PRELAB: COLLISIONS Collisions, p. 1/15 1. What is your Prediction 1-7 when the truck is accelerating? Explain the reasoning behind your prediction. 2. If you set up the force probes according to the instructions in Activity 1-1, will force probe 1 give positive or negative readings during the collision? What about force probe 2? Explain. Why did the developers do it this way? 3. How will you calculate the impulse from the force-time graph? 4. How do you determine the direction of the impulse vector? 5. Sketch Prediction 3-1 on the axes below. Explain the reasoning behind your prediction. force (N) velocity (m/s) time (s)
PRELAB: COLLISIONS Collisions, p. 2/15
Collisions, p. 3/15 COLLISIONS Topic: Forces, impulse, momentum in 1D, and Newton s 3 rd law Objectives: To study the forces between objects that interact with each other, especially in collisions To examine the consequences of Newton s third law as applied to interaction forces between objects. To formulate the law of conservation of momentum as a theoretical consequence of Newton s third law and the impulse-momentum law. To explore conservation of momentum in one-dimensional collisions Overview: In this lab, you will explore different aspects of collisions using carts, motion detectors and force probes. The collisions you will observe will all be in one dimension, but the results generalize to two dimensions. In the first investigation you will investigate the forces acting on two colliding objects compare during various collisions. This will lead you to a very general law known as Newton s third law, which relates the forces of interaction exerted by two objects on each other. Next you will examine how the area under a force-time graph is related to changes in the momentum. Then you will examine the consequences of this law and the impulse-momentum theorem when they are applied to collisions between objects. In doing so, you will arrive at one of the most important laws of interaction between objects, the conservation of momentum law. As usual you will be asked to make some predictions and then be given the opportunity to test these predictions. Writing it up: In this handout, you will be asked to perform calculations, analyze graphs and answer questions. It is strongly recommended that you do all the calculations and answer all the questions as you go through the experiment. This record is for your purposes only. You will not be graded on it. You will be graded on how well you understand the material covered in this lab, so you should do the entire activity, discuss the questions with your partners and take careful notes. Note for the TA: This lab requires a lot of small parts and accessories (hooks, rubber bumpers, etc.). Before and after each lab, check each lab station to make sure the required hardware is there. Notes about the force probes: Don t overload the force probes! The maximum push (or pull) the force probe can withstand is about 5 N. Zero the force probes before each data run. The force probe output drifts somewhat over time, so that the probe may not read zero when no force is applied.
Collisions, p. 4/15 Investigation 1: Forces between Interacting Objects There are many situations where objects interact with each other, for example, during collisions. In this investigation we want to compare the forces exerted by the objects on each other. In a collision, both objects might have the same mass and be moving at the same speed, or one object might be much more massive, or they might be moving at very different speeds. What factors might determine the forces the objects exert on each other? Is there some general law that relates the forces the objects exert on each other? Is there some general law that relates the forces? Activity 1-1: Collision Interaction Forces What can we say about the forces two objects exert on each other during a collision? Prediction 1-1: Suppose two objects have the same mass ( m 1 = m2 ) are moving toward each other with the same speed ( v 1 = v2 ). Predict the relative magnitudes of the forces between object 1 and object 2 during the collision. Place a check next to your prediction: Object 1 exerts a larger force on object 2. The objects exert the same size force on each other. Object 2 exerts a larger force on object 1. Prediction 1-2: Suppose the two objects have the same mass ( m 1 = m2 ) and that object 1 is moving toward object 2, but object 2 is at rest (i.e. vr 1, vr = 2 ). Prediction 1-3: Suppose the mass of object 1 is greater than that of object 2 ( m 1 > m2 ), and that it is moving toward object 2, which is at rest ( vr 1, vr = 2 ). Object 1 exerts a larger force on object 2. The objects exert the same size force on each other. Object 2 exerts a larger force on object 1. Object 1 exerts a larger force on object 2. The objects exert the same size force on each other. Object 2 exerts a larger force on object 1. Provide a summary of your predictions. What are the circumstances under which you predict that one object will exert a greater force on the other object?
