A stone projected with a velocity u at an angle (theta )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_1H_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 establish the relationships between the maximum heights \( H_1 \) and \( H_2 \) and the horizontal range \( R \) of the projectile.

### Step 1: Calculate Maximum Height \( H_1 \)
The formula for the maximum height \( H \) reached by a projectile is given by:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]
For the first case, where the stone is projected at an angle \( \theta \), the maximum height \( H_1 \) can be expressed as:
\[ H_1 = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 2: Calculate Maximum Height \( H_2 \)
For the second case, where the stone is projected at an angle \( \frac{\pi}{2} - \theta \), we can use the same formula for maximum height:
\[ H_2 = \frac{u^2 \sin^2\left(\frac{\pi}{2} - \theta\right)}{2g} \]
Using the identity \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta \), we can rewrite \( H_2 \) as:
\[ H_2 = \frac{u^2 \cos^2 \theta}{2g} \]

### Step 3: Calculate the Horizontal Range \( R \)
The formula for the horizontal range \( R \) of a projectile is given by:
\[ R = \frac{u^2 \sin 2\theta}{g} \]

### Step 4: Relate \( H_1 \), \( H_2 \), and \( R \)
Now we will multiply \( H_1 \) and \( H_2 \):
\[ H_1 H_2 = \left(\frac{u^2 \sin^2 \theta}{2g}\right) \left(\frac{u^2 \cos^2 \theta}{2g}\right) = \frac{u^4 \sin^2 \theta \cos^2 \theta}{4g^2} \]
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can express this as:
\[ H_1 H_2 = \frac{u^4}{4g^2} \cdot \frac{\sin^2 2\theta}{4} = \frac{u^4 \sin^2 2\theta}{16g^2} \]

### Step 5: Take the Square Root
Taking the square root of both sides gives:
\[ \sqrt{H_1 H_2} = \frac{u^2 \sin 2\theta}{4g} \]

### Step 6: Relate to the Range \( R \)
From the range formula, we can express \( R \) as:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
Dividing \( R \) by 4 gives:
\[ \frac{R}{4} = \frac{u^2 \sin 2\theta}{4g} \]

### Step 7: Final Relation
Thus, we can equate the two expressions:
\[ \sqrt{H_1 H_2} = \frac{R}{4} \]
Multiplying both sides by 4 gives:
\[ 4 \sqrt{H_1 H_2} = R \]

### Conclusion
The final relation between the horizontal range \( R \) and the maximum heights \( H_1 \) and \( H_2 \) is:
\[ 4 \sqrt{H_1 H_2} = R \]