A girl throws a ball with initial velocity u at an inclination of 45o. The ball strikes the smooth vertical wall at a horizontal distance d from girl and after rebounding returns to her hand. What is the coefficient of restitution between wall and ball?a)v2-gdb)gd/(v2-gd)c)gd/v2d)v2/gdCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

To solve this question, we can use the principles of projectile motion and the concept of coefficient of restitution.

Projectile Motion:
When an object is thrown at an angle, it follows a curved trajectory known as projectile motion. The horizontal and vertical components of motion can be analyzed separately.

Coefficient of Restitution:
The coefficient of restitution (e) is a measure of the elasticity of a collision between two objects. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach after the collision.

Given:
Initial velocity of the ball, u
Inclination angle, 45 degrees
Horizontal distance from the girl to the wall, d

1. Finding the Time of Flight:
Since the ball returns to the girl's hand, the total time of flight (T) would be twice the time taken to reach the wall.

Using the formula for time of flight in projectile motion:
T = 2 * (u * sinθ) / g

2. Finding the Horizontal Distance:
The horizontal distance (R) can be calculated using the formula:
R = u * cosθ * T

Substituting the value of T from step 1:
R = u * cosθ * 2 * (u * sinθ) / g
R = 2 * u^2 * sinθ * cosθ / g

3. Finding the Vertical Distance:
The vertical distance (H) can be calculated using the formula:
H = u^2 * sin^2θ / (2 * g)

4. Using the Coefficient of Restitution:
When the ball strikes the wall and rebounds, the vertical component of velocity changes direction while the horizontal component remains the same.

Since the ball returns to the girl's hand, the final velocity of the ball (v) would be equal to the initial velocity (u).

Using the formula for coefficient of restitution:
e = (v - u) / (u - (-u))

Substituting the value of v = u and simplifying:
e = (u - u) / (u + u)
e = 0

Thus, the coefficient of restitution between the wall and the ball is 0.
Hence, the correct answer is option B: gd / (v^2 - gd).

Answer

As the wall is smooth, the vertical component of the impulse it receives is zero. The total time of flight is given by

The time of flight t₁ from G to A =

The time of flight l₂ from A to G =

where e = coefficient of restitution.

Solving we get e =

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