A ball projected from ground vertically upward is at same height at time `t_(1) and t_(2)`. The speed of projection of ball is [Neglect the effect of air resistance ]

Options

A. `g[t_(2)-t_(1)]`
B. `(g[t_(1)+t_(2)])/(2)`
C. `(g[t_(2)-t_(1)])/(2)`
D. `g[t_(1)+t_(2)]`

Answer

To solve the problem of finding the speed of projection of a ball that is projected vertically upward and reaches the same height at times \( t_1 \) and \( t_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The ball is projected upwards from the ground with an initial speed \( V \).
- It moves upwards against the acceleration due to gravity \( g \) until it reaches its maximum height, where its velocity becomes zero.

2. **Time of Flight**:
- The ball reaches the same height at two different times \( t_1 \) and \( t_2 \). This implies that the time taken to go up to the maximum height and come back down to the same height is symmetrical.
- Let \( \Delta t \) be the time taken to go from the height \( h \) to the maximum height and back down to the same height \( h \). Thus, we can express the times as:
\[ t_2 = t_1 + 2\Delta t \]
- From this, we can derive:
\[ \Delta t = \frac{t_2 - t_1}{2} \]

3. **Using the First Equation of Motion**:
- The first equation of motion states:
\[ v = u - gt \]
- At the maximum height, the final velocity \( v = 0 \), and the time taken to reach the maximum height from the initial position is \( t_1 + \Delta t \).
- Therefore, we can write:
\[ 0 = V - g(t_1 + \Delta t) \]
- Rearranging gives:
\[ V = g(t_1 + \Delta t) \]

4. **Substituting for \( \Delta t \)**:
- Now, substitute \( \Delta t \) from step 2 into the equation:
\[ V = g\left(t_1 + \frac{t_2 - t_1}{2}\right) \]
- Simplifying this gives:
\[ V = g\left(\frac{2t_1 + t_2 - t_1}{2}\right) = g\left(\frac{t_1 + t_2}{2}\right) \]

5. **Final Expression for Speed of Projection**:
- Thus, the speed of projection \( V \) can be expressed as:
\[ V = \frac{g(t_1 + t_2)}{2} \]

### Conclusion:
The speed of projection of the ball is given by:
\[ V = \frac{g(t_1 + t_2)}{2} \]