Collisions, p. 5/15 To test these predictions, you can study collisions between two force probes attached to carts. You can add masses to one of the carts so that it has significantly more mass than the other. 1. Set up the apparatus shown at the right. The force probes should be securely fastened to the carts. The hooks should be replaced by rubber stoppers, which should be carefully aligned so that they will collide head-on with each other. If the carts have friction pads, these should be raised so that they don t rub on the ramp. 2. Set up the software to collect data from the two force probes at the fastest possible sampling rate (a few hundred samples per second). Set pull positive for probe 1, push positive for probe 2. (This choice makes all forces to right positive). Set up axes like the ones below for force-time graphs for the two probes. The time axes of the graphs should be aligned. +1 force probe 2 (N) -1 +1 force probe 1 (N) -1 time (s) 3. Use the probes to explore the various situations that correspond to the predictions you made about interaction forces. Your goal is to find out under what circumstances one cart exerts more force on the other. Test Prediction 1-1 by trying the appropriate collision (two carts of the same mass moving toward each other at about the same speed.) Q1-1: Did your observation agree with your prediction? How do the forces compare on a momentby-moment basis during the collision? How do the areas under the two force-time graphs compare?
Collisions, p. 6/15 Test Predictions 1-2 and 1-3 by trying the appropriate collisions. Zero both force probes before each collision. Q1-2: Did your observations agree with your predictions? What can you conclude about the forces of interaction during collisions? Under what circumstances does one object experience a different force than the other in a collision? How do the forces compare on a moment-bymoment basis during each collision? Q1-3: You have probably studied Newton s third law in lecture or in your text. Do your observations have anything to do with Newton s third law? Explain. Activity 1-2: More Collision Interaction Forces What can we say about the forces two objects exert on each other during a collision? Prediction 1-4: Suppose the mass of object 1 is greater than the Predict the relative magnitudes of the forces mass of object 2 ( m 1 > m2 ) and the objects are moving between object 1 and object 2 during the collision. toward each other at the same speed ( v 1 = v2 ). Prediction 1-5: Suppose the mass of object 1 is much greater than the mass of object 2 and that object 2 is moving in the same direction as object 1, but not as fast ( m 1 >> m2 and v 1 > v 2 ). Predict the relative magnitudes of the forces of interaction. Prediction 1-6: Suppose the mass of object 1 is much greater than the mass of object 2 and that both objects are at rest until an explosion occurs ( m 1 >> m2 and v1 = v2 = ). Predict the relative magnitudes of the forces of interaction.
Collisions, p. 7/15 4. Test Prediction 1-4 through 1-6 by trying the appropriate collisions. Record your observations here: Activity 1-3: Other Interaction Forces Interaction forces between two objects occur in many other situations besides collisions. For example, suppose a small car pushes a truck with a stalled engine, as shown in Prediction 1-7. The mass of object 1 (the car) is much smaller than object 2 (the truck). Prediction 1-7: A small car pushes a truck with a stalled engine. The mass of object 1 (the car) is much smaller than object 2 (the truck). Predict the relative magnitudes of the forces of interaction at the various times listed below: a) before the truck starts moving At first, the car doesn t push hard enough to make the truck move. Then as the driver pushes harder on the gas pedal, the truck begins to accelerate. Finally the car and truck are moving along at the same constant speed. b) while the truck is accelerating c) after the truck and car are moving at constant speed 1. Using the set up from the previous activities, test your predictions. 2. Add masses to cart 2 to make it more massive than cart 1. Zero both force probes. 3. Your hand will be the engine for cart 1. Move the carts so that the stoppers are touching and begin graphing. When graphing begins, push cart 1 toward the right. At first, hold cart 2 so that it cannot move, but then allow the push of cart 1 to accelerate cart 2, so that both cars move to the right, finally at constant velocity. Q1-4: How do youesults compare to your predictions? Is the force exerted by cart 1 on cart 2 (reading of force probe 2) significantly different from the force exerted by cart 2 on cart 1 (reading of force probe 1)? Explain any differences between your predictions and your observations.
