Momentum Conservation in Collisions: Equal Mass Carts

[nietzsche]

Homework Statement

When two carts collide, is momentum conserved at each instant before, during and after the collision? If the carts are of equal mass what happens to their individual momenta during and after the collision?

Homework Equations

The Attempt at a Solution

I'm actually unsure of what the question means. "If the carts are of equal mass what happens to their individual momenta during and after the collision?" During the collision? I'm confused because we normally just consider before the collision and after the collision, leaving out the complex in-between stuff.

As for after the collision, I think the answer is this. If cart 1 is moving at a velocity v relative to cart 2, and cart 2 is at rest, then after the collision, cart 1 will have 0 final momentum relative to cart 2, and cart 2 will have the same initial momentum that cart 1 had. Please correct me if I'm wrong.

 

[kuruman]

Momentum is conserved at each instant before, during and after a collision. This is why,

1. Before and after a collision Newton's 1st Law holds, "An object will maintain its state of motion unless an unbalanced force acts on it." Each cart experiences no unbalanced force.

2. During the collision momentum is conserved because of Newton's 2nd and 3rd Laws. Cart 1 experiences a force due to Cart 2 and exerts an equal and opposite force on Cart 2 (Newton's 3rd Law.) Now from Newton's 2nd Law, momentum change (impulse) is force times time interval over which the force acts. Since the forces that the carts exert on one another are equal and opposite and act for the same time (as long as the carts are in contact), then the changes in momentum are equal and opposite. This means that their sum is zero, i.e. momentum is conserved. The masses of the carts are irrelevant.

 

[nietzsche]

Thank you.

So does this necessarily mean that cart 1 will end up with cart 2's initial velocity, and cart 2 will end up with cart 1's initial velocity?

 

[kuruman]

No it does not necessarily mean that. The two carts will exchange velocities if (a) the masses are equal and (b) the collision is perfectly elastic.

 

[nietzsche]

Ah thank you. Makes sense.