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UNFORMATTED ATTACHMENT PREVIEW
Physics 4AL Lab 11: Collisions Lab 11: Collisions Introduction Collisions are categorized into two types: elastic and inelastic. In this lab, you will perform collisions of both types using low-friction carts that roll on a track, and test your predictions on how the total momentum and kinetic energy of each system changes during the collision Equipment (per group) (per class) Dynamics System (Carts, masses, and track) 2 Photogates LoggerPro with interface 2 Picket Fences Digital Balance Background I. Momentum and Kinetic Energy In this lab we are considering collisions between two carts, such as in the figure below: Figure 2 In this collision two carts are initially moving towards each other, collide, and then move apart. Alternatively, the carts could collide and stick together. The two parameters required for understanding collisions are momentum, p, and kinetic energy, K. 𝑝⃗ = 𝑚𝑣⃗ 1 𝐾 = 𝑚𝑣 2 2 Momentum is a vector, but in today’s lab we are only considering linear motion, so we can describe the velocity and momentum as + or – depending on the direction of the carts velocity on the track. Kinetic energy is a scalar, and can never be negative. We are often interested in the total momentum and kinetic energy of a system consisting of two or more objects. For example, in figure 1 the total momentum and kinetic energy of the carts before the collision are: 𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 93 Lab 11: Collisions Physics 4AL 1 1 2 2 𝐾𝑡𝑜𝑡𝑎𝑙 = 𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 2 2 The first rule governing collisions is Conservation of Momentum: If no external forces are acting on a system, then the total momentum is constant. As applied to the case above we consider the system to consist of the two carts, and we assume that there is no net external force acting on the carts. In one-dimension conservation of momentum for the carts in figure 1 boils down to: 𝑝1𝑖 + 𝑝2𝑖 = 𝑝1𝑓 + 𝑝2𝑓 The above equation links the momentum of the carts before the collision to the momentum of the carts after the collision. Note that this does not depend on any of the details of the collision itself, which may be quite complicated. The only requirement is that there is no net force from external objects acting on the carts. The other important parameter governing collisions is kinetic energy. The situation with kinetic energy and collisions is a bit more complicated, and how kinetic energy behaves determines the type of collision. We can categorize collisions as one of three basic types: • Elastic collisions: kinetic energy is conserved. In other words, the total kinetic energy of the system before the collision equals the total kinetic energy of the system after the collision. This can be thought of as a ‘perfect’ bounce. • Inelastic collision: kinetic energy is not conserved. Note that the total energy is still conserved, but in this type of collision some of the initial kinetic energy is converted to some other type of energy such as thermal energy (more properly internal energy). • Completely inelastic collision: objects collide and stick together. This is an inelastic collision so kinetic energy is again not conserved. Note that there is no general law of conservation of kinetic energy. If kinetic energy is conserved, we have an elastic collision, and if kinetic energy is not conserved we have an inelastic collision. However, the total energy must always be conserved: this is always true. In the case of an inelastic collision the kinetic energy lost must show up as some other form of energy such as internal energy, sound, etc. Completely inelastic collision with one cart initially at rest Consider the case of Cart 1, with mass 𝑚1 and initial velocity 𝑣1𝑖 colliding, and sticking together with an initially stationary Cart 2 of mass 𝑚2 . Figure 3 In this case the conservation of momentum requires that 94 Physics 4AL Lab 11: Collisions 𝑝𝑖 = 𝑝𝑓 where 𝑝𝑖 = 𝑚1 𝑣1𝑖 is the initial momentum of Cart 1, and 𝑝𝑓 = (𝑚1 + 𝑚2 )𝑣𝑓 is the final momentum of the carts after they have collided and stuck together. The situation with kinetic energy is a bit more complicated as this is an inelastic collision and kinetic energy is not conserved. However, we can find a relation between the initial and final kinetic energy. 