Two stones are projected with the same speed but making different angles with the horizontal. Their horizontal ranges are equal. The angle of projection of one is `pi/3` and the maximum height reached by it is 102 m. Then the maximum height reached by the other in metres is

Options

A. `3 h_1`
B. `2 h_1`
C. `h_1//2`
D. `h_1//3`

Explanation

To solve the problem, we need to find the maximum height reached by the second stone, given that both stones are projected with the same speed and have the same horizontal range, but at different angles.

### Step-by-Step Solution:

1. **Identify the angles of projection**:
- The angle of projection for the first stone is given as \( \theta_1 = \frac{\pi}{3} \) (or 60 degrees).
- Since the horizontal ranges are equal, the angle of projection for the second stone can be determined using the property that the angles that give the same range are complementary. Thus, \( \theta_2 = \frac{\pi}{2} - \theta_1 = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} \) (or 30 degrees).

2. **Use the formula for maximum height**:
- The maximum height \( H \) reached by a projectile is given by the formula:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]
- Here, \( u \) is the initial speed, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of projection.

3. **Calculate the maximum height for the first stone**:
- For the first stone:
\[ H_1 = \frac{u^2 \sin^2 \theta_1}{2g} = \frac{u^2 \sin^2 \left(\frac{\pi}{3}\right)}{2g} \]
- We know that \( H_1 = 102 \, \text{m} \).

4. **Set up the ratio of maximum heights**:
- The maximum height for the second stone can be expressed as:
\[ H_2 = \frac{u^2 \sin^2 \theta_2}{2g} = \frac{u^2 \sin^2 \left(\frac{\pi}{6}\right)}{2g} \]
- The ratio of the maximum heights can be expressed as:
\[ \frac{H_2}{H_1} = \frac{\sin^2 \theta_2}{\sin^2 \theta_1} \]

5. **Calculate the sine values**:
- We know:
\[ \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}, \quad \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
- Therefore:
\[ \frac{H_2}{H_1} = \frac{\left(\frac{1}{2}\right)^2}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]

6. **Find \( H_2 \)**:
- Now substituting \( H_1 = 102 \, \text{m} \):
\[ H_2 = \frac{1}{3} H_1 = \frac{1}{3} \times 102 = 34 \, \text{m} \]

### Conclusion:
The maximum height reached by the second stone is **34 meters**.