A particle of mass m moving at a velocity v strikes with the wall at an angle `alpha` with the wall and leaves ball making same angle with the wall. Choose the correct statemet (s) from the following.

Options

A. The impulsive reaction of wall is 2mV sin `alpha`
B. The impulsive reaction of wall is zero
C. The coefficient of restitution `ne 1`
D. There is no force by particle on the wall

Answer

To solve the problem, we need to analyze the collision of a particle with a wall at an angle and determine the impulse and the coefficient of restitution.

### Step-by-Step Solution:

1. **Understanding the Collision:**
- A particle of mass \( m \) is moving with velocity \( v \) and strikes a wall at an angle \( \alpha \).
- After the collision, the particle leaves the wall making the same angle \( \alpha \) with the wall.

2. **Components of Velocity:**
- Before the collision, the velocity components of the particle can be expressed as:
- \( v_x = v \cos \alpha \) (along the wall)
- \( v_y = v \sin \alpha \) (perpendicular to the wall)
- After the collision, the components of the velocity will be:
- \( v'_x = v' \cos \alpha \) (along the wall)
- \( v'_y = -v' \sin \alpha \) (since the direction changes)

3. **Conservation of Momentum:**
- Since there are no external forces acting along the line of impact, momentum is conserved.
- The initial momentum along the y-direction (perpendicular to the wall) is:
\[ p_{initial} = m v_y = m v \sin \alpha \]
- The final momentum along the y-direction after the collision is:
\[ p_{final} = -m v' \sin \alpha \]
- Setting initial momentum equal to final momentum:
\[ m v \sin \alpha = -m v' \sin \alpha \]
- This simplifies to:
\[ v' = -v \]

4. **Calculating Impulse:**
- Impulse is defined as the change in momentum.
- The change in momentum along the y-direction is:
\[ \Delta p = p_{final} - p_{initial} = -m v \sin \alpha - m v \sin \alpha = -2m v \sin \alpha \]
- The impulse reaction of the particle on the wall is:
\[ J = 2m v \sin \alpha \]

5. **Coefficient of Restitution:**
- The coefficient of restitution \( e \) is defined as the ratio of the velocity of separation to the velocity of approach.
- The velocity of separation (after collision) is:
\[ v_{separation} = -v' \sin \alpha = v \sin \alpha \]
- The velocity of approach (before collision) is:
\[ v_{approach} = v \sin \alpha \]
- Therefore, the coefficient of restitution is:
\[ e = \frac{v_{separation}}{v_{approach}} = \frac{v \sin \alpha}{v \sin \alpha} = 1 \]

6. **Conclusion:**
- The impulse reaction of the ball is \( 2mv \sin \alpha \).
- The coefficient of restitution is equal to 1.

### Final Answers:
- The correct statement is:
- The impulse reaction of the ball is \( 2mv \sin \alpha \) (Option 1 is correct).
- The coefficient of restitution is equal to 1 (Option 3 is incorrect).