Projectile motion is the motion of an object in the air that is subject to only the force of gravity and air resistance. In this scenario, a ball is projected with a speed v at an angle q to the horizontal towards a vertical smooth wall at distance d from the point of projection. The ball will follow a parabolic path until it hits the wall.



The coefficient of restitution is a measure of the "bounciness" of an object. It is defined as the ratio of the final velocity to the initial velocity of an object after a collision. In this scenario, the ball collides with the wall and then returns to the point of projection. Therefore, the coefficient of restitution can be calculated as follows:

Coefficient of Restitution = Final Velocity / Initial Velocity

However, we need to determine the final and initial velocities of the ball.



When the ball collides with the wall, it experiences a change in momentum. The change in momentum is equal to the impulse applied to the ball during the collision. Since the wall is smooth, there is no friction between the ball and the wall. Therefore, the total momentum of the system (ball and wall) is conserved.

Initial momentum of the system = mv

Final momentum of the system = -mv

Where m is the mass of the ball and v is the velocity of the ball just before the collision.

Since the total momentum of the system is conserved, we can use this to determine the final velocity of the ball after the collision.

Initial momentum of the system = Final momentum of the system

mv = -mvf

vf = -v/2

Therefore, the final velocity of the ball after the collision is -v/2.

The initial velocity of the ball is given by:

vx = v cos(q)
vy = v sin(q)

Where vx is the horizontal component of the velocity and vy is the vertical component of the velocity.

After the collision, the ball will travel back along the same path that it came from. Therefore, the horizontal component of the velocity will remain the same, while the vertical component will be reversed in direction.

vx = v cos(q)
vy' = -v sin(q)

Therefore, the initial velocity of the ball after the collision is:

vi = vx + vy' = v cos(q) - v sin(q)



Using the formula for the coefficient of restitution, we can now calculate its value:

Coefficient of Restitution = Final Velocity / Initial Velocity

Coefficient of Restitution = (-v/2) / (v cos(q) - v sin(q))

Coefficient of Restitution = -1 / (2 cos(q) - 2 sin(q))

Therefore, the coefficient of restitution depends on the angle q at which the ball is projected. For a given initial speed v, the coefficient of restitution will be maximum when q = 45 degrees.

Answer is 1.