7 = F 7 = F 8 = F 8 = F 9 = F 9 = F 10 = F 10 = t = 0.0400 ± 0.0001 s area = J 1 = ± kg m/sarea = J 2 = ± kg m/s The uncertainties in the impulse values should be based on your estimate of how well the data samples represent the force during the collision. Explain your estimate. 13. Compare the impulse on cart 1 to the impulse on cart 2 in magnitude and sign. 14. Compare the change in momentum of cart 1 obtained from 6 (p 1 ' - p 1 ) to the impulse from 12. The conservation of momentum is an absolute symmetry. That is, for any interaction between the members of an isolated system (no external forces) total momentum is conserved. Since we are observing interactions in one dimension, momentum in that dimension will be conserved provided there are no external forces with components in that dimension. Physics 117 6-5 Autumn 2006
15. You were asked to level the track at the beginning of the lab. Explain why this is important. 16. If you made your measurements carefully, you may have found in 7 and 8 that momentum in the horizontal direction was not conserved. (If your measurement showed momentum to be conserved, answer the question as though it were not). Did total momentum increase or decrease? Explain the origin and sign of the change. Collision #2 Turn the carts around so that the Velcro patches face each other. 17. Without taking data, observe what happens when Cart 1 collides with Cart 2 that is initially at rest. Describe how the final velocities of the two carts compare with the initial velocities. 18. From your observations in 17, predict how the total initial momentum of the system of the two carts compares to the final total momentum. Explain your prediction. 19. From your observations in 17, predict how the total kinetic energy of the system of the two carts compares to the final total kinetic energy. Explain your prediction. After answering these questions, use DataStudio with the file Lab 6 to take data. Collide the carts as you did for question 2 with the Velcro patches facing each other. Physics 117 6-6 Autumn 2006
20. On the graph of momentum vs. time, zoom-select the momentum data from a time just as the collision begins, t begin to a time just as it ends, t end. Maximize the graph and print a copy for both partners. Determine a time for which the collision is just beginning, t begin, and a time for which it is just ending, t end. Enter the two times and their uncertainties below. The uncertainties should be those of determining the start and end of the collision rather than that of reading the time from the graph. t begin = ± s t end = ± s 21. From the momentum vs. time graph, determine the momentum p 1 of cart 1 and the momentum p 2 of cart 2 at time t begin. Also determine the momenta of carts 1 and 2, p 1 ' and p 2 ', at time t end. Adopt a consistent sign convention for the momenta. p 1 = ± kg m/sp 2 = ± kg m/s p 1 + p 2 = ± kg m/s p 1 ' = ± kg m/sp 2 ' = ± kg m/s p 1 ' + p 2 ' = ± kg m/s 22. Determine the change in the total momentum, p, over the time of the collision. Show your work. p = ± kg m/s 23. Zoom-select the kinetic energy data from a time just as the collision begins, t begin to a time just as it ends, t end.. Maximize the graph and print a copy for both partners. From the kinetic energy vs. time graph, determine the kinetic energy K 1 of cart 1 and the kinetic energy K 2 of cart 2 at t begin. Also determine the kinetic energies of carts 1 and 2, K 1 ' and K 2 ', at t end. K 1 = ± J K 2 = ± J K 1 + K 2 = ± J K 1 = ± J K 2 = ± J K 1 ' + K 2 ' = ± J 24. Compare the total kinetic energy of the system before and after the collision. Does this agree with your prediction in 19? If it does not agree, what do you think is responsible for the discrepancy? Physics 117 6-7 Autumn 2006
Collision #3 Fig. 6.3 The spring-loadedplungeronthecartscanbe cocked halfwaysothatwhenthecartscolide the plunger will be released. Position the plunger so a notch in the plunger shaft is under the Plunger Bar Release and pull it upward slightly until the release engages. Position the carts so that one of them has the partially compressed plunger pointing toward the flat face of other cart. Theplungerdoesn talwaysreleasesucesfuly,soyoumaynedtorepeathecolisionseveral times until it does. 25. Without taking data, observe what happens when Cart 1 collides with Cart 2 that is initially at rest. Describe how the final velocities of the two carts compare with the initial velocities. 26. From your observations in 25, predict how the total initial momentum of the system of the two carts compares to the final total momentum. Explain your prediction. 27. From your observations in 17, predict how the total kinetic energy of the system of the two carts compares to the final total kinetic energy. Explain your prediction. After answering these questions, use DataStudio with the file Lab 6 to take data. Collide the carts you did for 25. Physics 117 6-8 Autumn 2006
28. On the graph of momentum vs. time, zoom-select the momentum data from a time just as the collision begins, t begin to a time just as it ends, t end. Maximize the graph and print a copy for both partners. Determine a time for which the collision is just beginning, t begin, and a time for which it is just ending, t end. Enter the two times and their uncertainties below. The uncertainties should be those of determining the start and end of the collision rather than that of reading the time from the graph. t begin = ± s t end = ± s 29. From the momentum vs. time graph, determine the momentum p 1 of cart 1 and the momentum p 2 of cart 2 at time t before. Also determine the momenta of carts 1 and 2, p 1 ' and p 2 ', at time t after. Adopt a consistent sign convention for the momenta. p 1 = ± kg m/sp 2 = ± kg m/s p 1 + p 2 = ± kg m/s p 1 ' = ± kg m/sp 2 ' = ± kg m/s p 1 ' + p 2 ' = ± kg m/s 30. From the momenta in 29, determine the change in the total momentum, p, over the time of the collision. p = (p 1 ' + p 2 ') - (p 1 + p 2 ) = ± kg m/s 31. Zoom-select the kinetic energy data for a time interval just before t before to a time just after t after. Maximize the graph and print a copy for both partners. From the kinetic energy vs. time graph, determine the kinetic energy K 1 of cart 1 and the kinetic energy K 2 of cart 2 at t before. Also determine the kinetic energies of carts 1 and 2, K 1 ' and K 2 ', immediately after the collision at t after. K 1 = ± J K 2 = ± J K 1 + K 2 = ± J K 1 ' = ± J K 2 ' = ± J K 1 ' + K 2 ' = ± J 32. Compare the total kinetic energy of the system before and after the collision. Does this agree with your prediction in 27? If it does not agree, what do you think is responsible for the discrepancy? Physics 117 6-9 Autumn 2006
Collision Summary 33. In which of the three types of collisions, #1, #2, and #3 that you observed is momentum conserved during the collision? 34. If the effect of the external force were removed, in which of the three types of collisions, #1, #2, and #3 would momentum be conserved. Explain, based on your data. (You see the effect of friction as a steady decrease in momentum. Look for deviation from this to indicate possible non-conservation of momentum during the collisions.) 35. Use your data from Collision #1 to arrive at the uncertainty in the conservation of momentum for a collision of the carts with no external forces acting. Explain your reasoning. (Again, look for the deviation from a smoothly sloping line.) p = ± kg m/s 36. In the absence of external forces, in which of the three types of collisions, #1, #2, and #3 would you expect that kinetic energy would be conserved. Explain. 37. The terms elastic, super-elastic and inelastic are frequently used to describe collisions. Fill in the table below that describes the characteristics of the three categories of collisions. K total = K final - K initial is the change in the total kinetic energy and p total = p final - p initial is the change in the total momentum. In the appropriate cell, enter > 0, = 0 or < 0. Your answers should assume no external forces are operating during the collision. collision type K total p total elastic inelastic super-elastic Physics 117 6-10 Autumn 2006