1 2 𝐾𝑖 = 𝑚1 𝑣1𝑖 2 1 𝐾𝑓 = (𝑚1 + 𝑚2 )𝑣𝑓2 2 We can find the final velocity,𝑣𝑓 , using conservation of momentum: 𝑣𝑓 = 𝑚1 𝑣1𝑖 𝑚1 + 𝑚2 Putting the final velocity into the formula for final kinetic energy gives: 1 𝑚1 𝑣1𝑖 2 1 𝑚1 2 𝐾𝑓 = (𝑚1 + 𝑚2 ) ( ) = 𝑚1 𝑣1𝑖 ( ) 2 𝑚1 + 𝑚2 2 𝑚1 + 𝑚2 𝑚1 𝐾𝑓 = 𝐾𝑖 ( ) 𝑚1 + 𝑚2 𝑚1 ) 𝑚1 +𝑚2 The factor of ( 𝑚1 ) 𝑚1 +𝑚2 𝛿( in the above equation has an uncertainty of = (𝑚 1 1 +𝑚2 ) 2 √(𝑚2 𝛿𝑚1 )2 + (𝑚1 𝛿𝑚2 )2 Note that the above uncertainty was calculated using the more accurate quadrature method. Elastic collision with one cart initially at rest Consider the case of Cart 1, with mass 𝑚1 and initial velocity 𝑣1𝑖 colliding elastically with an initially stationary Cart 2 of mass 𝑚2 . As with the perfectly inelastic collision discussed above momentum must be conserved. 𝑝𝑖 = 𝑝𝑓 𝑝𝑖 = 𝑚1 𝑣1𝑖 𝑝𝑓 = 𝑝1𝑓 + 𝑝2𝑓 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓 As this is an elastic collision, the kinetic energy is also conserved. 𝐾𝑖 = 𝐾𝑓 95 Lab 11: Collisions 1 2 𝐾𝑖 = 𝑚1 𝑣1𝑖 2 1 1 2 2 𝐾𝑓 = 𝐾1𝑓 + 𝐾2𝑓 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓 2 2 96 Physics 4AL Physics 4AL Lab 11: Collisions A. Perfectly inelastic collision with a light cart colliding with a stationary heavy cart. In part A you will collide a light cart against a stationary heavy cart. Velcro on each of the carts will cause the carts to stick together after the collision. 1. Predictions: Describe the plot of 𝑝𝑖 vs 𝑝𝑓 , and the plot of 𝐾𝑖 vs 𝐾𝑓 . What are the expected values from your graphs that you will be comparing to your experimental data? (You must provide the actual values for your predictions, with uncertainties where appropriate) 2. Experiment: Setup the carts so that the light cart will hit, and stick to the stationary heavier cart using Velcro tabs at the end of each cart. The sequence of events is shown in figure 3. In this sequence of events the light cart with mass m1 and an initial velocity v1 passes through a photogate that measures the time over which an infrared light beam is blocked by a piece of plastic. By measuring the time over which the photogate is blocked, and the distance over which the light beam is blocked, it is possible to measure v1. After the light cart completely passes through photogate 1 it will undergo a completely inelastic collision with the heavier stationary cart. After the collision, the cart will pass through the second photogate to measure the final velocity v2. LoggerPro will collect this data, and calculate the velocities. Here are the basic steps for this experiment: a) Weight the heavier cart with 4 of the metal masses that fit along the sides. b) Before taking data carefully choreograph the sequence of triggering the photogates to match figure 3. Determine the positions of the photogates, and where the collision occurs so that photogate 1 measures the blocked time before the collision, and photogate 2 measures the blocked time after the collision. Make sure the collision does not occur while one of the photogates is blocked since this will throw the calculations off. Likewise, don’t move the photogates too far apart, or friction will slow the carts down too much between measurements. A distance of about 10 cm between the photogates works well, and select a consistent position for the stationary cart. c) You will take 20 measurements over a range of speeds from about 0.4 m/s to 1.0 m/s. Avoid pushing the carts too fast or too slow. Make a few practice runs to get a feel for how hard you must push the carts to get a certain velocity. d) The configuration file for LoggerPro is 4AL collision setup.cmbl. Each run will produce a data table with a column for the speed. 97 Lab 11: Collisions Physics 4AL e) To assist you a template spreadsheet has been created for you: 4AL collision analysis.xlsx. You will have to enter the data and formulas, but this should help you get started. f) As you take data watch the graph, and try to fill in gaps. Figure 4 3. Analysis: Complete your data analysis on a spread sheet to determine whether your data agrees with your prediction or not. This should include both quantitative analysis using linear regression, and qualitative analysis using graphing. Your instructor will tell you how to submit the data analysis. 4. Conclusion: Discuss whether your data supports your prediction or not. You must provide quantitative reasoning. 98 Physics 4AL Lab 11: Collisions B. Elastic collision with a light cart colliding with a stationary heavy cart. In part B you will collide a light cart against a stationary heavy cart. Repulsive magnets on the front of the carts will cause the carts to bounce of each other. 