A stone projected with a velocity u at an angle q with the horizontal reaches maximum heights `H_(1)`. When it is projected with velocity u at an angle `(pi/2-theta)` with the horizontal, it reaches maximum height `H_(2)`. The relations between the horizontal range R of the projectile, `H_(1) and H_(2)`, is

Options

A. `R=4sqrt(H_(1)H_(2))`
B. `R=4(H_(1)-H_(2))`
C. `R=4(H_(1)+H_(2))`
D. `R=(H_(1)^(2))/(H_(2)^(2))`

Explanation

To solve the problem, we need to find the relationship between the horizontal range \( R \) of a projectile and the maximum heights \( H_1 \) and \( H_2 \) when projected at angles \( \theta \) and \( \frac{\pi}{2} - \theta \) respectively.

### Step-by-Step Solution:

1. **Identify the maximum height for the first projectile**:
The maximum height \( H_1 \) for a projectile launched at an angle \( \theta \) with initial velocity \( u \) is given by the formula:
\[ H_1 = \frac{u^2 \sin^2 \theta}{2g} \]

2. **Identify the maximum height for the second projectile**:
The maximum height \( H_2 \) for a projectile launched at an angle \( \frac{\pi}{2} - \theta \) is given by:
\[ H_2 = \frac{u^2 \sin^2\left(\frac{\pi}{2} - \theta\right)}{2g} \]
Since \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta \), we can rewrite \( H_2 \) as:
\[ H_2 = \frac{u^2 \cos^2 \theta}{2g} \]

3. **Calculate the horizontal range for both projectiles**:
The horizontal range \( R \) for a projectile launched at an angle \( \theta \) is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
For the second projectile launched at an angle \( \frac{\pi}{2} - \theta \):
\[ R' = \frac{u^2 \sin(2(\frac{\pi}{2} - \theta))}{g} = \frac{u^2 \sin(2\theta)}{g} \]
Thus, both projectiles have the same range \( R \).

4. **Relate \( H_1 \) and \( H_2 \) to \( R \)**:
From the expressions for \( H_1 \) and \( H_2 \):
\[ H_1 = \frac{u^2 \sin^2 \theta}{2g} \]
\[ H_2 = \frac{u^2 \cos^2 \theta}{2g} \]
We can express \( R \) in terms of \( H_1 \) and \( H_2 \):
\[ R = \frac{u^2 \sin(2\theta)}{g} = \frac{u^2 (2\sin \theta \cos \theta)}{g} \]

5. **Final Relationship**:
We can express \( R \) in terms of \( H_1 \) and \( H_2 \):
\[ R = 2gH_1H_2 \]
This shows that the horizontal range \( R \) is related to the maximum heights \( H_1 \) and \( H_2 \).

### Conclusion:
The relationship between the horizontal range \( R \) and the maximum heights \( H_1 \) and \( H_2 \) can be summarized as:
\[ R = 2gH_1H_2 \]