LK-14 Conservation of Linear Momentum in 1-D

LK-14: CONSERVATION OF LINEAR MOMENTUM Rev 04-04-2022 Objective Experimentally demonstrate that linear momentum and kinetic energy are conserved in an elastic collision, and that linear momentum is conserved but kinetic energy is not conserved in an inelastic collision. Also study the Recoil when one object splits up into two. Materials and Equipment  PASCO Smart Cart, blue, with magnetic bumper  PASCO Track End Stop (2)  PASCO Smart Cart, red, with magnetic bumper  PASCO Cart Masses, 250-g (2)  PASCO Track with feet Theory The momentum of an object is equal to the product of its mass m and velocity : (1) The total momentum of a system of objects is equal to the linear sum of each object's individual momentum: (2) For a linear system that is not influenced by outside forces, the total momentum of the system is conserved – the total system momentum before the collision is equal to the total system momentum after the collision. This extends to objects, or systems of objects, experiencing collisions, both elastic and inelastic; and also to a system of objects experiencing an explosion, i.e. breaking up into many parts. As an example, consider the red cart moving to the right, and colliding with the blue cart. Using the notation: M R = Mass of Red Cart M B = Mass of Blue Cart U R = Initial Velocity of Red Cart U B = Initial Velocity of Blue Cart 1

V R = Final Velocity of Red Cart V B = Final Velocity of Blue Cart By the Law of Conservation of Linear Momentum: ∑ p i = ∑ p f Or p iR + p iB = p fR + p fB Or M R U R + M B U B = M R V R + M B V B As an example, if the Blue cart is initially stationary, Red cart is moving to the right at 4.00 m/s before the collision, and both carts stick together due to the Velcro. Mass of Red cart = 500 g, mass of Blue cart = 250 g. Due to sticking together, their final speeds will be the same, which we write as V. Then: 0.500 ∗ 4.00 + 0.250 ∗ 0.0 = 0.500 ∗ V + 0.250 ∗ V 2.0 = 0.750 ∗ V V = 2.67 m / s If, in the same conditions, there is no Velcro, and the Blue cart is seen to move to the right at 3.00 m/s after the collision, what is the velocity of the Red cart? 0.500 ∗ 4.00 + 0.250 ∗ 0.0 = 0.500 ∗ V R + 0.250 ∗ 3.00 2.0 = 0.500 ∗ V R + 0.750 V R = 2.50 m / s If you calculate the kinetic energies before and after the collision for both of these examples, you will note a loss of kinetic energy in the collision. This is the case for most collisions that we can see or hear. In some collisions, especially those of atomic particles, the collisions also conserve kinetic energy. Hence we have three types of collisions: Elastic, Inelastic and Perfectly Inelastic. Momentum is conserved in each of these. Elastic collisions occur when two objects bounce off each other without any loss of kinetic energy. They would not stick or change shape. Collision of air molecules with each other are elastic, as are all collisions between atomic and molecular particles. Collision of two billiards balls is almost an elastic collision. Momentum is conserved and kinetic energy is conserved. Inelastic collisions occur when two objects don’t bounce off each other perfectly, such as two objects sticking fully or partially to each other, like two clay balls colliding and then moving as one object; or two objects colliding and deforming as a result of the collision, like a two-car accident. Momentum is conserved and kinetic energy is lost. Perfectly Inelastic collisions occur when the two colliding objects stick together and move off as one. Momentum is conserved and kinetic energy is lost. Explosions are when one object splits up into two or more, e.g. when a gun is fired, the bullet and the gun move in opposite directions. If the object was stationary before the explosion, its momentum was zero. After the explosion, the momentum to the right must be equal to the momentum to the left, so that they add up to zero. Momentum is conserved and kinetic energy is increased. In this activity, you will demonstrate that the total momentum of a two-cart system is conserved in both elastic and inelastic collisions, and in explosions, but the total kinetic energy of the system is only conserved in elastic collisions. We will do this be noting the percent loss of Momentum and Kinetic Energy. This is done by using: 2

