Question

A ball of mass m approaches a moving wall of infinite mass with a speed 'v' along the normal to the wall. The speed of the wall is 'u' toward the ball. The speed of the ball after 'elastic' collision with wall is:

• A. u + v away from the wall
• B. 2u + v away from the wall
• C. | u- v | away from the wall
• D. | v-2u | away from the wall

Answer 1

Answer: (B)

2u + v away from the wall

Answer 2

The problem describes an elastic collision between a ball of mass 'm' and a wall of infinite mass.

Let's define a coordinate system. Let the direction away from the wall be positive.

1. Initial Velocities:

   • The ball approaches the wall with speed 'v'. So, its initial velocity is $v_{ball,i} = -v$ (negative sign indicates it's moving towards the wall).
   • The wall moves towards the ball with speed 'u'. So, its initial velocity is $v_{wall,i} = u$ (positive sign indicates it's moving in the positive direction, i.e., towards the ball from the perspective of the ball being on the negative side).

2. Final Velocities:

   • Since the wall has infinite mass, its velocity does not change during the collision. Therefore, its final velocity is $v_{wall,f} = v_{wall,i} = u$.
   • Let the final velocity of the ball be $v_{ball,f}$.

3. Coefficient of Restitution (e): For an elastic collision, the coefficient of restitution $e=1$. The formula for 'e' is: $e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}}$ $e = \frac{v_{ball,f} - v_{wall,f}}{v_{wall,i} - v_{ball,i}}$

4. Substitute and Solve: Substitute the known values into the equation: $1 = \frac{v_{ball,f} - u}{u - (-v)}$ $1 = \frac{v_{ball,f} - u}{u + v}$

   Now, solve for $v_{ball,f}$: $u + v = v_{ball,f} - u$ $v_{ball,f} = u + v + u$ $v_{ball,f} = 2u + v$

5. Interpret the Result: Since $v_{ball,f}$ is positive ($u$ and $v$ are speeds, hence positive), the ball's final velocity is in the positive direction, which means it is moving away from the wall. The speed of the ball after the collision is the magnitude of its final velocity, which is $|2u + v| = 2u + v$.

Therefore, the speed of the ball after the elastic collision with the wall is $2u + v$ away from the wall.