1. Predictions: Describe the plot of 𝑝𝑖 vs 𝑝𝑓 , and the plot of 𝐾𝑖 vs 𝐾𝑓 . What are the expected values from your graphs that you will be comparing to your experimental data? (You must provide the actual values for your predictions, with uncertainties where appropriate) 2. Experiment: Similar to the Experiment A setup the two carts so that you will produce an elastic collision with the light cart hitting the stationary heavier cart. Then set up the picket fence and photogate such that you will be able to measure the initial velocity of the light cart, and the final velocities after the collision. This collision is more involved since you will need the initial and final velocity of the light cart, as well as the final velocity of the heavy cart. Again, test everything to make sure it works the way you expect. Here are the basic steps for this experiment: a) Weight the heavier cart with 4 of the metal masses that fit along the sides. b) Carefully determine where to position the photogates for this experiment, as well as where to position the stationary cart relative to the photogates. This is going to be trickier than the perfectly inelastic collision since the light cart will be bouncing backwards through the first photogate. Some points: o Make sure that the first cart is all the way through the first photogate before the collision occurs. o The magnets have a fairly long-range force, and the collision actually starts when the carts are still widely separated. Play around with the carts to estimate when the collision realistically starts. c) Take 20 measurements over a range of speeds from about 0.4 m/s to 1.0 m/s. Avoid pushing the carts too fast or too slow. Make few practice runs to get a feel for how hard you must push the carts to get a certain velocity. d) The configuration file for LoggerPro is 4AL collision setup.cmbl. Each run will produce a data table with a column for the speed. e) In the spreadsheet you used for the perfectly inelastic collision there is a sheet labeled Elastic Collision. Complete your analysis in this sheet. Note that you must set this spreadsheet up yourself. 99 Lab 11: Collisions Physics 4AL 3. Analysis. Complete your data analysis in the provided spread sheet. Be sure to include proper labels and units for all your data and graphs, and your work should be well organized. One caution: the photogates measure the speed of the carts, not the velocity. You will need to consider this in your calculations. Your instructor will tell you how to submit the data analysis. 4. Conclusion: Discuss whether your data supports your prediction or not. You must provide quantitative reasoning. 100 Physics 4AL Lab 11: Collisions C. Systematic Errors For applying conservation of momentum to collisions, are there any conditions that are required? Did the collisions you studied meet these conditions, or was there a systematic error affecting your results? If there was a systematic error, how would it affect your predictions? 101 Lab 12: Collisions in 2-dimensions Objective To analyze a two-dimensional collision between two objects, and determine if the collision is elastic or inelastic. Equipment A computer Background The momentum of an object is defined as the product of its mass and velocity i.e.𝑝⃗ = 𝑚𝑣⃗, and is a vector. 1 2 The Kinetic Energy of an object is defined as 𝐾 = 𝑚𝑣 2 , and is a scalar. The law of conservation of momentum states that the total momentum in a system is conserved, if there are no external forces acting on the system. For such a system, the velocity of the center of mass of the system 𝑣𝐶𝑀 remains constant. Collisions are one of the following kinds: (i) Elastic ● Kinetic Energy is conserved (no loss of KE) ● Momentum is conserved (ii) (a) Inelastic ● Kinetic Energy is not conserved (part of the KE is converted to some other form of energy) ● Momentum is conserved (b) Completely/Perfectly Inelastic ● Kinetic Energy is not conserved (maximum amount of KE is lost) ● Momentum is conserved ● Colliding objects stick together after collision In this lab, you will analyze a video of a collision between two pucks that have the same mass (48 g or 0.048 kg). The pucks are released on an air hockey table that has negligible friction. It is therefore a reasonable approximation that there are no external forces acting on the system. Lab Exercises Pre-lab activity Two pucks are released on a frictionless air hockey table and follow the path shown in the following diagram: 1. Predict what the x-t and y-t graphs would look like for the blue and red pucks before and after collisions. (You should have four graphs in all) 2. Predict what the x-t and y-t graphs would look like for the center of mass of the system of pucks over the entire time period (You should have two graphs in all) Procedure 1. Download the video Puck Collisions and import it into Tracker. 