9. Momentum and Collisions in One Dimension* The motion of objects in collision is difficult to analyze with force concepts or conservation of energy alone. When two objects collide, Newton s third law says that the force exerted by one on the other is equal and opposite to the force the other exerts on the one. That may sound strange. Imagine a head-on collision between a small car and a truck. It may be hard to believe that the force of the car acting on the truck has the same magnitude as the force of the truck acting on the car. But according to the Newton s third law this must be the case. Why is it then the small car may get damaged seriously but perhaps not the truck? What is going on? In this lab, you will use the concept of linear momentum to analyze one dimensional collisions. You are going to analyze two types of head-on collisions in one dimension using video capture for data taking. Learning Objectives: 1. Know what momentum is, be able to give a formula for it, and know why it has to be a vector quantity. 2. Know what it means for the total momentum of an isolated system to be conserved. 3. Be able to say under what conditions momentum is conserved and to explain why collisions always meet this criterion. 4. Know what in means for a collision to be elastic, inelastic or totally inelastic, including what is or isn t conserved in each case. 5. Be able to use conservation ideas to solve collision problems in a realistic context. Reading Assignment: Before you start the lab you should be familiar with the concepts of linear momentum, conservation of momentum, elastic collisions and inelastic collisions. Read these sections in your text. Knight, Jones & Field (161): Chapter 9 sections 9.1 through 9.6. Serway and Vullie (211): 6.1 Momentum and Impulse, 6.2 Conservation of Momentum and 6.3 Collisions. Serway and Jewett (251): 9.1 Linear Momentum, 9.2 Isolated System, 9.3 Non Isolated System, 9.4 Collision in One Dimension Before you come to the lab answer Prediction Questions below and all Prediction questions in Problems 1 through 3. You must have the solutions with you when you enter lab. 1. Here are some review questions. Answer based on your reading. (a) What is the momentum of a small object of mass m? Keep in mind that it s a vector, so you need to indicate that it is in your definition. Give a formula. (b) Consider a system of five small objects with masses labeled from m 1 to m 5. What do I mean by the total momentum of the system? Give a formula. * William A Schwalm 2012 9-1
(c) A process, such as a collision takes place during some time interval. For a collision that interval is short. Consider a time t i before the process and a time t f after the process. Explain precisely what it means to say, The total momentum of the system is conserved during the process. Assuming this is true, explain exactly what it means, including an equation plus discussion. (d) Now here comes the main thing: From your reading, under what conditions is the total momentum of a system of objects conserved? (e) Thus, why is the total momentum conserved during collisions? (f) What is an elastic collision? What other thing is conserved, in addition to momentum, during an elastic collision? (g) An inelastic collision is one that is not elastic. However, what is a totally inelastic collision? 2. A ball is dropped to the ground. On the diagram to the right, draw all the force(s) acting on the ball as it is falling. Is there a net force acting on the ball as it falls? Is the momentum of the ball conserved (constant) as It falls? If you think momentum is conserved, state why. If not, see if you can describe a larger system that includes the ball plus something else, in which the total momentum is conserved. 3. A ball is dropped to the ground. The ball hits the ground and bounces up. It has the same speed just before it hits the ground and just after it bounces. Is the momentum of the ball conserved (constant) between just before it hits the ground and just after it bounces up from the ground? On the diagram to the right, draw arrows representing all the force(s) acting on the ball while it is in contact with the ground. 9-2
The following four questions refer to an elastic collision involving two low-friction carts, A and B. Before the collision B is at rest and A comes from left to right with speed v 0. Afterward, A exits with velocity v 1 and B, leaves with velocity v 2 (points toward the right). 4. What does it mean when we say that a collision is elastic? 5. Use the two conservation equations that operate during an elastic collision to solve for the final velocity of cart B when the masses of carts are the same. Your equation that determine the final velocity of cart B should only depend on the initial velocity of cart A and the masses of the two cart s assuming there is no energy dissipation. Hint: You may find, looking at your energy equation, that there is a difference of squares. Notice that 2 2 x y x y x y. This may be of help. 6. Repeat problem 5 for the case when the masses of the carts are different. 9-3
7. Now let s see what would happen in an elastic collision between two objects of very different mass. As a sort of step 1 you should develop an estimate of what you think should happen, without calculating. Let s suppose as before that B is at rest before the collision and A comes in from the left and hits B. Ignoring the calculation you did, what do you reckon would happen if the moving object A is much lighter than the heavy object B? How about if A is much heavier than B? Write your predictions in two sentences. 8. So much for predicting. Now from your actual calculation in the problem 5 above determine the direction of motion of cart A and B after the collision in each of three cases: When cart A has a larger mass than cart B (ma > mb), when the carts have equal masses (ma = mb) and when cart A has a smaller mass than cart B (ma < mb). Is your calculation consistent with the prediction you made in the step 1 prediction in 7? Problem 1: Totally inelastic collisions Equipment: Dynamic cart (2) and track, weights, pc with logger pro, USB webcam. You will use the track and carts with which you are already familiar. For this problem, cart A is given an initial velocity toward a stationary cart B. Velcro pads at the end of each cart are used to get the carts to stick together after the collision. You will need video camera and a meter stick, a stopwatch, two end stops and extra masses to load on the carts. In the figure below the cart A is moving toward the cart B that is stationary. moving A stationary B Using symbols, calculate the final velocities of the carts as a function of the initial velocity of cart A and the masses of the two carts. 9-4
