Lab 5 - Conservation of Momentum - F23

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Elastic and Inelastic Collisions Physics Topics If necessary, review the following topics and relevant textbook sections from Serway / Jewett “Physics for Scientists and Engineers”, 10th Ed. • Kinetic Energy (Serway 7.5) • Linear momentum and its conservation (Serway 9.1, 9.2) • One dimensional collisions (Serway 9.4) Introduction You have been hired to investigate a car accident which occurred when the driver of one car was stopped at a stoplight. The driver claims that she was idling at the stoplight and her car was in neutral when she was rear-ended by another car. You have data on the tire skid marks of the two cars, which can help you determine their velocities. You also know the make and model of each car, which helps you determine their masses. To get a handle on the basic physics involved, you want to model the situation with a simple laboratory experiment. Given the masses of the cars, and the initial velocity of one car, can you predict the final velocities of the two cars in an elastic collision and a perfectly inelastic collision? Pre-Lab Questions Please complete the following questions prior to coming to lab. They will help you prepare for both the lab and the pre-lab quiz (Found on D2L). Read through the entire lab manual before beginning 1.) What is the specific goal of this lab? Exactly what specific question are you trying to answer? Be as specific as possible. (“To learn about topic X...” is not specific!) 2.) What specific measurements or observations will you make in order to answer this question? 3.) Perfectly Inelastic Collision (a) A perfectly inelastic collision is one where the two objects stick together after the collision. Draw two diagrams of the situation, one before the collision and one after the collision. Your diagram should show both cars. Let’s call the car which is initially moving “1”, while the car initially at rest is “2”. Page 1 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited (b) What quantity or quantities are conserved in this problem? Explain how you know. (c) Using the conservation law(s) you identified in the previous part, write one or more equations relating the masses of the cars m 1 and m 2 , the initial velocity of car 1 ( v 1 ) and the final velocity of the two cars v f . (d) Solve your equation(s) for v f . You should now have a prediction for the final velocity of the two cars if you know the initial velocity of car “1” and both masses. 4.) Elastic Collision (a) In an elastic collision, the cars do not stick together, but bounce off each other. In addition to this fact, what specifically characterizes an elastic collision? Make the most precise statement possible. (In other words, what is an elastic collision?) (b) Prediction: If the car which is initially at rest is twice as massive as the initially moving car, what direction will each car move after the collision? Try to answer this question before completing the derivation below. It is not important that you get this prediction right, it is important that you think about it first! (c) What quantity or quantities are conserved in this problem? Explain how you know. (d) It may not be so obvious at first sight, but in one dimensional elastic collisions, the following relationship is satisfied: ⃗v 1 i + ⃗v 1 f = ⃗v 2 f + ⃗v 2 i . (1) Here ⃗v 1 i is the initial velocity of cart 1, and ⃗v 1 f is the final velocity of cart 1. The notation is similar for cart 2. Note that in our problem, ⃗v 2 i = 0 Your goal is to write two equations, one for v 1 f in terms of m 1 , m 2 and v 1 i , and another one for v 2 f in terms of m 1 , m 2 and v 1 i . Combine equation (1) with the conservation law(s) you identified in the previous part to acheive this. Apparatus • LabQuest Mini • Graphical Analysis Software • Vernier Carts (2) with inserted magnets and velcro attachments • Aluminum Track • Vernier Motion Sensor (2) • Additional masses Page 2 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Procedure 1.) Measure and record the mass of each cart and any weights that will be used as attach- ments. As always, include an estimate of the uncertainty in these measurements. 2.) Setup (a) Insert the two motion sensors into CH-1 and CH-2 of the LabQuest mini. (b) Place a sensor at each end of the track. Rotate the pivoting head of the sensor so it is perpendicular to the table. Make sure that the switch on each sensor is set to “cart” mode. (c) Open Graphical Analysis. The sensors should be recognized automatically. (d) Click the ‘Data Collection Setting’ button in the bottom left corner, set the sam- pling rate to 10 samples / second (0.1 seconds per sample). Set the total duration to 5 seconds. Once you have made these changes click Done . (e) Zero your motion sensors and set them to measure in the same direction. Hold your hand or a piece of paper on the left side of the track near motion sensor 1. Press the “Position 1” button in the lower right corner and then click ”Zero” in the pop up (as seen in the image below). Now press the ”Position 2” button in the lower right corner, click zero and also set ”Reverse” to be on/green. Now both of your motion sensors are set to have + x in the same direction. (f) Try taking data by clicking . While the sensors are taking data, move the cart back and forth with your hand. Make sure that the graphs behave as you expect. If you are not seeing the appropriate response, try repositioning the sensor(s) and ask your TA for assistance if necessary. 