Collisions, p. 8/15 Q1-5: Many students find the outcome of this experiment troubling. One of the questions that often arise is, How is it possible for the truck to speed up if the force the car exerts does not exceed the force exerted on the truck? Develop an explanation for this apparent contradiction. (You may find it helpful to draw two well labeled force diagrams, one for each vehicle). Investigation 2: Momentum and Momentum Change In this investigation, you are going to develop the idea of momentum to predict the outcome of collisions. Even the lab has not yet officially defined momentum yet, you probably already have a good intuitive sense of what it is. Consider the following example: A light cart (mass = m) travels to the left at some initial speed (call it v ). A heavier cart, three times as massive as the light cart, travels to the right. The two carts collide and stick together. Which way will the two carts (now stuck together) go? Will they stop? Clearly, the final outcome depends on the initial speed of the heavy cart. Activity 2-1: Can you stop the car? In this activity, you will investigate the example described above. You will need two carts and some masses, so you can make the second cart three times as massive as the other. Prediction 2-1: How fast will the heavy cart have to be going to completely stop the light cart? Explain the reasoning behind your prediction. 1. Try some head on collisions with the carts of different mass to simulate the example above. Be sure the carts stick together after the collision. 2. Observe qualitatively what combinations of velocities cause the two carts to be at rest after the collision. Q2-1: What happens when the less massive cart is moving much faster than the more massive cart? much slower?
Collisions, p. 9/15 Q2-2: Based on your prediction and your observations, what mathematical definition might you use for the momentum needed to stop an oncoming cart with known mass and velocity? Should it depend on the mass, the velocity, or both? Explain your choice. Just to double check youeasoning, you should have come to the conclusion that momentum is defined by the equation p mv where the symbol is used to designate defined as. Notice that momentum is a vector quantity. The direction of an object s momentum is the same as the direction of its velocity. Activity 2-2: Momentum change and the force on an egg You can see the relationship between force and momentum changes from a simple experiment tossing raw eggs. To avoid the mess, we will do this as a thought experiment (what physicists like to call a gedanken experiment). What is the relationship between the force you have to exert on the to stop it, the time it takes to stop it and the momentum change that the egg experiences? You probably already have some intuition about this matter. In more ordinary language, would you want to catch the egg slowly (by relaxing your hands and pulling them back slowly) or quickly (by holding your hands rigidly)? Q1-8: If the egg of mass m that is heading toward your hand at speed v, what is the magnitude of the momentum change the egg undergoes? What is the direction of the momentum change? Q1-9: Does the total momentum change depend on whether egg comes to a stop quickly or slowly? Q1-1: Suppose you the time you take to bring the egg to a stop is egg in such a way that t is large or small? t. Would you rather catch the Q1-11: What do you suspect might happen to the average force you exert on the egg while catching it when t is small? Explain.
Collisions, p. 1/15 A quantity called impulse may have been defined in lecture and/or in the textbook. It combines the applied force and the time interval over which it acts. In general, the impulse delivered by a force F r acting over the time interval from time t 1 to time t 2 is defined by r J = t2 t1 r Fdt As you can see, a large force acting over and short time and small force acting over a short time can have the same impulse. In one dimension, the magnitude of the impulse is equal to the area under the force-time graph. For a constant force acting over a time interval, the magnitude of the impulse is J = F t. Notice that F t is the area of the rectangle, i.e., the area under the force vs. time curve. Even if the applied force is not constant, the impulse is still equal from the area under the force vs. time graph (though it is harder to calculate). In the egg example, the force on the egg is small when the time interval is short (and vice versa), so it seems plausible that the impulse delivered to the egg does not depend on how quickly the egg comes to a stop (provided the egg starts with the same initial momentum). However, it seems likely that the impulse delivered to the egg might be larger if the egg underwent a larger change in momentum. In the next investigation, you explore the connection between impulse and changes in momentum. Area = Impulse Investigation 3: Impulse, Momentum and Collisions Let s first see qualitatively what an impulse curve might look like in a real collision in which the forces change over the time of the collision. Activity 3-1: Observing Collision Forces That Change with Time Release the catch on the cart s spring plunger. Collide the cart with a fixed wall (your hand will do) several times and observe what happens with the spring plunger. Q3-1: If friction is negligible, what is the net force applied on the cart just before it starts to collide? Q3-2: When is the magnitude of the force on the cart maximum? Q3-3: Roughly how long does the collision process take? Half a second? Less time? Several seconds?
