Two particles are projected obliquely from ground with same speed such that their range `'R'` are same but they attain different maximum heights `h_(1)` and `h_(2)` then relation between `R, h_(1)` and `h_(2)` is:

Options

A. `R = sqrt(h_(1)h_(2))`
B. `R = sqrt(2h_(1)h_(2))`
C. `R = 2 sqrt(h_(1)h_(2))`
D. `R = 4 sqrt(h_(1)h_(2))`

Explanation

To solve the problem, we need to establish a relationship between the range \( R \) and the maximum heights \( h_1 \) and \( h_2 \) of two particles projected obliquely with the same speed. Here’s a step-by-step solution:

### Step 1: Understand the Range Formula
The range \( R \) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

### Step 2: Identify Angles of Projection
Since the two particles have the same range and initial speed, we can denote their angles of projection as \( \theta_1 = \theta \) and \( \theta_2 = 90^\circ - \theta \).

### Step 3: Maximum Height Formula
The maximum height \( h \) achieved by a projectile is given by:
\[ h = \frac{u^2 \sin^2(\theta)}{2g} \]
Thus, for the two particles, we can write:
- For particle 1:
\[ h_1 = \frac{u^2 \sin^2(\theta)}{2g} \]
- For particle 2:
\[ h_2 = \frac{u^2 \sin^2(90^\circ - \theta)}{2g} = \frac{u^2 \cos^2(\theta)}{2g} \]

### Step 4: Multiply the Heights
Now, we multiply \( h_1 \) and \( h_2 \):
\[ h_1 \cdot h_2 = \left(\frac{u^2 \sin^2(\theta)}{2g}\right) \cdot \left(\frac{u^2 \cos^2(\theta)}{2g}\right) \]
This simplifies to:
\[ h_1 \cdot h_2 = \frac{u^4 \sin^2(\theta) \cos^2(\theta)}{4g^2} \]

### Step 5: Use the Identity for Sine
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we can express \( \sin^2(\theta) \cos^2(\theta) \) as:
\[ \sin^2(\theta) \cos^2(\theta) = \frac{1}{4} \sin^2(2\theta) \]
Thus, we can rewrite \( h_1 \cdot h_2 \):
\[ h_1 \cdot h_2 = \frac{u^4}{4g^2} \cdot \frac{1}{4} \sin^2(2\theta) = \frac{u^4 \sin^2(2\theta)}{16g^2} \]

### Step 6: Relate to the Range
From the range formula, we know:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Squaring both sides gives:
\[ R^2 = \frac{u^4 \sin^2(2\theta)}{g^2} \]

### Step 7: Substitute Back
Now, substituting \( R^2 \) back into the equation for \( h_1 \cdot h_2 \):
\[ h_1 \cdot h_2 = \frac{R^2}{16} \]

### Step 8: Final Relation
Rearranging gives us:
\[ R^2 = 16 h_1 h_2 \]
or
\[ R = 4 \sqrt{h_1 h_2} \]

### Conclusion
The relationship between the range \( R \) and the maximum heights \( h_1 \) and \( h_2 \) is:
\[ R = 4 \sqrt{h_1 h_2} \]