4. Use the Multi Coordinate Tool in Capstone to find the velocity of both carts just before the collision (initial velocity) and just after the collision (final velocity) in each trial. (The Multi- Coordinates Tool ensures that the data for both carts is at the same instant of time). Enter the velocity values in Table 1. Calculate the Momenta and Kinetic Energies for the individual Carts, then add them up to get the sum of Momenta and sum of KE before the collision and after the collision. Hence find the percent loss of Momentum and Kinetic Energy. Part 2 – In-elastic Collision 1. Remove the Cart Masses from the Blue Cart. 2. Remove the magnetic bumpers from both carts and attach the rubber bumpers. 3. Place the blue cart in the middle of the track with its rubber bumper facing left, and the red cart at the left-side end of the track with its rubber bumper facing right. (i.e. bumpers facing each other). 4. Start recording data, and then gently push and release the red cart toward the blue cart, allowing them to collide. Stop recording data soon after the carts have collided. The Red cart should move some clear distance before reaching the Blue Cart. 5. Repeat the same data collection steps two additional times, each time adding one of the 250-g cart masses to the blue cart; keep the mass of the red cart the same for trials 2 and 3. 6. Use the Multi Coordinate Tool in Capstone to find the velocity of both carts just before the collision (initial velocity) and just after the collision (final velocity) in each trial. Enter the velocity values in Table 1. Calculate the Momenta and Kinetic Energies for the individual Carts, then add them up to get the sum of Momenta and sum of KE before the collision and after the collision. Hence find the percent loss of Momentum and Kinetic Energy. Part 3 – Perfectly Inelastic Collision 1. Remove the cart masses from the blue cart. 2. Remove the rubber bumpers from both carts, and then rotate both carts on the track so that the blue cart is still on the right side of the track and the red cart is on the left side, with the Velcro bumpers of the two carts facing each other. This will change the sign of the velocities, so go into Cart settings and set “Sign Change” so that the velocities of both carts is positive when motion is to the right. 3. Repeat the same data collection steps from Part 1. Push the red cart to the right, so that it hits the blue cart. When the carts collide they will hit Velcro bumper to Velcro bumper and stick together, resulting in a perfectly inelastic collision. The final velocity of both carts should be the same. 4. Perform three trials, adding one 250-g cart mass to the blue cart in trial 2, and two cart masses in trial 3. 4

5. Record the cart masses, initial and final velocities for each trial into Table 2. Calculate the Momenta and Kinetic Energies of individual carts, and for both of them, and calculate the percent loss of Momenta and Kinetic Energy. Part 4 – Explosion 1. Remove the cart masses from the blue cart. 2. Set the red cart on the left side of the track, and the blue cart on the right, with both plungers facing to the left. Again make sure that positive velocity is to the right for both carts by setting sign change in cart settings. 3. Press the plunger on the blue cart until it loads at its second stage (two clicks), and then set both carts in the middle of the track, plungers facing left. Adjust the carts so they sit motionless with the plunger on the blue cart touching the red cart. 4. Start recording data, and then gently tap the plunger release trigger on the blue cart to simulate an explosion within the two-cart system. Stop recording data before either cart hits the end stop. 5. Repeat the same data collection steps two additional times, each time adding one of the 250-g cart masses to the blue cart. Keep the mass of the red cart the same for trials 2 and 3. 6. Record the data in Table 3. Use the tools in Capstone to find the velocity of both carts just before (initial velocity) and after the explosion (final velocity) in each trial. Then do the required calculations. Note: In all cases, the Blue Cart is on the right, and Red Cart is on the left, and extra masses are added to the Blue Cart. In all cases the Blue Cart is stationary before the collision / explosion. In first two cases we push the red cart to the stationary blue cart. In all cases the velocity must be positive when the cart moves towards your right side (for both carts), and negative when it moves to your left. This is set in the Cart Settings. The data recording rate must be the same for both carts. Use 100 Hz for both. Make sure that the velocities for both carts are measured at the same instants of time. Use the Multi-Coordinates Tool to do this. Attach at least one screenshot of your Capstone Graphs for each of the three parts. Attach photographs of your setup with your report / Dataset Get the Origin of both Y-axes and their scales to be same by using: Display – All Axes Share Origin – All Axes Scale Together. To save time in calculations, you may enter the data is the excel file in the canvas module, then copy-paste the tables in your report. 5

Data Analysis Part 1 – Elastic Collision: Both carts have Magnetic Bumpers that collide. Red Cart moving to the right and colliding with a stationary Blue Cart. Masses added to Blue Cart. Cap. Run No Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Percent Change in Momentum Percent Change in Kinetic Energy m U V m U m V ½ m U 2 ½ m V 2 100 |Δp| / pi 100 |ΔK| / Ki Red 250 g X X Blue 250 g X X Both X X X Red 250 g X X Blue 500 g X X Both X X X Red 250 g X X Blue 750 g X X Both X X X 7

Part 2 – Inelastic Collision: Both carts have Rubber Bumpers that collide. Red Cart moving to the right and colliding with a stationary Blue Cart. Masses added to Blue Cart. Cap. Run No Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Percent Change in Momentum Percent Change in Kinetic Energy m U V m U m V ½ m U 2 ½ m V 2 100 |Δp| / pi 100 |ΔK| / Ki Red 250 g X X Blue 250 g X X Both X X X Red 250 g X X Blue 500 g X X Both X X X Red 250 g X X Blue 750 g X X Both X X X 8

