Two stones are projected with the same speed but making different angles with the horizontal. Their reanges are equal. If the angles of projection of one is `pi//3` and its maximum height is `h_(1)` then the maximum height of the other will be:

Options

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

Answer

To solve the problem, we need to find the maximum height of the second stone when both stones are projected with the same speed but at different angles, and their ranges are equal.

### Step-by-Step Solution:

1. **Understanding the Angles of Projection**:
- Let the angle of projection of the first stone be \( \theta_1 = \frac{\pi}{3} \).
- Since the ranges of the two stones are equal and they are projected with the same speed, the angles of projection \( \theta_1 \) and \( \theta_2 \) are complementary. Therefore, we have:
\[ \theta_1 + \theta_2 = \frac{\pi}{2} \]
- Substituting \( \theta_1 \):
\[ \frac{\pi}{3} + \theta_2 = \frac{\pi}{2} \]
- Solving for \( \theta_2 \):
\[ \theta_2 = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} \]

2. **Formula for Maximum Height**:
- The maximum height \( H \) of 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. **Finding the Maximum Heights**:
- For the first stone (angle \( \theta_1 = \frac{\pi}{3} \)):
\[ h_1 = \frac{u^2 \sin^2\left(\frac{\pi}{3}\right)}{2g} \]
- For the second stone (angle \( \theta_2 = \frac{\pi}{6} \)):
\[ h_2 = \frac{u^2 \sin^2\left(\frac{\pi}{6}\right)}{2g} \]

4. **Calculating the Sine Values**:
- We know:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
- Therefore:
\[ h_1 = \frac{u^2 \left(\frac{\sqrt{3}}{2}\right)^2}{2g} = \frac{u^2 \cdot \frac{3}{4}}{2g} = \frac{3u^2}{8g} \]
\[ h_2 = \frac{u^2 \left(\frac{1}{2}\right)^2}{2g} = \frac{u^2 \cdot \frac{1}{4}}{2g} = \frac{u^2}{8g} \]

5. **Finding the Ratio of Maximum Heights**:
- We can find the ratio of the maximum heights:
\[ \frac{h_1}{h_2} = \frac{\frac{3u^2}{8g}}{\frac{u^2}{8g}} = \frac{3u^2}{u^2} = 3 \]
- Thus:
\[ h_1 = 3h_2 \implies h_2 = \frac{h_1}{3} \]

6. **Final Result**:
- If we denote the maximum height of the first stone as \( h_1 \), then the maximum height of the second stone is:
\[ h_2 = \frac{h_1}{3} \]

### Conclusion:
The maximum height of the second stone is \( \frac{h_1}{3} \).