In collisions which involve a larger body and a smaller body momentum conservation doesn't hold good (although it is conserved) because of the much higher mass of larger body. Here the concept of coefficient of restitution can be applied on the assumption that the velocity of the larger body (here wall surface) doesn't change after collision. If a ball falls freely with velocity u strikes a wall or ground then its velocity after collision will be v = eu. This is the case if it falls normally. If it falls obliquely we have to resolve the vectors. Remember that in any collision velocities change only change along common normal (CN) and in any other direction velocities remain unchanged.

Explanation:

When a ball strikes a wall with a velocity u at an angle a and rebounds from it at an angle b, the coefficient of restitution can be determined. The coefficient of restitution, denoted by e, is a measure of the elasticity of a collision. It represents the ratio of the relative velocity of separation to the relative velocity of approach between two objects after a collision. In this case, the two objects are the ball and the wall.

Key Points:

1. Coefficient of restitution (e) is defined as the ratio of the final relative velocity of separation (v_s) to the initial relative velocity of approach (v_a):

e = v_s / v_a

2. Since the wall is smooth, there is no friction or rotation involved in the collision. Therefore, the tangential component of velocity remains unchanged.

3. The normal component of velocity changes direction due to the rebound. The angle of rebound (b) can be determined using the angle of incidence (a) and the law of reflection.

4. The law of reflection states that the angle of incidence is equal to the angle of reflection. Therefore, a = b.

5. The coefficient of restitution can be calculated by considering the change in normal velocity. As the wall is smooth, the normal velocity before and after the collision remains the same.

6. The initial normal velocity (v_n1) can be determined using the initial velocity (u) and the angle of incidence (a):

v_n1 = u * sin(a)

7. The final normal velocity (v_n2) can be determined using the angle of rebound (b):

v_n2 = u * sin(b)

8. Since a = b, the final normal velocity can also be expressed as:

v_n2 = u * sin(a)

9. Since the normal velocity remains the same before and after the collision, the coefficient of restitution can be calculated as:

e = v_n2 / v_n1 = (u * sin(a)) / (u * sin(a)) = 1

Conclusion:

In the given scenario, where a ball strikes a smooth wall and rebounds from it at an angle equal to the angle of incidence, the coefficient of restitution is equal to 1. This indicates a perfectly elastic collision, where the relative velocity of separation is equal to the relative velocity of approach.