Part 4 – Explosion: Plunger in Blue Cart pushes both carts apart. Red Cart on left, Blue Cart on right. Plunger of Blue Cart pushes both Carts apart. Plunger in 2 nd click. Cap. Run No. Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Difference in Momentum (kg.m/s) Difference in Kinetic Energy (J) m U V m U m V ½ m U 2 ½ m V 2 Δp ΔK Red 250 g Blue 250 g Both X X X Red 250 g Blue 500 g Both X X X Red 250 g Blue 750 g Both X X X Note: Initial velocities should be zero, but they may not be if the carts are not completely steady on the track before pressing the plunger button. RESULTS Elastic Collision In-elastic Collision Perfectly Inelastic Collision Explosion Mass of Carts (kg) % change in Momentum % change in Kinetic Energy % change in Momentu m % change in Kinetic Energy % change in Momentum % change in Kinetic Energy change in Momentum change in Kinetic Energy Red 0.250 0.0 0.0 1.6 23.5 0.0 50.0 0.000 0.024 Blue 0.250 Red 0.250 8.5 2.8 1.2 28.1 2.3 65.1 0.001 0.024 Blue 0.500 Red 0.250 0.0 0.0 1.7 44.7 0.5 74.7 0.001 0.025 Blue 0.750 10

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SAMPLE DATA Part 1 – Elastic Collision: Both carts have Magnetic Bumpers. Run No. Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Percent Change in Momentum Percent Change in Kinetic Energy m U V m U m V ½ m U 2 ½ m V 2 100 |Δp| / pi 100 |ΔK| / Ki 1 Red 250 g 0.633 - 0.183 0.15825 - 0.04575 0.05009 0.00419 X X Blue 500 g 0.00 0.435 0.00 0.21750 0.00 0.04731 X X Both X X X 0.15825 0.17175 0.05009 0.05149 8.5 2.8 Sample Calculations for Run 1: Red Cart: Pi = 0.250 kg*0.633 m/s = 0.15825 kg.m/s Pf = 0.250*(-0.183) = – 0.04575 kg.m/s KEi = ½ (0.25)(0.633) 2 = 0.05009 J KEf = ½ (0.25)(-0.183) 2 = 0.00419 J Blue Cart: Pi = 0.500 kg*0.00 m/s = 0.00 kg.m/s Pf = 0.500*(0.435) = 0.21750 kg.m/s KEi = ½ (0.50)(0.00) 2 = 0.00 J KEf = ½ (0.50)(0.435) 2 = 0.04731 J Both Carts: ∑Pi = 0.15825 + 0.00 = 0.15825 kg.m/s ∑Pf = – 0.04575 + 0.21750 = 0.17175 kg.m/s ∑KEi = 0.05009 + 0.00 = 0.05009 J ∑Kef = 0.00419 + 0.04731 = 0.05149 J % loss in momentum = 100*(0.17175 - 0.15825) / 0.17175 = 8.5 % % loss in KE = 100*(0.05149 - 0.05009) / 0.05149 = 2.8 % 13

Part 2 – Inelastic Collision: Both carts have Rubber Bumpers. Run No Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Percent Change in Momentum Percent Change in Kinetic Energy m U V m U m V ½ m U 2 ½ m V 2 100 |Δp| / pi 100 |ΔK| / Ki 13 Red 250 g 0.323 -0.053 0.08075 -0.01325 0.013041 0.000351 X X Blue 500 g 0.000 0.190 0.000 0.09500 0.000 0.009025 X X Both X X X 0.08075 0.08175 0.013041 0.009376 1.2 28.1 14

Part 4 – Explosion: Plunger in Blue Cart pushes both carts apart. Red Cart on left, Blue Cart on right. Plunger of Blue Cart pushes both Carts apart. Plunger in 1 st click. Cap. Run No. Cart Mass Initial Velocity (m/s) Final Velocity (m/s) Initial Momentum (kg · m/s) Final Momentum (kg · m/s) Initial Kinetic Energy (J) Final Kinetic Energy (J) Difference in Momentum (kg.m/s) Difference in Kinetic Energy (J) m U V m U m V ½ m U 2 ½ m V 2 ½ m U 2 ½ m V 2 19 Red 250 g 0.000 -0.356 0.000 -0.08900 0.000 0.015842 X X Blue 500 g 0.000 0.180 0.000 0.09000 0.000 0.008100 X X Both X X X 0.000 0.00100 0.000 0.023942 0.001 0.024 Fig: Capstone Graph for Run 19: Explosion. Red cart 250 g, Blue Cart 500 g. 16

RESULTS: Elastic Collision In-elastic Collision Perfectly Inelastic Collision Explosion Mass of Carts (kg) % change in Momentum % change in Kinetic Energy % change in Momentu m % change in Kinetic Energy % change in Momentum % change in Kinetic Energy change in Momentum change in Kinetic Energy Red 0.250 Blue 0.250 Red 0.250 8.5 2.8 1.2 28.1 2.3 65.1 0.001 0.024 Blue 0.500 Red 0.250 Blue 0.750 17