Copy of PHYS182A_195L-NewtonsThirdLaw_Plus_CoM

PHYSICS 182A/195L LAB REPORT - LAB 6+7 Lab 6+7: Newton’s Third Law + Conservation of Momentum San Diego State University Department of Physics Physics 182A/195L TA: Alvin Yassuiae Lab partner 1: Matthew Ying, Olivia Sekimoto Lab partner 2: Tyler Shonnard, Emily Gerken Date: October 10, 2023 Score: Theory Newton’s 3 rd Law Newton’s third law of motion explains the relationship between forces on bodies interacting within a system. “If Body A exerts a force on Body B (an action), then Body B exerts a force on Body A (a reaction). These two forces have the same magnitude but are opposite in direction.” This pairing of forces between two bodies, A and B, is called an Action-Reaction Pair . We can state the relation between the forces acting on bodies in an action-reaction pair as: . Note that the magnitudes are the same, but the signs are opposite. This will be the case with the lack of external forces; the pair isolated with only their own forces will always produce this pattern. What is Momentum? Momentum is the product of a body's mass and it’s velocity : . Of the fundamental kinematic quantities, mass, position, velocity, acceleration, why does the product of mass and velocity deserve its own name? It turns out that the product of mass and velocity is what’s known as a conserved quantity . Consider Newton’s third law, , for the forces experienced by two interacting masses and . By replacing each force by , we can show the following: 1 Department of Physics

This equation says that the quantity in the parenthesis does not change with time. Another way to say this is that the term in parenthesis is a constant : This fact is so important that we give its own name and symbol, : This important result shows us that the total momentum of a system is constant . We say: Momentum is always conserved. Newton’s 3 rd Law and its relation to Momentum In this lab we will be plotting the forces that two bodies exert on each other as a function of time. According to Newton’s Third Law, these forces should be equal and opposite for all time . The 𝑡 figure below shows an example of the Force versus Time graph for two objects A and B which collide with each other. We can see that . You might also notice that the area under each curve is also the same. It turns out that the area under a force versus time graph is momentum. To see this is the case, consider the derivative (slope) of momentum. Recall that momentum is : Since is just a constant, we have, But the derivative (rate of change) of velocity is just acceleration:

. After a few lines of algebra, we can solve these two equations for both and , which are the final speeds of Cart 1 and Cart 2 after the collision, respectively. The details for solving these equations are shown in the Appendix. The final results are: . Inelastic Collision While an elastic collision maintains both the conservation of momentum and kinetic energy, an inelastic collision only conserves momentum. This creates a problem for our equations because we no longer have two sets of equations to work with. There is one special case where we can still find and , and that’s when . What would this mean? It implies that the two carts stick together after the collision. This results in a perfectly inelastic collision . If is known and , conservation of momentum tells us , like before. If we plug-in , we get . Now we can easily solve for the unknown : . Procedure (Parts A, B, and C) Setup 1. Connect one Blue and one Red Smart Cart (one at a time) via the Hardware Setup tab. 2. The track should have already been leveled with a Torpedo Level, so do not adjust the screws on the bottom of the track. 3. To check if level: Set one Smart Cart on the track and if it remains at rest when placed in a few positions along the track, the track is level. If this is not the case, ask your instructor for assistance.

