If `H_(1) and H_(2)` be the greatest heights of a projectile in two paths for a given value of range, then the horizontal range of projectile is given by

Options

A. `(H_(1)+ H_(2))/(2)`
B. `(H_(1) + H_(2))/(4)`
C. `4sqrt(H_(1) H_(2))`
D. `4 [ H_(1) + H_(2) ] `

Answer

To find the horizontal range of a projectile given the maximum heights \( H_1 \) and \( H_2 \) for two different paths with the same range, we can follow these steps:

### Step 1: Understand the relationship between range and height
For a projectile launched at an angle \( \theta \) with an initial speed \( u \), the range \( R \) and maximum height \( H \) can be expressed as:
- Range:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
- Maximum Height:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 2: Express \( H_1 \) and \( H_2 \) in terms of \( R \)
Given that \( H_1 \) corresponds to the angle \( \theta \) and \( H_2 \) corresponds to the complementary angle \( 90^\circ - \theta \):
- For \( H_1 \):
\[ H_1 = \frac{u^2 \sin^2 \theta}{2g} \]
- For \( H_2 \):
\[ H_2 = \frac{u^2 \sin^2 (90^\circ - \theta)}{2g} = \frac{u^2 \cos^2 \theta}{2g} \]

### Step 3: Relate \( H_1 \) and \( H_2 \) to \( R \)
From the range formula, we can express \( u^2 \) in terms of \( R \):
\[ u^2 = \frac{Rg}{\sin 2\theta} \]

### Step 4: Substitute \( u^2 \) into the height equations
Substituting \( u^2 \) into the equations for \( H_1 \) and \( H_2 \):
- For \( H_1 \):
\[ H_1 = \frac{Rg \sin^2 \theta}{2g \sin 2\theta} = \frac{R \sin^2 \theta}{2 \sin 2\theta} \]
- For \( H_2 \):
\[ H_2 = \frac{Rg \cos^2 \theta}{2g \sin 2\theta} = \frac{R \cos^2 \theta}{2 \sin 2\theta} \]

### Step 5: Combine the expressions for \( H_1 \) and \( H_2 \)
Now, we can express the horizontal range \( R \) in terms of \( H_1 \) and \( H_2 \):
\[ H_1 = \frac{R \sin^2 \theta}{2 \sin 2\theta} \quad \text{and} \quad H_2 = \frac{R \cos^2 \theta}{2 \sin 2\theta} \]

### Step 6: Final expression for the range \( R \)
By adding these two equations, we can derive the expression for the range \( R \):
\[ R = \frac{2H_1 H_2}{H_1 + H_2} \]

### Conclusion
Thus, the horizontal range \( R \) of the projectile can be expressed in terms of the maximum heights \( H_1 \) and \( H_2 \) as:
\[ R = \frac{2H_1 H_2}{H_1 + H_2} \]