A ball with initial momentum P collides with rigid wall elastically. If `P^(1)` be it's momentum after collision then

Options

A. `P^(1)=P`
B. `P^(1)=-P`
C. `P^(1)=2P`
D. `P^(1)=-2P`

Answer

To solve the problem of a ball colliding elastically with a rigid wall, we can follow these steps:

### Step 1: Understand the Initial Conditions
The ball has an initial momentum \( P \). We can express this in terms of mass \( m \) and initial velocity \( u_1 \):
\[ P = m \cdot u_1 \]

### Step 2: Identify the Type of Collision
The collision is elastic, which means both momentum and kinetic energy are conserved. The coefficient of restitution \( e \) for an elastic collision is equal to 1.

### Step 3: Apply the Coefficient of Restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
In this case, the wall is rigid and stationary, so its velocity \( u_2 = 0 \). The velocities involved are:
- Velocity of approach: \( u_1 - u_2 = u_1 - 0 = u_1 \)
- Velocity of separation: \( v_2 - v_1 \) (where \( v_1 \) is the velocity of the ball after the collision and \( v_2 = 0 \) for the wall)

Thus, we have:
\[ e = \frac{0 - v_1}{u_1 - 0} = \frac{-v_1}{u_1} \]
Since \( e = 1 \):
\[ 1 = \frac{-v_1}{u_1} \]

### Step 4: Solve for the Final Velocity
From the equation above, we can rearrange to find \( v_1 \):
\[ -v_1 = u_1 \implies v_1 = -u_1 \]

### Step 5: Calculate the Momentum After Collision
Now, we can calculate the momentum after the collision \( P_1 \):
\[ P_1 = m \cdot v_1 \]
Substituting \( v_1 \):
\[ P_1 = m \cdot (-u_1) = -m \cdot u_1 \]
Since \( P = m \cdot u_1 \), we can substitute:
\[ P_1 = -P \]

### Final Answer
The momentum of the ball after the collision is:
\[ P_1 = -P \]

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