p. 1/7 PRELAB: COLLISIONS IN TWO DIMENSIONS 1. In the collision described in Prediction 1-1, what is the direction of the change in momentum vector D p r for the less massive puck? for the more massive puck? Explain how you determined your answers.. Make Prediction 1- in the space below. Explain the reasoning behind your prediction. 3. The diagram below shows the initial setup for the collision in Prediction -1. In the space to the right of the diagram, sketch a diagram which shows the final momentum vector for each puck in the collision described in Prediction -1. Explain the reasoning behind your diagram. 4. Sketch the change in momentum vector for the slower puck in the collision of Prediction -1. Explain why the Dp r vector points in the direction you have shown.
p. /7 Topic: Momentum, impulse and vector addition COLLISIONS IN TWO DIMENSIONS Overview: In this lab, you will observe collisions between two pucks as they travel on an air table. The pucks are connected to a spark timer. As the pucks travel over a sheet of newsprint, the spark timer records the position of both pucks at regular time intervals. You can use the position information from the marks on the paper to estimate the x- and y-components of the velocity vector for each puck. In class, you have probably studied momentum and used the concept to analyze collisions. In this lab, you will use momentum to analyze collisions and explore the limitations of the principle of momentum conservation. Writing it up: Throughout this handout, you will be asked to answer questions, sketch graphs and diagrams, and do calculations. Write these things in your lab notebook as you go through the experiment. Label each answer/graph/calculation/diagram so that you (or your lab TA) can find things quickly. If you have any computer printouts (such as graphs), remember to affix them to your lab notebook. After lab, write a short (<300 words) conclusion of the experiment that summarizes what you did and the major findings of the experiment. Safety/Equipment Tips Shock hazard: It is possible to get an (unpleasant, but not dangerous) electric shock from the apparatus if you touch the metal part of one of the pucks while the spark timer is operating. Avoid getting shocked: Do not touch the metal part of the pucks when the spark generator is on! Make sure both pucks are on the air table when the spark timer is on! Turn the spark generator off after each run. Protect the equipment: Press the foot pedal only while the pucks are moving. A series of sparks in one position will burn a hole in the specialty carbon paper and/or mar the smooth surface of the air table. Leave the pucks on the air table when the apparatus is idle (to avoid stretching the fragile air tubes). Room setup notes (for TA s) Given the limited number of stations available for this experiment (five working air tables in October, 003), students should stay at the air tables only when they are collecting data. Students can do the Investigations in either order, so you may want to have half the class work on Investigation 1 while the other half works on Investigation. Two or three stations should have unequal puck masses (for the elastic collision in Investigation 1). The remaining two or three stations should have pucks with about equal masses and Velcro (for the completely inelastic collision in Investigation ).
p. 3/7 Procedure: Some preliminaries 1. There should be a sheet of carbon paper (carbon side up) on the surface of the air table. (If there isn t, consult your TA). Place a sheet of newsprint on top of the carbon paper.. Turn on the air source. Check to make sure the table is level by placing the pucks in the middle of the table. Ideally, the pucks should remain motionless. If the table is not level, level the table by adjusting the legs. 3. Set the spark timer to 30 Hz so that there is 1/30 of a second between the sparks. Note: The two investigations in this experiment can be done in either order and the analysis of the data is essentially identical for the two Investigations. The questions in Investigation 1 provide more guidance about how to analyze the data than do the questions in Investigation. Investigation 1: A nearly elastic collision: In elastic collisions (like those between billiard balls), the objects bounce off each other cleanly without sticking together at all. In this investigation, you will investigate changes in velocity and momentum before, during and after a nearly elastic collision. 