Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1210001     Different systems for colliding carts on a track     1210001
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Question

 

Cart A has a speed of 1.5 m/s in the +x direction and Cart B is at rest when they collide. After the collision, Cart A rebounds in the –x direction with a speed of 0.75 m/s, and Cart B travels in the +x direction with a speed of 0.75 m/s. The mass of Cart A is 0.50 kg, and the mass of Cart B is 1.5 kg.

  1. Define Cart A to be the system. What is the change in momentum of the system due to the collision?
  2. Define Cart B to be the system. What is the change in momentum of the system due to the collision?
  3. Define Cart A and Cart B, together, to be the system. What is the change in momentum of the system due to the collision?

 



Solution

 

(a) Begin by drawing a picture of the situation.


Figure: The momentum of Carts A and Be before and after the collision.

Define the system. In this case, the system is Cart A. The initial momentum of Cart A is

The final momentum of Cart A is

To find the change in momentum of Cart A, begin by sketch a picture, with its initial momentum and final momentum vectors drawn tail to tail and the change in momentum drawn from the head of the initial momentum to the head of the final momentum, as shown below.


Figure: The change in momentum of Cart A.

The change in momentum can also be calculated algebraically.

(The result was rounded to two significant figures.) Note that the change in the momentum of Cart A is to the left, which is consistent with the picture.

(b) Now, define the system to be Cart B and calculate the change in momentum of the system. The initial momentum of Cart B is zero, therefore the change in momentum of Cart B is equal to its final momentum.

Check the answer by sketching the change in momentum of Cart B.


Figure: The change in momentum of Cart B.

Note that the change in momentum is in the +x direction which is consistent with the calculation.

(c) Now, define the system to be Carts A and B together. The total initial momentum of the system is the sum of the initial momentum of Cart A and Cart B.

The total final momentum is the sum of the final momentum of Cart A and Cart B.

The change in momentum of the system is

Note that the change in momentum of the system (including Cart A and Cart B) is zero! Though the collision took place and Cart A slowed down and Cart B sped up, the change in momentum of the system of both carts is zero.

This can be computed another way. The change in momentum of the system consisting of both carts is the sum of the change in momentum of Cart A and the change in momentum of Cart B. That is,

Though each cart's momentum changed, the sum of their changes in momentum is zero. Cart B's positive change in momentum is exactly balanced by Cart A's negative change in momentum so that the sum is zero.