A ball approaches a moving wall of infinite mass with speed v along the normal to the wall. The speed of the wall is u away from the ball and u

To analyze the scenario of a ball colliding elastically with a moving wall, we can apply the principles of conservation of momentum and kinetic energy. In this case, the wall has an infinite mass, which means it will not change its speed significantly due to the collision. Let's break down the problem step by step.

Understanding the Collision Dynamics

In an elastic collision, both momentum and kinetic energy are conserved. However, since the wall is moving away from the ball, we need to consider the relative velocities of both objects involved in the collision.

Initial Conditions

• The speed of the ball before the collision is \( v \) (towards the wall).
• The speed of the wall is \( u \) (moving away from the ball).

Since the wall has infinite mass, its velocity remains effectively constant, and we can treat it as a fixed reference point for our calculations.

Relative Velocity Concept

To find the speed of the ball after the collision, we first need to determine the relative velocity of the ball with respect to the wall. The relative velocity \( v_{rel} \) before the collision is given by:

Relative Velocity Before Collision: \( v_{rel} = v + u \)

This equation accounts for the fact that the ball is approaching the wall while the wall is moving away from it.

Calculating the Speed After Collision

In an elastic collision, the relative velocity after the collision is equal in magnitude and opposite in direction to the relative velocity before the collision. Therefore, we can express the speed of the ball after the collision \( v' \) as follows:

Relative Velocity After Collision: \( v'_{rel} = -(v_{rel}) = -(v + u) \)

Now, to find the actual speed of the ball after the collision, we need to convert this relative velocity back to the ground frame (the frame of reference where the wall is moving). The speed of the ball after the collision can be expressed as:

Speed of the Ball After Collision: \( v' = -v'_{rel} + u = (v + u) + u = v + 2u \)

Final Result

Thus, the speed of the ball after the elastic collision with the moving wall is:

Final Speed: \( v' = v + 2u \)

This result shows that the speed of the ball increases by twice the speed of the wall due to the elastic nature of the collision and the wall's motion. This example illustrates how the principles of momentum and energy conservation apply in a dynamic system involving moving objects.