To test if momentum is the product of an object's mass and its velocity, if impulse is a change in momentum, and if it is a conserved quantity, we studied the motion of two near frictionless carts on a track as they interacted with each other. If momentum is indeed conserved and only changes when an impulse acts on the system, its momentum should be able to be represented by a bar chart with the initial momentum, final, momentum, and impulse. The initial momentum and the impulse should add up to be the final momentum.
Bouncy, One Moving, One Stationary
For the first test, one cart rolls at a constant velocity toward a second stationary cart. The carts have magnets that repel each other, so the carts 'bounce' off of each other with practically no loss of kinetic energy.
As a result, the red cart transfers its velocity to the blue cart as seen in the graphs:
Blue Solid: Position (m) vs. Time (s) - Blue Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Blue Dashed: Velocity (m/s) vs. Time (s) - Blue Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Blue Dotted: Momentum (kg*m/s) vs. Time (s) - Blue Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Green Solid: Momentum (kg*m/s) vs. Time (s) - Both Carts
Both carts have the same mass of 252 grams, so if momentum is the product of mass and velocity, this transfer of velocity is really a transfer of momentum. This makes sense because the blue cart's velocity after the collision is nearly identical to the red cart's velocity before the collision. The carts' behavior also points to the conclusion that momentum is conserved. While the red cart's momentum decreases to zero, the blue cart's momentum increases to nearly match that of the red cart before the collision.
Red cart: 0.66m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0.166kg*m/s
Red cart: 0m/s * 0.252kg + Blue Cart: 0.60m/s * 0.252kg = Total Final Momentum: 0.151kg*m/s
(0.151 - 0.166)/0.166 = 9.1% error
Because the same amount of mass is moving the same velocity before and after the collision, the total momentum of the two-cart system should remain the same. The 9.1% difference between the two values is most likely because of an impulse created by friction as the carts roll. Because there was no impulse from outside of the system, this points to the conclusion that momentum is conserved in a system. This means the momentum of the system can be represented with a bar chart consisting of each cart's momentum before and after the collision as well as any impulse.
Red cart: 0.66m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0.166kg*m/s
Red cart: 0m/s * 0.252kg + Blue Cart: 0.60m/s * 0.252kg = Total Final Momentum: 0.151kg*m/s
(0.151 - 0.166)/0.166 = 9.1% error
Because the same amount of mass is moving the same velocity before and after the collision, the total momentum of the two-cart system should remain the same. The 9.1% difference between the two values is most likely because of an impulse created by friction as the carts roll. Because there was no impulse from outside of the system, this points to the conclusion that momentum is conserved in a system. This means the momentum of the system can be represented with a bar chart consisting of each cart's momentum before and after the collision as well as any impulse.
This conservation of momentum means that the initial momentum should be equal to the final momentum of the system (p_i = p_f), which it nearly is. This data also points to the fact that impulse is a change in momentum. While the total momentum of the system did not change, the momentum in each cart did. Because impulse is a force over a period of time and this force acted on the two carts, the carts form a Newton's Third Law pair. As this force causes the red cart to decelerate from 0.66m/s to 0m/s, it also causes the blue cart to accelerate from 0m/s to 0.60m/s over the same period of time. The impulse changes the momentum of each cart nearly the same amount.
Bouncy, Both Moving
For the second test, we rolled the two carts toward each other using the same magnets to bounce off of each other.
Blue Solid: Position (m) vs. Time (s) - Blue Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Blue Dashed: Velocity (m/s) vs. Time (s) - Blue Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Blue Dotted: Momentum (kg*m/s) vs. Time (s) - Blue Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Green Solid: Momentum (kg*m/s) vs. Time (s) - Both Carts
Though both carts change velocity, momentum is still conserved. Initially, the red cart travels at 0.33m/s and the blue cart -0.38m/s, and after the collision travel at -0.35m/s and 0.29m/s respectively. Note that the speed of the red cart increases after the collision while the speed of the blue cart decreases. This is because, similarly to the first experiment, some of the blue cart's momentum was transferred to the red cart via an impulse. However, the total momentum of the system remains nearly the same.
