Introduction:
In this scenario, a ball of mass m collides perpendicularly with a smooth stationary wedge of mass M, which is placed on a smooth horizontal plane. We are given the coefficient of restitution (e), and we need to determine the velocity of the wedge after the collision.

Understanding the Problem:
To solve this problem, we need to analyze the collision between the ball and the wedge. The coefficient of restitution (e) is a measure of how elastic the collision is. It represents the ratio of the final relative velocity of separation to the initial relative velocity of approach.

Applying Conservation of Momentum:
To determine the velocity of the wedge after the collision, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.

Initial Momentum:
The initial momentum of the system can be calculated as the sum of the momentum of the ball and the momentum of the wedge before the collision. Since the wedge is stationary, its initial momentum is zero. The initial momentum of the ball can be calculated using the equation:

Initial momentum of the ball = mass of the ball (m) * velocity of the ball before the collision (u)

Final Momentum:
After the collision, the ball will rebound off the wedge and move in the opposite direction. The wedge will acquire a certain velocity in the direction opposite to the ball's initial velocity. Let's denote the final velocity of the ball as v and the final velocity of the wedge as V.

The final momentum of the system can be calculated as the sum of the momentum of the ball and the momentum of the wedge after the collision.

Final momentum of the system = (mass of the ball * velocity of the ball after the collision) + (mass of the wedge * velocity of the wedge after the collision)

Applying Coefficient of Restitution:
The coefficient of restitution (e) can be used to relate the velocities of separation and approach. According to the definition of the coefficient of restitution, we have:

e = (velocity of separation) / (velocity of approach)

For a collision perpendicular to the surface, the velocity of approach is the initial velocity of the ball (u), and the velocity of separation is the final velocity of the ball (v) minus the final velocity of the wedge (V).

e = (v - V) / u

Simplifying this equation, we can express the final velocity of the ball (v) in terms of the initial velocity of the ball (u), the final velocity of the wedge (V), and the coefficient of restitution (e):

v = e * u + V

Using Conservation of Momentum:
Now, we can substitute the values of the initial and final momenta into the conservation of momentum equation:

m * u = m * v + M * V

Substituting the expression for v derived from the coefficient of restitution equation:

m * u = m * (e * u + V) + M * V

Simplifying this equation will give us the value of V, the velocity of the wedge after the collision.

Conclusion:
By applying the principles of conservation of momentum and using the equation derived from the coefficient of restitution, we can determine the velocity of the wedge after the collision. The final equation obtained can be solved to find