2. Set your coordinate system by clicking on in the top menu of Tracker. 3. Drag your axes so that the x-axis is lined up with the motion of the blue puck. It should look like the image below. (The pink lines are the coordinate axes) 4. Set the calibration using the meter stick on the left side of the video. You will have greater accuracy if you use 5 of the 10-cm segments for your calibration. In other words, stretch the blue calibration arrow across 5 segments for a total length of 0.5 m. 5. Create a point mass for the blue puck and name it as Blue Puck. To mark the position of the puck, use shift-click. It will the advance to the next frame, and you repeat the process. 6. Create a point mass for the red puck and name it as Red Puck. Mark the path of the red puck like what you did for the blue puck. 7. You should be able to see when the collision occurred by looking at the x-t graph. For instance, the x-t graph might look like what is shown in the adjoining image. You can see a distinct point where the velocity changed. 8. Right click on the graph, click Analyze, and select the area before the collision occurred. Using a linear curve fit, calculate the velocity of the blue puck before collision. Remember that it is the slope of the line i.e. the parameter A in the linear fit. Repeat for after collision. 9. Using the same method, calculate the velocity of the red puck before and after the collision. 10. Calculate the initial and final momentum in the x and y direction for each puck (by multiplying the mass in kg) 11. Calculate the initial and final kinetic energy for each puck. 12. Tabulate your data in a table like the one shown below: Puck Mass Velocity 𝑣𝑖𝑥 𝑣𝑖𝑦 𝑣𝑓𝑥 Momentum 𝑣𝑓𝑦 𝑝𝑖𝑥 𝑝𝑖𝑦 𝑝𝑓𝑥 Kinetic Energy 𝑝𝑓𝑦 𝐾𝑖 𝐾𝑓 Blue Red 13. Calculate the total initial x-momentum in the system, the total initial y-momentum in the system, and the total initial KE in the system. Next calculate the total final x-momentum in the system, the total final y-momentum in the system, and the total final KE in the system. Calculate the percent difference between the final and initial values. Tabulate your calculations in a table like the one shown below: Total X-momentum in system 𝑝𝑖𝑥 𝑝𝑓𝑥 % difference Total Y-momentum in system 𝑝𝑖𝑦 𝑝𝑓𝑦 % difference Kinetic Energy in system 𝐾𝑖 𝐾𝑓 % difference 14. Is momentum conserved in this collision? Is kinetic energy conserved in this collision? Is the collision elastic or inelastic? Center-of-mass velocity You will now determine the velocity of the center of mass of the system of pucks before and after collision. Tracker can track the center of mass for you. 1. We need to define the masses of the pucks. Click the tab for the Blue Puck in the Track Control toolbar. In the drop-down menu, select Define… . In the resulting pop-up window, enter the mass of the puck for the parameter m as shown in the adjoining figure. 2. Repeat the previous step for the red puck and enter its mass. 3. Click the create button and select Center of Mass. 4. A window asking you to select the masses pops up. Select blue puck and red puck and click on OK. 5. What is the velocity of the center of mass in the x and y directions? What do you notice about the x-t and y-t graphs for the CM as compared to before? Can you tell where the collision occurred? Lab 12: Collisions in 2-Dimensions Lab Activity Report Group members: Date: Lab activity 1. Attach a screenshot of the x-t and y-t graphs obtained on Tracker for the red and blue pucks. Do the graphs agree with your predictions? 2. Data: (You may create a table similar to the sample one in step 12 or create your own) 3. Calculations: (You may create a table similar to the sample one in step 12 or create your own) 4. Is momentum conserved in this collision? 5. Is kinetic energy conserved in this collision? 6. Is this collision elastic or inelastic? 7. Attach a screenshot of the x-t and y-t graphs obtained on Tracker for the center of mass. Do the graphs agree with your predictions? 8. What is the velocity of the center of mass in the x and y directions before and after the collision? Is the center of mass velocity constant? In this collision, the laws of conservation of linear momentum and energy applied to “before” and “after” collision, so I used it to predict the outcome of a collision. Predict x-t graphs for the blue puck Predict y-t graphs for the blue buck Predict x-t graphs for the Red puck Predict y-t graphs for the Red puck Predict x-t and y-t graphs for the center of mass of the system of pucks over the entire time.
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