Collisions, p. 11/15 Prediction 3-1: What will the force-time and velocity-time graphs look like from the moment just after the cart is released until after the collision with the wall? Sketch your predictions on the axes provided. During the collision the force is not constant. To Prediction 3-1 measure the impulse and compare it to the change in momentum of the cart, you must (1) plot a force-time graph and find the area under it, and (2) measure the velocity of the cart before and after the collision with the wall. This can be done with force probe, motion detector and the software. Activity 3-1: Impulse and momentum change In this activity, you will investigate how impulse is related to changes in momentum when only one force acts on an object. In this collision (as in most collisions), the only noticeable forces acting on the cart arise from the interaction between the wall and the cart. Other interactions (such as friction between the cart and the track) are so small that they can be time (s) ignored. 1. Set up to test Prediction 3-1 using the setup shown. Fasten the force probe to the bracket. Fasten the bracket securely to the track so that the rubber stopper on the force probe faces the cart. The force probe attached to the bracket will play the role of the wall. Make sure that the spring plunger is released (so that the collision extends over a reasonably long time interval). Place the motion detector at the opposite end of the track from the force probe. Be sure that the ramp is level. (Why is this precaution needed?) 2. Set the force probe to collect data at the maximum sample rate and push positive.* Set up the motion to collect velocity data at about 5 data points per second. (*The motion detector and the force probe need to be set up with consistent signs. Since the motion detector is at the right, leftward is the positive direction for the motion detector. The force probe must also be set up with positive to the left). 3. Set up axes for force-time and velocitytime graphs, with the time axes aligned vertically, as shown. 4. Test Prediction 3-1. When you are ready, zero the force probe and take data. Begin graphing and, when you hear the clicks of the motion detector, give the cart a push time (s) toward the wall and let the cart collide force (N) velocity (m/s) force (N) velocity (m/s)
Collisions, p. 12/15 with the wall. Repeat until you get a good set of graphs (i.e. graphs in which the motion detector saw relatively constant velocities of the cart as it moved toward and away from the wall and also the maximum force was no more than about 4 N).You may need to take a few runs before you get good data. (Unless you are careful, the motion detector will see your hand when you release the cart). When you get a good graph, adjust the axes to focus in on the time interval of the collision and record the results on the axes provided. Q2-5: Do the shapes of your graphs match your predictions? Explain. 5. Use the software to measure the velocity of the cart just before and just after the collision with the wall. Measure the mass of the cart. Use the results to calculate the initial momentum, final momentum and change in momentum of the cart. Don t forget to indicate the direction of each momentum value by including the sign. p = (Don t forget the units or the signs!) initial p = final p = 6. Use the statistics feature in the software to find the area under the force-time graph the impulse. (The area under the curve is the same thing as the integral of force vs. time). Record the value of the impulse in below (with units and sign). Explain how you determined the sign of the impulse. (Note: The statistics tool in DataStudio only gives the area to one significant figure. A better estimate of the impulse can be attained from the force-time graph by approximating the spike in the graph as a triangle and finding the area of the triangle from its dimensions. Q2-6: J = (Don t forget the units or the sign) How do the change in momentum and the measured impulse delivered to it during the nearly elastic collision compare? Explain. Are the units of p and J the same (or mathematically equivalent)? Extension 3-4: Examining impulse in other collisions How would the impulse change if the initial momentum of the cart were larger than in Activity 2-3? What if the collision took longer (or less time)? What if the collision were inelastic (i.e. the cart stuck to the wall) or partially elastic? If you have time, perform experiments to answer these questions. Be careful to set up your experiments so that the conclusions you draw make sense. For instance, if you want to investigate the effect of the duration of the collision, compare two collisions with the same initial momentum.
