A ball is projected from ground with a velocity `V` at an angle `theta` to the vertical. On its path it makes an elastic collison with a vertical wall and returns to ground. The total time of flight of the ball is

Options

A. `(2vsintheta)/(g)`
B. `(2vcostheta)/(g)`
C. `(vsin2theta)/(g)`
D. `(vcostheta)/(g)`

Explanation

To solve the problem of finding the total time of flight of a ball projected from the ground with a velocity \( V \) at an angle \( \theta \) to the vertical, we can follow these steps:

### Step 1: Understand the Components of Velocity
When the ball is projected at an angle \( \theta \) to the vertical, we can resolve the initial velocity \( V \) into two components:
- The vertical component \( V_y = V \cos \theta \)
- The horizontal component \( V_x = V \sin \theta \)

### Step 2: Analyze the Motion
The motion of the ball can be analyzed in two dimensions:
- The vertical motion is influenced by gravity.
- The horizontal motion remains constant since there are no horizontal forces acting on the ball (assuming no air resistance).

### Step 3: Time of Flight Calculation
For projectile motion, the time of flight \( T \) can be calculated using the formula:
\[ T = \frac{2V_y}{g} \]
where \( g \) is the acceleration due to gravity.

Substituting the vertical component of the velocity:
\[ T = \frac{2(V \cos \theta)}{g} \]

### Step 4: Consider the Collision
Since the ball makes an elastic collision with a vertical wall, the horizontal component of the velocity \( V_x \) will reverse direction but maintain its magnitude. However, this does not affect the vertical component of the motion, which is solely responsible for the time of flight.

### Step 5: Final Expression for Time of Flight
Thus, the total time of flight remains:
\[ T = \frac{2V \cos \theta}{g} \]

### Conclusion
The total time of flight of the ball is given by:
\[ T = \frac{2V \cos \theta}{g} \]

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