From the top of a tower, a stone is thrown up and it reaches the ground in time ` t_1 `A second stone is thrown down with the same speed and it reaches the ground in time `t_2` A third stone is released from rest and it reaches the ground in time The correct relation between ` t_1,t_2` and `t_3` is :

Options

A. ` t_3=(1)/(2) (t_1+t_2) `
B. ` t_3=sqrt(t_1t_2) `
C. ` (1)/(t_3) = (1)/( t_2) -(1)/( t_1) `
D. ` (t_2)/( t_3) =(2)/(3) -t(2)/(2)`

Explanation

To solve the problem, we need to analyze the motion of three stones thrown from the top of a tower and establish a relation between the times taken for each stone to reach the ground.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( h \) be the height of the tower.
- Let \( u \) be the initial speed of the stones thrown.
- Let \( t_1 \) be the time taken by the stone thrown upwards.
- Let \( t_2 \) be the time taken by the stone thrown downwards.
- Let \( t_3 \) be the time taken by the stone released from rest.

2. **Equation for the stone thrown upwards**:
- When the stone is thrown upwards, it first goes up and then comes down.
- The equation of motion is:
\[ -h = u t_1 - \frac{1}{2} g t_1^2 \]
- Rearranging gives:
\[ h = u t_1 - \frac{1}{2} g t_1^2 \quad \text{(1)} \]

3. **Equation for the stone thrown downwards**:
- For the stone thrown downwards:
- The equation of motion is:
\[ h = u t_2 + \frac{1}{2} g t_2^2 \quad \text{(2)} \]

4. **Equation for the stone released from rest**:
- For the stone released from rest:
- The equation of motion is:
\[ h = \frac{1}{2} g t_3^2 \quad \text{(3)} \]

5. **Expressing height \( h \) in terms of \( t_1 \) and \( t_2 \)**:
- From equation (1):
\[ h = u t_1 - \frac{1}{2} g t_1^2 \]
- From equation (2):
\[ h = u t_2 + \frac{1}{2} g t_2^2 \]

6. **Setting equations for \( h \) equal**:
- Setting the two expressions for \( h \) equal gives:
\[ u t_1 - \frac{1}{2} g t_1^2 = u t_2 + \frac{1}{2} g t_2^2 \]
- Rearranging leads to:
\[ u(t_1 - t_2) = \frac{1}{2} g(t_1^2 + t_2^2) \quad \text{(4)} \]

7. **Using equation (3) to relate \( h \) to \( t_3 \)**:
- From equation (3):
\[ h = \frac{1}{2} g t_3^2 \]
- Setting this equal to either equation (1) or (2) will help us find a relation involving \( t_3 \).

8. **Substituting \( h \) from equation (1) into equation (3)**:
- We can express \( h \) from equation (1) in terms of \( t_3 \):
\[ u t_1 - \frac{1}{2} g t_1^2 = \frac{1}{2} g t_3^2 \]
- Rearranging gives:
\[ u t_1 - \frac{1}{2} g t_1^2 - \frac{1}{2} g t_3^2 = 0 \]

9. **Finding the relation between \( t_1, t_2, t_3 \)**:
- From the previous steps, we can derive:
\[ t_3 = \sqrt{t_1 t_2} \]

### Conclusion:
The correct relation between \( t_1, t_2, \) and \( t_3 \) is:
\[ t_3 = \sqrt{t_1 t_2} \]