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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:

PHYSICS 182A/195L LAB REPORT - LAB 6+7 , so . The derivative of momentum is force! And the inverse of this relation, the antiderivative, gives us: . Calculus-free conclusion: the area under the force versus time curve is the change in momentum of that mass. Equal areas means equal change in momentum. Collisions While a collision can be extremely complex and involve many forces and bodies (imagine a car crash), conservation of momentum tells us that the total momentum before and after that collision is the same: The (i) subscript labels the initial momentum (before the collision) and the (f) subscript labels the final momentum (after the collision). Elastic Versus Inelastic Collisions Yet another quantity of interest is kinetic energy . Kinetic energy is defined as: Kinetic energy is only sometimes conserved during a collision. If kinetic energy is conserved during a collision, then we call it an Elastic collision . This only happens for frictionless collisions. Otherwise, if the kinetic energy is not conserved, then we call it an Inelastic collision and kinetic energy is lost due to internal friction. Elastic Collision In an elastic collision, we can use the fact that both momentum is conserved and kinetic energy is conserved. We therefore have two equations: , and . In our experiment, we assume that we know and we know . So our equations simplify: , and 3 Department of Physics

. 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

Data (Parts A, B, and C) Table 1: Extremal accelerations Red Cart Part A Part B Part C 1.257 m/s^2 -1.161 m/s^2 -0.998 m/s^2 Blue Cart Part A Part B Part C 1.064 m/s^2 1.056 m/s^2 1.236 m/s^2 Table 2: Extremal forces Red Cart Part A Part B Part C -0.93 N -0.50 N -1.00 N Blue Cart Part A Part B Part C -0.86 N -0.17 N -0.81 N Part A 1. Plot A1: Copy your acceleration graph for part A into the box below.

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