The question is: What is the final, mutual velocity of the two carts stuck together after the collision as a function of the initial velocity of A and of the two masses? More than just answering the equation algebraically by finding the final velocity we want to look into the situation in more detail. We want to know what is really happening during the collision process in a light of conservation of momentum or energy. Before starting the measurements answer the Prediction questions below. Prediction questions: In order to answer this question let s first work through the following prediction steps, based on your intuition, your reading and on prior knowledge. 1. Make a sketch that predicts how the carts move after the collision. Indicate the direction motion of the carts using the velocity vectors on your sketch. Explain your reasoning. 2. Write down the momentum conservation equation and identify all of the terms in the equation. There should be vector quantities involved, so be sure to indicate this. 3. Write down expressions for the total mechanical energy before and after the collision. Identify all the terms in the equation. Does the potential energy change during the collision? How about the kinetic energy? Is mechanical energy conserved here? 4. What conservation principle should you use to predict the final velocity of the stuck-together carts, or do you need two different equations combined? Why? (In other words, each conservation law comes with a set of conditions when it holds. Your choice of conservation laws should be based on conditions.) 5. You can now complete your prediction question for the problem. What are the direction and magnitude of the final velocity of the carts? 9-5
6. During an inelastic collision mechanical energy is lost. Why cannot all of the kinetic energy be lost in a totally inelastic collision? Exploration: It s important to get the track as level as possible. A carpenter s level is provided, which is somewhat useful, but whether a cart rolls is a more sensitive indication. Practice setting the A cart into motion in such a way that the two carts stick together after the collision. Also, after the collision carefully observe the carts to determine whether or not either cart leaves the grooves in the track. Adjust your procedure to minimize this effect so that your results are reliable. Try giving the cart A various amounts of initial velocity over the range that will give reliable results. Note the outcomes qualitatively. Keep in mind also that you want to choose an initial velocity that gives you a good video. Try varying the masses of the carts by transferring block of masses from one car to the other while keeping the total mass of the carts the same. Be sure the carts still move freely over the track. Decide what masses you will use in your final measurement. Measurement Plan: You should devise a plan for making the measurements you need in order to answer the question. This plan should say what data you need to collect, why these data are relevant, referring to the theory, and how you will make the measurements. You should design and draw a data table. You will use the video capture ideas you have learned. Record your plan here. Include details. For instance, it is best to use only several points just before the collision and several points just after the collision, rather than lots of points. Why is this? Measurement: Start the Logger Pro program and take a video of the collision. Examine your video and decide if you have enough frames to determine the velocities you need. Collect enough data to convince yourself (and others) of your conclusion about how the final velocity of both carts in this type of collision depends on the velocity of the initially moving cart, and on the masses of the carts. Analysis: Determine when the collision occurred by liking at the data plotted on the position versus time graph. Find the velocity and momentum of the carts just before and of the combined system just after the collision. Unfortunately there are several sources of error that enter in this experiment so the agreement may be only quantitative. However, you know how to get an estimate of the size of the error. 9-6
Figure out the initial and the final kinetic energy for each case. Calculate the mechanical energy dissipation during the collision. In other words, how much mechanical energy is lost? What fraction of the initial energy does this represent? Where do you figure the energy goes? Compare your findings with your predictions and find % difference for each case. This would be the percentage of what you find minus what you expect, with respect to what you expect. Problem 2: Almost-elastic collision In an elastic collision, both momentum and kinetic energy are conserved. Question: Magnets are used as magnetic bumpers to get the carts bounce apart after the collision, hopefully without loosing much energy. To what extent can you create an elastic collision using carts? moving A stationary B Before starting the measurements answer the Prediction questions and go through the planning activities. Describe the final velocity of the cart B as a function of the initial velocity of cart A and the masses of the two carts for an elastic collision in each of the following 3 cases: (a) the moving cart m A has a larger mass than the stationary cart (ma > mb), (b) the masses of the cart s are equal (ma = mb), and (c) the moving cart has a smaller mass than the stationary cart (ma < mb). Discuss both magnitude and direction. 9-7
Prediction Questions: In order to answer this question do the following steps. 1. Make a sketch that shows the situation before the collision and another one for the situation after the collision for each of the three cases. Draw and label velocity vectors on your sketch. 2. Write down the momentum conservation equation for this situation. 3. Write down the energy conservation equation for this situation. Identify all the terms in the equation. Is any energy transferred into or out of the system? If so, how? Are you making terms in the equation. Is any energy transferred into or out of the system? If so, how? Are you making any approximations? 9-8
Measurement Plan: In group discussion, come up with a measurement plan. You know that you have to compare such-and-such before and after the collision. To do that, what are all the data you need? For instance, do you need velocities? How do you plan to get them and so on? Design a data table to contain all the data you will need to solve this problem. Measurement: Collect enough data to convince yourself and others of your conclusion about how the final velocities of both carts in this type of collision depends on the velocity of the initially moving cart, the masses of the carts, and the energy efficiency of the collision. Also, you want to know about three cases, one with m A > m B, one with m A = m B, and one with m A < m b. Record the masses of the two carts. Make a video of their collision. Examine your video and decide if you have enough frames to determine the velocities you need. Are there any peculiarities in the data that might suggest that the data are unreliable? Analysis: Carry out the analysis of the graphical data as described in your measurement plan. You need to show how close (as a percent difference) the initial kinetic energy is to the final. You need to do this for each case, m A > m B, m A = m B, and m A < m b. 9-9
Discussion: If you find that energy is lost, where do you think it went? Also, from your analysis, to what extent can you tell, if the energies don t seem to be conserved, if they are really not conserved or if there is just a lot of uncertainty in the measurement? How can you tell? You need to respond to this. Conclusion: At the end of the lab period there will be a brief class discussion of how the lab activities relate to the learning objectives. Record here the most relevant points of that discussion. In addition, add your comments and your group s comments on each of the learning objectives. What are the objectives, in your words, and how do the activities get at each of them? 9-10