3.) Perfectly Inelastic collision (a) Set up the two carts with velcro pads facing each other so that they will stick together upon collision. Do not add any mass to either cart. Practice rolling cart “1” towards cart “2” slowly so that the two stick together after the collision. One member of your team should be ready to stop the two carts before hitting the sensor at the end of the track. Page 4 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited (b) When you think you have a reasonable technique, click to start data collection, and then start cart “1” rolling on its collision course with cart “2”. (c) After data collection is finished, you should be able to see from the graph cart 1’s velocity before and after the collision. If it is not obvious where the collision occurred, try taking data again. (d) Use Graphical Analysis to measure cart 1’s velocity before the collision and the final velocity of both carts after the collision. To do this: i. Click and drag on the velocity graph to select the range of data you want to fit ii. Click the data options button and select ‘View Statistics’. iii. Record the mean velocity value for the highlighted region with uncertainty. In your report you should note what values you used to assign uncertainty (mean, max value, min value, standard deviation). (e) Rename your latest run as something significant (ex. Inelastic EqualMass). Click the View Options button in the top right and turn on ‘Data Table’. Click the three dots beside the data set and select ‘Rename Data Set’. (f) Add mass to cart “1”. You should add enough mass so that the total mass of cart “1” increases by at least 50%. Record the new mass of each cart; then go back to step 3b and repeat all steps to measure the initial and final velocities for this set of masses. Note that you can show or hide different data sets on the graph by clicking the label for the y-axis and changing the selection. (g) Remove the mass from cart “1”, and add it to cart “2” so that cart “2” becomes the heavier cart. Record the new mass of each cart; then go back to step 3b and repeat all steps to measure the initial and final velocities for this set of masses. (h) You should now have 3 data sets for the three different mass carts in a single Graphical Analysis file. Save your Graphical Analysis file to the desktop of the computer with a descriptive name like PCS211 Firstname LastName Inelastic.gmbl . Also export your data as a .txt file, upload this .txt file to your Lab 5 submission folder on D2L. Then copy the files to a USB key or e-mail it to yourself. All files will be deleted from the computers when you log off! 4.) Nearly Elastic Collision (a) Now that you have your “inelastic” data saved in a safe place (or e-mailed to yourself), start a fresh file by clicking the file menu in the top left corner and selecting ‘New Experiment’. Once the data is cleared, save the file to the desktop with a new name, for example PCS211 Firstname LastName Elastic.gmbl . (b) Setup the two carts so that the magnets on the ends of the carts face each other. Check that when the carts are close, they repel each other. Do not add any mass to either cart. Practice rolling cart “1” towards cart “2” slowly so that the two bounce off of each other without touching . Page 5 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited (b) Calculate the momentum with uncertainty for the system of two carts before and after the collision. Is the momentum conserved? Do the two momentum values agree within uncertainty? If not, what percentage of the initial momentum was lost in the collision? (c) Calculate the kinetic energy with uncertainty for the system of two carts before and after the collision. Is the kinetic energy conserved? Do the two momentum values agree within uncertainty? If not, what percentage of the initial kinetic energy remained after the collision? Wrap Up The following questions are designed to make sure that you understand the physics impli- cations of the experiment and also to extend your knowledge of the concepts covered. Your report should seamlessly answer these questions in their noted sections. 1.) [Theory] For the collisions that did not conserve (kinetic) energy, what happened to that energy? Give a few possibilities. 2.) [Theory] Two particles collide: one of which is initially at rest. Is it possible for both of the particles to be at rest after the collision? If so, give the conditions under which this might occur. If not, explain why it is not possible. 3.) [Discussion] For the elastic collision: under what conditions will cart “1” bounce back- ward after the collision? Explain in words using your experimental data and your predicted equations. Page 7 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Report Here is a brief guide for writing the report for the lab. The report should include the following sections: • Title Page – Include: Report Title, Your Name, Course, Section Number, Instructor, TA Name, and Date of Submission. • Introduction – What is the experiment’s objective? • Theory – Derivations of the physics being investigated, or reference to a source that provides a description/equation representing the physics being investigated. – Providing graphs that illustrate or predict how the system under study is expected to behave. • Procedure – Briefly explain the systematic steps taken for the experiment. • Results and Calculation – Tabulate the measurements in an organized manner. – Based on the procedure, one should have a sense of how the tables will look like prior to taking measurements. – Graph the main results. – Provide examples of any calculations. • Discussion and Conclusion – Discuss the main observations and outcomes of the experiment. – Summarize any significant conclusions. • References – Very important to include to avoid plagiarism claims. • Appendices – If large quantities of data are obtained, or lengthy details are needed, but break the overall flow of the report, they should be referenced and placed in the appendix. Page 8 of 10

Toronto Metropolitan University - PCS211 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Vernier Graphical Analysis™ and Graphical Analysis Pro — User Manual www.vernier.com/ga 13 Use the x- and y-axis labels to change what data are plotted on your graph. Customize the graph appearance by changing the default point symbol and/or the trace color. Access data analysis tools from the Graph Tools menu including curve fits, statistics, and integral tools. Scale your graph to zoom to all data or zoom to a selection of data. Set your data-collection mode and modify data- collection parameters such as rate and duration. Access a sensor meter for changing units and sensor calibration. Access Sensor Setup to view which sensors are connected, connect to wireless sensors, and modify sensor channels (select Go Direct sensors). Page 10 of 10