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Conservation of Momentum in 1-D & 2-D Systems

Instructor: Matthew Bergstresser
Momentum is the resistance to the change in velocity, and momentum is always conserved in any collision or explosion. This lesson will explore the conservation of momentum for multiple masses colliding in one and two dimensions, and for a single mass exploding.

Conservation of Momentum

Momentum is mass times velocity (mv), and is always conserved when objects collide as long as there is no external force(s) acting on the system. In multiple-mass systems, the center-of-mass is the location where the total mass of the system can be thought of being concentrated. The center-of-mass velocity is very useful because this velocity never changes even after a collision occurs, and can be calculated by dividing the total momentum by the total mass of the system.


This equation is for two masses. It can be expanded for additional masses.
v_CM

We can determine the final velocities of the masses involved in the collision by following a few steps which involve leaving the original frame of reference, entering the center-of-mass reference frame, and then converting back to the original frame of reference. The procedure depends on whether the objects collide and bounce off of each other or if they collide and stick together.

For scenarios where the masses bounce off of each other:

1. Determine the vCM.

2. Determine the individual mass's velocities relative to the vCM by subtracting the vCM from each original velocity. This is changing our frame of reference to the center-of-mass reference frame.

3. Reverse the signs of the center-of-mass reference-frame velocities after the masses collide.

4. Add the vCM to each of the new velocities determined in step 3. Now we are back in the original reference frame with the final velocities of the masses.

For scenarios where the masses stick together after the collision:

When masses collide and stick together, the center-of-mass is located in the new clump of combined masses. This means that the vCM is the final velocity of the new combined-mass.

Exploding masses:

A mass exploding can be thought of as the opposite of a collision where the masses colliding all stick together. You start with one mass that breaks apart into multiple masses, each with their own velocities, after the explosion.

We will go through three examples involving the conservation of momentum; a 1-D collision, a 2-D collision, and an explosion scenario.

1-D Collision

Prompt:

A 5 kg mass sliding along a frictionless air track at 3 m/s collides with a 4 kg mass sliding at 5 m/s in the opposite direction. They bounce off of each other after the collision. What is the velocity of each mass after the collision?

Solution:

We will follow the steps for a collision where the masses do not stick together.

Step 1: Calculating the center-of-mass velocity.


prob1_step1

Step 2: Converting velocities to center-of-mass reference frame.


prob1_step2

Step 3: Switching the signs of the center-of-mass reference frame velocities.


prob1_step3

Step 4: Converting final velocities back to original reference frame.


prob_1_step4

Notice that the masses are moving in the opposite direction compared to their original directions.

2-D Collision

Prompt:

A 2 kg mass moving at 4 m/s, at 40o collides with, and sticks to a 1 kg mass moving at 2 m/s, at 270o. What is the final velocity of the combined masses?


problem2_diagram

Solution:

The first thing we need to do is to put the initial velocities into unit-vector notation.


problem_2_velocities

Now we can use the center-of-mass approach to solve this problem.

Step 1:


prob_2_soln

We don't need to go beyond step 1 because the center of mass velocity does not change, and since the masses are stuck together their final velocity is the vCM.

Explosion

Prompt:

A 1 kg firecracker is launched straight up at 15 m/s. 5 seconds into its flight it is traveling at 10 m/s and explodes into three pieces of equal mass. Two parts move horizontally and one part moves vertically. One of the pieces moving horizontally was measured to have an initial velocity of 20 m/s to the right. What are the velocities of the other two pieces right after the explosion?

Solution:

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