Red cart: 0.33m/s * 0.252kg + Blue Cart: -0.38m/s * 0.252kg = Total Initial Momentum: -0.013kg*m/s
Red cart: -0.35m/s * 0.252kg + Blue Cart: 0.29m/s * 0.252kg = Total Final Momentum: -0.015kg*m/s
(-0.013 - (-0.015))/-0.013 = 15.4% error
This increase in error is most likely because both carts are moving throughout the experiment so friction will have a greater impact on the results. The values before and after the collision are still nearly identical, suggesting that the momentum of the two-cart system was conserved. The momentum of the system can also be modeled with a similar bar chart:
Red cart: 0.33m/s * 0.252kg + Blue Cart: -0.38m/s * 0.252kg = Total Initial Momentum: -0.013kg*m/s
Red cart: -0.35m/s * 0.252kg + Blue Cart: 0.29m/s * 0.252kg = Total Final Momentum: -0.015kg*m/s
(-0.013 - (-0.015))/-0.013 = 15.4% error
This increase in error is most likely because both carts are moving throughout the experiment so friction will have a greater impact on the results. The values before and after the collision are still nearly identical, suggesting that the momentum of the two-cart system was conserved. The momentum of the system can also be modeled with a similar bar chart:
There is no impulse on the two cart system, so the total momentum remains nearly the same. The two cart's momentum nearly cancel out as the carts were rolled toward each other at similar speeds. This means that the total momentum of the system is low because the velocities average to nearly zero. This fits with the definition of momentum as a product of mass and velocity of the system.
Sticky, One Moving, One Stationary
For the third test, we rolled one cart at a constant velocity toward a stationary cart. However, instead of bouncing off of each other with magnets, the carts have velcro which makes then stick together. As a result, the two carts will have the same movement after they collide.
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Blue Solid: Position (m) vs. Time (s) - Blue Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Blue Dashed: Velocity (m/s) vs. Time (s) - Blue Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Blue Dotted: Momentum (kg*m/s) vs. Time (s) - Blue Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Green Solid: Momentum (kg*m/s) vs. Time (s) - Both Carts
The motion of these carts provides strong evidence that momentum is the product of mass and velocity. The red cart has a velocity of 0.43m/s initially, but slows to 0.20m/s after the collision. This is because the mass of the red cart essentially doubles when it collides with the blue cart. Because the momentum of the system remains constant and mass and velocity are inversely proportional, the velocity will be halved when the mass doubles. The carts move at slightly less than half the red cart's initial velocity because some of its kinetic energy was converted into sound as the two carts collided.
Red cart: 0.43m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0.11kg*m/s
Red cart: 0.20m/s * 0.252kg + Blue Cart: 0.20m/s * 0.252kg = Total Final Momentum: 0.10kg*m/s
(0.10 - 0.11)/0.11 = 9.1% error
Red cart: 0.43m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0.11kg*m/s
Red cart: 0.20m/s * 0.252kg + Blue Cart: 0.20m/s * 0.252kg = Total Final Momentum: 0.10kg*m/s
(0.10 - 0.11)/0.11 = 9.1% error
Momentum is conserved throughout this collision, as the red cart's momentum was distributed to the blue cart as well. However, the total momentum remains nearly the same because the carts together at about half the red cart's initial velocity.
Sticky, Both Moving
For the fourth test, we rolled the two carts with velcro toward each other at a constant velocity. When they collide, the carts stick together and their motion is the same.