1. Data taking throughout this lab takes a little coordination. You must release the pucks so that they have the velocities you want and then, as soon as the pucks are released, press down and hold the spark generator s foot switch. (The spark generator only makes sparks when the foot switch is depressed). Just before the first puck hits the edge of the air table, release the switch. To avoid confusing data, practice each run at least once with the spark generator turned off. Before tackling a two dimensional collision, it might be helpful to consider a one dimensional example first. Prediction 1-1: Two pucks with different masses approach each other head-on, as shown below. The more massive puck is traveling faster than the less massive one before the collision. The two pucks collide elastically. Sketch a qualitative prediction of the movement of the pucks after the collision. What is the direction of the change in momentum vector Dp r for the less massive puck? What is the direction of the change in momentum vector for the more massive puck? Explain how you determined your answers. Prediction 1-: Which puck will experience a larger change in velocity? Which puck will experience a larger change in momentum? Explain your reasoning. (Keep in mind that velocity and momentum are both vector quantities).. Set up to test Prediction 1-1: Turn on the air source. Make sure the spark generator is off. (You will do a practice data run before turning on the spark generator). Start both pucks at opposite ends of the table. Push the pucks toward each other. Each puck should travel at low
p. 4/7 to moderate velocity. Press the foot switch just after you release the pucks. Hold the switch down as the pucks travel across the table. Release the switch just before either puck hits the side of the air table. Once you are confident of your timing, turn on the spark generator and take data to test Prediction 1-1. Turn the spark generator off. (Your collision does not have to be exactly head-on. In practice, it is almost impossible to get a 1-dimensional collision with this apparatus.) 3. Pick up the newsprint and turn it over. (You should see a series of dots that traces the paths of the two pucks). Place the newsprint into the frame of the measuring apparatus. Use the measuring frame to record the x- and y- coordinates of one of the two pucks. Construct graphs of x-position versus time and y-position versus time for that puck. The graph should have 10 to 0 data points. (Note: Use Excel or some other spreadsheet program to make the graph). Q1-1: Is the x-component of the puck s velocity constant before the collision? Is the y- component? Explain how you can tell from your graphs. Is this what you expected? Is the puck s velocity vector constant after the collision? Q1-: Determine the velocity vector for each puck before and after the collision. If you use the definition of average velocity for your calculation, clearly show which two data points you used for the calculation. Explain why you chose those two points. Express r your answers as vectors in component form (e.g. v = [3.1ˆ x 0. ˆ] cm/spark). puck 1, initial + y Q1-3: Use the calculations you have done so far to find the change in velocity r r r D v = v final - v ) of each puck as a result of the collision. Compare the result with ( int ial your prediction. Also, find the change in the momentum of each puck during the r r r collision. ( D p = p after - p ). Express your answers as vectors in component form. before Compare the results with your prediction. (Even though it is unlikely that your experiment produced the one-dimensional collision described in the Prediction, you should still be able to make some legitimate and insightful comparisons). 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 one dimension, for a constant force F acting over a time interval D t, the magnitude of the impulse is J = 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. Notice FD t that is the area of the rectangle, i.e., the area under the force vs. time curve.
p. 5/7 In general, the impulse delivered by a force F r acting over the time interval from time t 1 to time t is a vector quantity defined by r t r J Ú Fdt t1 If the force F r in the integral equals the net force F r net acting on the object, then the impulse equals the change in momentum of the object during the time interval from clock reading t 1 to clock reading t. r r J = Dp This result is called the impulse-momentum theorem. You might notice the impulse-momentum theorem is equivalent to Newton s nd law. (Simply take the time derivative of both sides to recover Newton s nd law). Q1-4: Estimate the impulse delivered to each puck during the collision. Express your answers as vectors. Explain how you determined your answer. (Yes, the answer to this question is very easy). You may have used the conservation of momentum principle to analyze collisions like the one in this experiment. The principle simply states that the total momentum of all objects in a system before a collision equals the total momentum after the collision. While you have probably used the principle in homework problems, you may not be aware of its mathematical basis. Under what conditions does the principle apply? In order to find out, let s take a careful look at its derivation. Consider a collision between two objects. The argument starts will the impulsemomentum theorem. The impulse acting on each puck equals the change in momentum of that puck: r r r r J 1 = Dp 1 and J = Dp where J r 1 is the impulse delivered to puck 1 by the net force acting on puck 1 and J r is the impulse delivered to puck by the net