A particle is projected vertically upwards from a point A on the ground. It takes `t_(1)` time to reach a point B but it still continues to move up. If it takes further `t_(2)` time to reach the ground from point B then height of point B from the ground is :-

Explanation

To find the height of point B from the ground, we can use the equations of motion and the information given in the problem. Let's break down the solution step by step.

### Step 1: Understand the motion of the particle
The particle is projected upwards from point A and takes time \( t_1 \) to reach point B. After reaching point B, it continues to move upwards and then comes back down to the ground, taking an additional time \( t_2 \).

### Step 2: Determine the total time of flight
The total time of flight from point A to the ground can be expressed as:
\[ T = t_1 + t_2 \]

### Step 3: Analyze the motion to find the initial velocity
At point B, the particle reaches its maximum height, where its velocity becomes zero. The time taken to reach the maximum height from point A is \( t_1 \). The time taken to go from point B to the ground is \( t_2 \).

Using the symmetry of projectile motion, the time taken to go up from point B to the maximum height is equal to the time taken to come down from the maximum height to point B. Therefore, the time taken to go from point B to the maximum height is:
\[ t_{\text{up}} = t_2 - t_1 \]

### Step 4: Use the equations of motion
Using the first equation of motion:
\[ v = u + at \]
At the maximum height, the final velocity \( v = 0 \), and the acceleration \( a = -g \) (where \( g \) is the acceleration due to gravity). Thus, we can write:
\[ 0 = u - g t_1 \]
From this, we can express the initial velocity \( u \) as:
\[ u = g t_1 \]

### Step 5: Calculate the height of point B
Now, we can use the second equation of motion to find the height \( H \) from point A to point B:
\[ H = ut_1 - \frac{1}{2} g t_1^2 \]
Substituting \( u = g t_1 \):
\[ H = (g t_1) t_1 - \frac{1}{2} g t_1^2 \]
\[ H = g t_1^2 - \frac{1}{2} g t_1^2 \]
\[ H = \frac{1}{2} g t_1^2 \]

### Step 6: Relate \( t_1 \) and \( t_2 \)
Since the time taken to go from point B to the ground is \( t_2 \), we can also express the height \( H \) in terms of \( t_2 \):
\[ H = \frac{1}{2} g t_2 (t_2 - t_1) \]

### Final Answer
Thus, the height of point B from the ground can be expressed as:
\[ H = \frac{1}{2} g t_1 t_2 \]