Blue Solid: Position (m) vs. Time (s) - Blue Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Blue Dashed: Velocity (m/s) vs. Time (s) - Blue Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Blue Dotted: Momentum (kg*m/s) vs. Time (s) - Blue Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Green Solid: Momentum (kg*m/s) vs. Time (s) - Both Carts
Similar to the previous test, the velocity of the carts changes as their masses combine. Before the collision, the red cart rolls at 0.18m/s and the blue cart at -0.72m/s. Adding the two cart's velocities yields -0.54m/s. After the carts collide, their velocity is -0.26m/s, about half of their added velocity before the collision. This is because the mass of the each cart essentially doubles as they collide, as momentum is the product of an object's mass and its velocity.
Red cart: 0.18m/s * 0.252kg + Blue Cart: -0.72m/s * 0.252kg = Total Initial Momentum: -0.14kg*m/s
Red cart: -0.26m/s * 0.252kg + Blue Cart: -0.26m/s * 0.252kg = Total Final Momentum: -0.13kg*m/s
(-0.13 - (-0.14))/-0.13 = 7.7% error
This error is likely the result of some of the carts' kinetic energy being converted into sound during the collision. There is no impulse outside of the two cart system, so the total momentum is conserved. Though the velocities of both carts change during the collision, the values before and after the collision add to nearly the same velocity. Because the two carts together have the same mass throughout the experiment, the total momentum of the system remains the same.
Red cart: 0.18m/s * 0.252kg + Blue Cart: -0.72m/s * 0.252kg = Total Initial Momentum: -0.14kg*m/s
Red cart: -0.26m/s * 0.252kg + Blue Cart: -0.26m/s * 0.252kg = Total Final Momentum: -0.13kg*m/s
(-0.13 - (-0.14))/-0.13 = 7.7% error
This error is likely the result of some of the carts' kinetic energy being converted into sound during the collision. There is no impulse outside of the two cart system, so the total momentum is conserved. Though the velocities of both carts change during the collision, the values before and after the collision add to nearly the same velocity. Because the two carts together have the same mass throughout the experiment, the total momentum of the system remains the same.
Explosion
For the fifth test the two stationary carts were placed together near the center of the track. At the one second mark we released a spring between the two carts which send them rolling in opposite directions.
Blue Solid: Position (m) vs. Time (s) - Blue Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Red Solid: Position (m) vs. Time (s) - Red Cart
Blue Dashed: Velocity (m/s) vs. Time (s) - Blue Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Red Dashed: Velocity (m/s) vs. Time (s) - Red Cart
Blue Dotted: Momentum (kg*m/s) vs. Time (s) - Blue Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Red Dotted: Momentum (kg*m/s) vs. Time (s) - Red Cart
Green Solid: Momentum (kg*m/s) vs. Time (s) - Both Carts
While a force acted on the two carts, it was only between the two carts and a part of a system. The two carts did not gain or lose momentum as their equal and opposite velocities add to zero. The carts move at the same speed because they form an Newton's Third Law pair, so the same force pushes each cart. Because the carts have the same mass, they both accelerate in opposite directions at the same rate for the same amount of time, making the final velocities equal but opposite. Because these carts have the same mass and opposite velocities, their net momentum is zero, just as they were when stationary. This points to the conclusion that the total momentum in a system is conserved.
Red cart: 0m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0kg*m/s
Red cart: -0.77m/s * 0.252kg + Blue Cart: 0.77m/s * 0.252kg = Total Final Momentum: 0kg*m/s
(0 - 0)/0 = 0% error
Red cart: 0m/s * 0.252kg + Blue Cart: 0m/s * 0.252kg = Total Initial Momentum: 0kg*m/s
Red cart: -0.77m/s * 0.252kg + Blue Cart: 0.77m/s * 0.252kg = Total Final Momentum: 0kg*m/s
(0 - 0)/0 = 0% error
There is a zero percent error between the initial and final total momentums of the carts because friction and air resistance act on both carts the same way, as their motion is the same but in opposite directions. As shown in the bar chart, the momentum of each cart changes, but because the carts move in opposite directions, the net momentum remains zero.
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