PHYSICS 182A/195L LAB REPORT - LAB 6+7 4. With each Smart Cart sitting on the track, go to the Calibration tab and select Force as the measurement you want to calibrate. Click Next. 5. Check the box that says Force Measurements which will allow you to calibrate both Smart Carts at the same time, click Next. 6. Select Restore Factory Calibration and click Next to finish. 7. Go to the individual Smart Cart Force Sensor settings (Gear icon) in the Hardware Setup tab and click Zero Sensor Now. If your Smart Cart disconnects during the experiment, you will need to redo these calibration setups. Part A Excess force will generate incorrect values from the force and acceleration sensors! Do not ‘ram’ the carts together! i.e. Do not exceed 0.2 Newtons! 1. Practice pushing the carts toward each other with their magnetic bumpers facing each other. Conduct a trial measurement to make sure that the forces are properly zeroed out and that acceleration of either cart does not exceed 1m/s 2 . Remember to delete the trial measurements afterward. 2. Open a page with the Force and Acceleration graphs, which should be titled Newton’s Third Law. 3. Start with one cart at each end of the track. Click RECORD. Push the carts together with similar speeds. After the collision click STOP. You should see plots of both carts’ accelerations and forces as a function of time. 4. Find the maximum accelerations experienced by each cart during the collision (what does during the collision imply? How can you tell from the graph where the collision takes place?) using the coordinate tool (crosshair-looking button in toolbar) and record the values in Table 1. 5. Find the maximum forces experienced by each cart during the collision (this should line up vertically in time with your acceleration graph) using the coordinate tool and record the values in Table 2. 6. Include your acceleration and force graphs in the appropriate Analysis section. Right click on the edge of the PASCO graph object and select “Copy Display”. Paste into this document with “Ctrl+v”. Part B 1. Place the red cart in the center of the track and the blue cart on the left end of the track. 2. Repeat Part A with the red cart at rest before the collision, and the blue cart in motion. Part C 1. Place a 250g Stackable Mass on the red cart such that its mass is approximately double that of the blue cart. 2. Place the red cart in the center of the track and the blue cart on the left end of the track. 3. Repeat with the red cart at rest before the collision and the blue cart in motion. 5 Department of Physics

PHYSICS 182A/195L LAB REPORT - LAB 6+7 2. Plot A2: Copy your force graph for part A into the box below. 7 Department of Physics

2. Plot B2: Copy your force graph for part B into the box below.

PHYSICS 182A/195L LAB REPORT - LAB 6+7 11 Department of Physics

PHYSICS 182A/195L LAB REPORT - LAB 6+7 2. Plot C2: Copy your force graph for part C into the box below. 13 Department of Physics

Yes, the carts experienced equal and opposite forces. This occurs for acceleration as well since it was equal to the magnitude in the opposite direction. 2. In this case, the blue cart changed from a state of motion into a state of rest , losing all of its momentum . The red cart did the opposite, changing from a state of rest into a state of motion . Where did the momentum of the blue cart go? During the collision, all the blue cart’s momentum that it lost is transfered over to the red cart Part C: Unequal masses 1. Did the carts experience equal and opposite forces? What about their accelerations? There was no equal and opposite force since the carts’ masses weren’t equal. This goes the same for acceleration. 2. Compute the ratio of the maximum accelerations experienced by the red cart versus the blue cart in Part C of the procedure: 0.998/1.236= 0.8074 3. How does the ratio of the accelerations of the carts compare to the ratio of their masses? Each cart has the same mass so the only difference is the acceleration of each cart Procedure (Parts D and E) Setup 1. Make sure that both carts have magnetic bumpers on them. 2. Use 0.276 kg for the mass of each cart. This is roughly the average of all the carts (with the magnetic bumper), and therefore not as precise as measuring them individually, but should only contribute a slight amount of experimental error. Part D: (Perfectly) Inelastic Collision 1. Place the red and blue carts at rest with the Velcro ® bumpers facing each other. The blue cart should be in the center of the track and the red cart should be on the left end. 2. Start recording and give the red cart a push toward the blue cart. Stop recording before either cart reaches the end of the track. 3. On the velocity vs. time graph, find the velocity of the red cart just before and just after the collision. You can accomplish this using the coordinate tool. The time just before the

