A ball of mass m approaches a heavy wall of mass M with speed 4 m/s along the normal to the wall. The speed of wall before collision is 1m/s towards the ball. The ball collides elastically with the wall. What can you say about the speed of the ball after collision? Will it be slightly less than or slightly higher than 6 m/s ?

Answer

To solve the problem, we need to analyze the situation using the principles of elastic collision and the concept of relative velocity.

### Step-by-Step Solution:

1. **Identify the Initial Velocities**:
- The ball has a mass \( m \) and approaches the wall with a speed of \( 4 \, \text{m/s} \) towards the wall.
- The wall has a mass \( M \) and moves towards the ball with a speed of \( 1 \, \text{m/s} \).

2. **Define the Direction**:
- Let's assume the direction towards the wall is positive. Therefore, the initial velocities can be defined as:
- Velocity of the ball, \( v_b = 4 \, \text{m/s} \) (towards the wall)
- Velocity of the wall, \( v_w = -1 \, \text{m/s} \) (towards the ball)

3. **Calculate the Relative Velocity of Approach**:
- The relative velocity of approach is given by:
\[ v_{\text{approach}} = v_b - v_w = 4 - (-1) = 4 + 1 = 5 \, \text{m/s} \]

4. **Apply the Elastic Collision Principles**:
- For an elastic collision, the relative velocity of separation after the collision equals the relative velocity of approach before the collision.
- The formula for the relative velocity of separation is:
\[ v_{\text{separation}} = v'_{b} - v'_{w} \]
- Here, \( v'_{b} \) is the final velocity of the ball and \( v'_{w} \) is the final velocity of the wall.

5. **Assume the Wall is Fixed (Infinite Mass)**:
- Since the wall is much heavier than the ball, we can approximate its mass as infinite. Thus, the wall's final velocity remains approximately \( 1 \, \text{m/s} \) after the collision.
- Therefore, \( v'_{w} \approx -1 \, \text{m/s} \).

6. **Calculate the Final Velocity of the Ball**:
- Now, substituting into the relative velocity of separation:
\[ v'_{b} - (-1) = 5 \]
\[ v'_{b} + 1 = 5 \]
\[ v'_{b} = 5 - 1 = 4 \, \text{m/s} \]

7. **Determine the Final Speed of the Ball**:
- The ball's final speed after the collision will be:
\[ v'_{b} = 4 \, \text{m/s} \]
- However, since the wall is moving towards the ball, the ball will rebound with a speed greater than \( 4 \, \text{m/s} \) due to the wall's motion.

8. **Final Calculation**:
- The final speed of the ball after the collision will be:
\[ v'_{b} = 4 + 1 = 5 \, \text{m/s} \]
- The speed of the ball after the collision will be slightly less than \( 6 \, \text{m/s} \) because the wall is moving towards the ball.

### Conclusion:
The speed of the ball after the collision will be slightly less than \( 6 \, \text{m/s} \).