Collisions, p. 13/15 Investigation 4: Newton s Laws and Momentum Conservation Your previous work should have shown that interaction forces between two objects are equal in magnitude and opposite in direction on a moment by moment basis for all the interactions you have studied. This is a testimonial to the seemingly universal applicability of Newton s third law to interactions between objects. Your previous work also should have shown that the total impulse delivered to an object during a collision is equal to the change in the object s momentum. These two observations can be combined into a new physical principle. The logic is outlined below. Consider two objects interacting with each other. The total impulse J r 1 delivered to cart 1 during the collision equals the change in momentum of cart 1, and the impulse J r 2 acting on cart 2 during the collision equals the change in momentum of cart 2: r v r v J 1 = p 1 J 2 = p2 If the only forces on each object are the interaction forces between them, then the impulses delivered to object 1 will be exactly equal and opposite to the impulse delivered to object 2: J 1 = J 2. (Remember that magnitude of the impulse equals the area under the force-time graph, so that if the force-time graphs for the two objects are identical, the impulses will be identical in magnitude). Thus, by simple algebra: v v v v p 1 = p 2 or p1 + p2 = That is, the change in momentum for the one object equals the opposite of the change in momentum of the other. Stated another way, there is no change in the total momentum of the system (the two objects) provided that the only forces on each object are the interaction forces between them. If the momenta of the two carts before (initial subscript i) and after (final subscript f) the collision are represented in the diagram below, then where r pi m1v1 i + m2v2i p i = p f r = and p f = m1v1 f + m2v2 f In the next activity you will examine whether momentum is conserved in a simple inelastic collision between two carts of unequal mass. Activity 4-1: Inelastic collision 1. Set up the carts, ramp and motion detector as shown below. Remove the force probe from the cart. Add masses to cart 1 so that it is about twice as massive as cart 2. 2. Measure the masses of the carts. M (car1) = M (car2) = (don t forget units) 3. Set up the motion detector to take velocity data at about 5 points per second. Set up axes to graph velocity-time like those below.
Collisions, p. 14/15 +2 velocity (m/s) -2 time (s) Prediction 4-1: You are going to give the more massive cart 1 a push and collide it with cart 2, which is initially at rest. The carts will stick together after the collision. Suppose you measure the total momentum of cart 1 and cart 2 before and after the collision. How do you think the total momentum after the collision will compare to the total momentum before the collision? Explain the basis for your prediction. 4. Test your prediction. Begin with cart 1 at least 2 cm from the motion detector. 5. Use the software to measure the velocity of cart 1 just before the collision and the velocity of both carts together just after the collision. (You will want to find the average velocities over the short time intervals just before and just after but not during the collision.) Record the values (with units) below. r r v 1i = v 2i = r r v 1 f = v 2 f = 6. Calculate the total momentum of carts 1 and 2 before the collision and after the collision. Show the calculations below. Don t forget units for youesults. Q4-1: p r i p r f = = Was momentum conserved during the collision? Did youesults agree with your prediction? Explain. Q4-2: What are the difficulties in doing this experiment that might cause the momentum before the collision to be slightly different that the momentum after the collision? Explain.
Collisions, p. 15/15 Q4-3: Momentum is conserved even when the carts do not stick together. Such collisions are called partially elastic collisions. This lab will not test whether momentum is conserved in partially elastic collisions because of equipment limitations. What additional equipment would be needed to test whether momentum is conserved in such a collision? Why is only one motion detectoequired to analyze the inelastic collision? Extension 4-2: Consider other collisions you can examine with the available apparatus. Describe each physical situation carefully. Draw a diagram. Predict how the total momentum after the collision will compare to the momentum before the collision. Then set up the apparatus and test your predictions. Describe your observations. Include copies of any graphs you make. Compare your observations to your predictions. Historical note on momentum: Originally, Newton did not use the concept of acceleration or velocity in his laws. Instead he used the term motion, which he defined as the product of mass and velocity (the quantity we now call momentum). Let s examine a translation from Latin of Newton s first two laws with some parenthetical changes for clarity. Newton s First Two Laws of Motion* 1. Every body continues in a state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it. 2. The (rate of) change of motion is proportional to the motive force impressed: and is made in the direction of the right line in which that force is impressed. * I. Newton, Principia Mathematica, Florian Cajori, ed. (Berkeley: University of California Press, 1974), p. 13. * L.W. Taylor, Physics: The Pioneer Science, Vol. 1 (New York: Dover, 1959), pp. 129-131. The more familiar contemporary statement of the second law is that the net force on an object can be calculated as the product of its mass and acceleration where the direction of the force and direction of the resulting acceleration are the same. Newton s statement of the second law and the more modern statement are mathematically equivalent. It would be nice to be able to use Newton s formulation of the second law of motion to find collision forces, but it is difficult to measure the rate of change of momentum during a rapid collision without special instruments. However, measuring the momenta of objects just before and just after a collision is not too difficult. This led scientists to in the seventeenth and eighteenth centuries to concentrate on overall changes in momentum that resulted from collisions. They then tried to relate changes in momentum to the forces experienced by an object during a collision.