PHYSICS 182A/195L LAB REPORT - LAB 6+7 collision is most easily identified by finding the time when the blue cart first begins to move. Record these velocities in Table D.2. 4. The initial velocity of the blue cart is zero and its final velocity is the same as the red cart because they stick together. Record the blue cart’s final velocity in Table D.2. 5. Add together the sum of the initial velocities, as well as the sum of the final velocities, and record these values in Table D.2. 6. Using the masses in Table D.1, multiply your carts’ respective masses with their initial and final velocities to find corresponding momentums. Record the values in Table D.3. Part E: Elastic Collision Part E1: m 1 = m 2 1. Record the masses of each cart in Table 1.E1. 2. Place the red and blue carts at rest on the track, with the magnetic bumpers facing each other. The blue cart should be in the center of the track and the red cart should be on the left end. 3. Start recording and give the red cart a push toward the blue cart. Stop recording before either cart reaches the end of the track. 4. On the velocity vs. time graph, find the velocity of the red cart just before and just after the collision. The time just before the collision is most easily identified by finding the time when the blue cart first begins to move. Record these values in Table 2.E1. 5. The initial velocity of the blue cart is zero. Find the final velocity blue cart just after the collision, then record this value in Table 2.E1. 6. Add together the sum of the initial velocities, as well as the sum of the final velocities, and record these values in Table 2.E1. 7. Using the masses in Table 1.E1, multiply your carts’ respective masses with their initial and final velocities to find corresponding momentums. Record the values in Table 3.E3. 8. Calculate the final and initial Kinetic Energies and record them in Table 4.E1 Part E2: 2m 1 = m 2 1. Add mass to Cart 2 (blue cart) until it weighs twice as much as Cart 1 (red cart). To accomplish this, you can use the mass bar. 2. Record these new mass values in Table 1.E2. 3. Repeat steps 2-8 from Part E1, except now use tables 1.E2-4.E2. Part E3: m 1 =2m 2 1. Remove the extra mass on Cart 2 (blue cart) that you added in Part E2. 2. Add mass to Cart 1 (red cart) until it weighs twice as much as Cart 2 (blue cart). To accomplish this you can use the mass bar. 3. Record these new mass values in Table 1.E3. 4. Repeat steps 2-7 from Part E1, except now use tables 2.E3-3.E3. 17 Department of Physics

PHYSICS 182A/195L LAB REPORT - LAB 6+7 Part E: Elastic Collisions Part E1: m 1 = m 2 Table 1.E1: Cart masses m 1 (red cart) mass (kg) 0.276 m 2 (blue cart) mass (kg) 0.276 Table 2.E1: Velocities Cart 1 (red) Cart 2 (blue) Sum (1+2) v i (m/s) 0.238 0 0.238 v f (m/s) 0.016 -0.217 -0.201 Table 3.E1: Momentums (p=mv) Cart 1 (red) Cart 2 (blue) Sum (1+2) p i (kg m/s) 0.0657 0 0.0657 p f (kg m/s) 0.00442 -0.0599 -0.0555 Table 4.E1: Kinetic Energies (KE=0.5mv^2=0.5p^2/m) Cart 1 (red) Cart 2 (blue) Sum (1+2) KE i (kg m 2 /s 2 ) 0.0078 0 0.0078 KE f (kg m 2 /s 2 ) 3.533 x 10^-5 -0.0065 -0.006454 Part E2: 2m 1 = m 2 Table 1.E2: Cart masses m 1 (red cart) mass (kg) 0.276 m 2 (blue cart) mass (kg) 0.276 Table 2.E2: Velocities Cart 1 (red) Cart 2 (blue) Sum (1+2) 19 Department of Physics

v i (m/s) 0.243 0 0.243 v f (m/s) -0.076 -0.152 -0.228 Table 3.E2: Momentums Cart 1 (red) Cart 2 (blue) Sum (1+2) p i (kg m/s) 0.0671 0 0.0671 p f (kg m/s) -0.0291 -0.0799 -0.1009 Part E3: m 1 =2m 2 Table 1.E3: Cart masses m 1 (red cart) mass (kg) 0.526 m 2 (blue cart) mass (kg) 0.276 Table 2.E3: Velocities Cart 1 (red) Cart 2 (blue) Sum (1+2) v i (m/s) 0.282 0 0.282 v f (m/s) 0.103 -0.342 -0.239 Table 3.E3: Momentums Cart 1 (red) Cart 2 (blue) Sum (1+2) p i (kg m/s) 0.148 0 0.148 p f (kg m/s) 0.054 -0.0944 -0.0401 Analysis (D and E) Part D: (Perfectly) Inelastic Collision We will use the tables from the data section to answer questions about which sum of variables is conserved and which is not conserved . Fill out the following table “Is it Conserved”, by