force acting on puck. If the only force acting on puck 1 during the collision is due to puck, the total impulse delivered to puck 1 is equal to the impulse delivered by puck : r t r J = F dt 1 Ú t1 Æ1 Similarly, if puck 1 is the only object exerting a force on puck : r t r J = F dt Ú t1 The final step of the argument is to apply Newton s 3 rd law. At all instants during the collision, r r r r v v FÆ 1 = -F1 Æ. Therefore, J 1 = -J and D p 1 = -Dp. Simple algebra gives the principle of v v v v momentum conservation: Dp1 + Dp = 0 (i.e. there is no change in the total momentum p 1 + p of the system). Q1-5: Is it possible that the net force experienced by puck 1 during the collision does not equal the force puck exerts on puck 1? Explain. Q1-6: In practice, you will find that the impulse delivered to puck 1 is not equal and opposite to the impulse delivered to puck. Is this evidence that one of the 1Æ
p. 6/7 assumptions in the derivation above is being violated during the collision? Or is this evidence of inexact measurements? (Note: If you argue that the observed difference is due to the violation of one of the assumptions of the derivation, clearly identify which assumption is likely to have been violated and explain how the assumption might have been violated. If you argue that the observed difference is simply due to inexact measurements, back up your assertion with calculations! You will need to convincingly show that the difference is most likely due to inexact measurements.) Q1-7: Estimate the magnitude of average force exerted on puck 1 during the collision. Explain how you arrived at your estimate. How reliable is this estimate? Explain. Is this estimate likely to be too low or too high? Explain. Investigation : An inelastic collision When two objects stick together after a collision, the collision is called a completely inelastic collision. In this section, you will use two pucks with Velcro sides to examine a completely inelastic collision. Prediction -1: Two pucks approach each other at an angle, as shown below. The pucks stick together and travel together after the collision. Sketch a qualitative prediction of the movement of the pucks after the collision. Sketch and label the change in momentum vector D p r for the faster puck on your diagram. Explain how you determined your answer. Prediction -: Which puck will experience a larger change in velocity? Which puck will experience a larger change in momentum? Explain your reasoning. 1. Put the Velcro bands around each puck.. Place the newsprint you used for Investigation 1 on the carbon paper. The dots you made in Investigation 1 should visible. (Remember that the new carbon paper marks will appear on the underside of the newsprint). Turn on the air source. (Make sure the spark generator is off for the practice runs). Push the pucks toward each other. Start the pucks so that they will collide. Press the foot switch just after you release the pucks. Hold down the foot switch as the pucks travel, collide and then travel after the collision. Release the foot switch before either puck hits the far edge of the table. Once you are confident of the timing, turn on the spark generator and take data. Note the general direction of each puck s initial velocity, so that you can identify which spark track goes with which puck when you analyze the data. 3. Pick up the newsprint and turn it over. (You should see a series of dots that traces the paths of the two pucks). Place the newsprint into the frame of the measuring apparatus.
p. 7/7 Q-1: Estimate the momentum of each puck just before the collision. Express your answer as a vector in component form. Explain how you arrived at your answer. If you use the definition of average velocity for your calculation, clearly show which two data points you used for the calculation. Explain why you chose those two points. Q-: Estimate the momentum of each puck just after the collision. Express your answer as a vector in component form. Explain how you arrived at your answer. If you use the definition of average velocity for your calculation, clearly show which two data points you used for the calculation. Explain why you chose those two points. Q-3: Use the results of the previous two questions to calculate the change in velocity Dv r for each puck during the collision. Express your answers as vectors. Q-4: Calculate the change in momentum Dp r for each puck during the collision. Again, express your answers as vectors. Q-5: Compare the change in velocity for puck 1 with the change in velocity for puck. Are they in the same direction? Do they have the same magnitude? Is this what you expect? Explain. Q-6: Compare the change in momentum for puck 1 with the change in momentum for puck. Are they in the same direction? Do they have the same magnitude? Is this what you expect? Explain. Q-7: Estimate the magnitude of the average force exerted on puck during the collision. Explain how you arrived at your estimate. How reliable is this estimate? Explain. Is this estimate likely to be too low or too high? Explain.