Two stones are projected with same velocity v at an angle `theta` and `(90^(@)-``theta)` . If H and `H_(1)` are
the greatest heights in the two paths. Then what
is the relation between R, H and `H_(1)`?

Options

A. `R = 4 sqrt(HH_(1)`
B. `R = sqrt(HH_(1))`
C. `R = HH_(1)`
D. None of these

Explanation

To solve the problem, we need to find the relationship between the range \( R \), the maximum height \( H \), and the maximum height \( H_1 \) of two stones projected at angles \( \theta \) and \( (90^\circ - \theta) \) with the same initial velocity \( v \).

### Step 1: Calculate the maximum height \( H \) for the angle \( \theta \)

The formula for the maximum height \( H \) reached by a projectile is given by:

\[ H = \frac{v^2 \sin^2 \theta}{2g} \]

where:
- \( v \) is the initial velocity,
- \( g \) is the acceleration due to gravity.

### Step 2: Calculate the maximum height \( H_1 \) for the angle \( (90^\circ - \theta) \)

For the angle \( (90^\circ - \theta) \), we can use the same formula for maximum height:

\[ H_1 = \frac{v^2 \sin^2(90^\circ - \theta)}{2g} \]

Using the identity \( \sin(90^\circ - \theta) = \cos \theta \), we can rewrite \( H_1 \):

\[ H_1 = \frac{v^2 \cos^2 \theta}{2g} \]

### Step 3: Calculate the range \( R \) for the angle \( \theta \)

The range \( R \) of a projectile is given by the formula:

\[ R = \frac{v^2 \sin(2\theta)}{g} \]

Using the double angle identity \( \sin(2\theta) = 2 \sin \theta \cos \theta \), we can rewrite \( R \):

\[ R = \frac{v^2 \cdot 2 \sin \theta \cos \theta}{g} \]

### Step 4: Relate \( R \), \( H \), and \( H_1 \)

Now, we can express \( R^2 \) in terms of \( H \) and \( H_1 \):

1. From the expression for \( H \):
\[ H = \frac{v^2 \sin^2 \theta}{2g} \implies v^2 = 2gH \sin^{-2} \theta \]

2. From the expression for \( H_1 \):
\[ H_1 = \frac{v^2 \cos^2 \theta}{2g} \implies v^2 = 2gH_1 \cos^{-2} \theta \]

3. Substitute \( v^2 \) into the range formula:
\[ R^2 = \frac{(2gH \sin^2 \theta)(2gH_1 \cos^2 \theta)}{g^2} = 4H H_1 \]

Thus, we have the relationship:

\[ R^2 = 4H H_1 \]

### Final Relation

The final relationship between the range \( R \), the maximum height \( H \), and the maximum height \( H_1 \) is:

\[ R = 2 \sqrt{H H_1} \]