A ball moving with a velocity `v` hits a massive wall moving towards the ball with a velocity a. An elastic impact lasts for time `/_\t`

Options

A. The average elastic force acting on the ball is `[m(u+v)]///_\t`
B. The average elastic force acting on the ball is `[2m(u+v)\]///_\t`
C. The kinetic energy of the ball increases by `2"mu" (u + v)`.
D. The kinetic energy of the ball remains the same after the collision.

Explanation

To solve the problem, we need to analyze the elastic collision between a ball and a massive wall. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a ball moving with velocity \( v \) towards a massive wall that is moving towards the ball with velocity \( a \). We need to find the average elastic force acting on the ball during the collision and the change in kinetic energy.

### Step 2: Determine the Relative Velocity Before Collision
The relative velocity of the ball with respect to the wall before the collision is given by:
\[ v_{\text{relative}} = v + a \]
This is because both the ball and the wall are moving towards each other.

### Step 3: Determine the Velocity After Collision
For an elastic collision, the relative velocity after the collision is equal in magnitude and opposite in direction to the relative velocity before the collision. Thus, we can write:
\[ v_{\text{relative after}} = -(v + a) \]
Let \( x \) be the velocity of the ball after the collision. Then:
\[ x - a = -(v + a) \]
Solving for \( x \):
\[ x - a = -v - a \implies x = v + 2a \]

### Step 4: Calculate Change in Momentum
The momentum of the ball before the collision is:
\[ p_{\text{before}} = mv \]
The momentum of the ball after the collision is:
\[ p_{\text{after}} = m(v + 2a) \]
The change in momentum (\( \Delta p \)) is:
\[ \Delta p = p_{\text{after}} - p_{\text{before}} = m(v + 2a) - mv = 2ma \]

### Step 5: Calculate the Average Force
The average force (\( F \)) acting on the ball during the collision can be calculated using the formula:
\[ F = \frac{\Delta p}{\Delta t} \]
Substituting the values we found:
\[ F = \frac{2ma}{\Delta t} \]

### Step 6: Calculate Kinetic Energy Before and After Collision
The initial kinetic energy (\( KE_{\text{initial}} \)) of the ball is:
\[ KE_{\text{initial}} = \frac{1}{2} mv^2 \]
The final kinetic energy (\( KE_{\text{final}} \)) after the collision is:
\[ KE_{\text{final}} = \frac{1}{2} m(v + 2a)^2 \]
Expanding this:
\[ KE_{\text{final}} = \frac{1}{2} m(v^2 + 4av + 4a^2) \]

### Step 7: Calculate Change in Kinetic Energy
The change in kinetic energy (\( \Delta KE \)) is:
\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = \frac{1}{2} m(v^2 + 4av + 4a^2) - \frac{1}{2} mv^2 \]
Simplifying this:
\[ \Delta KE = \frac{1}{2} m(4av + 4a^2) = 2m(a(v + a)) \]

### Conclusion
- The average elastic force acting on the ball is:
\[ F = \frac{2ma}{\Delta t} \]
- The change in kinetic energy is:
\[ \Delta KE = 2m(a(v + a)) \]