# Elastic Collision: Momentum and Kinetic Energy Conservation

## What happens during an elastic collision between two carts on an air track?

In an elastic collision between two carts on an air track, both momentum and kinetic energy are conserved. What are the implications of this conservation law?

## Answer:

During an elastic collision between Cart A (mass m) and Cart B (mass 2m) on an air track, the following scenarios occur:

- Cart A moves towards Cart B at a speed +v until they collide elastically.
- After the collision, Cart A continues moving towards Cart B at the same speed, while Cart B remains at rest.
- Both momentum and kinetic energy are conserved during the collision.

## Analysis of the Scenario:

Let's analyze the conservation of momentum and kinetic energy in the given elastic collision scenario:

## Initial Momentum:

The initial momentum of Cart A is given by: p_{Ai} = m * v (since it has mass m and velocity v).

The initial momentum of Cart B is zero since it is at rest: p_{Bi} = 0.

## Final Momentum:

After the collision, Cart A moves with velocity v_{Af} and Cart B moves with velocity v_{Bf}.

The final momentum of Cart A is given by: p_{Af} = m * v_{Af}.

The final momentum of Cart B is given by: p_{Bf} = 2m * v_{Bf} (due to its mass being 2m).

## Conservation of Momentum:

Since momentum is conserved, the initial and final momenta must be equal:

m * v = m * v_{Af} (equation 1)

0 = 2m * v_{Bf} (equation 2)

From equation 2, Cart B remains at rest (v_{Bf} = 0) after the collision.

From equation 1, Cart A continues moving with the same speed v after the collision.

## Conservation of Kinetic Energy:

Since the collision is elastic, kinetic energy is conserved:

The initial kinetic energy (KE_{i}) is given by: KE_{i} = (1/2) * m * v^2.

The final kinetic energy (KE_{f}) is given by: KE_{f} = (1/2) * m * v_{Af}^2.

By equating KE_{i} to KE_{f}, we find that both carts have the same speed v after the elastic collision.

Therefore, Cart A continues moving towards Cart B at the same speed, while Cart B remains at rest.