If `T_1 and T_2` are the times of flight for two complementary angles, then the range of projectile R is given by

Options

A. `R = 4gT_1T_2`
B. `R=2gT_1T_2`
C. `R=1/4 gT_1T_2`
D. `R = 1/2gT_1T_2`

Explanation

To find the range \( R \) of a projectile in terms of the times of flight \( T_1 \) and \( T_2 \) for two complementary angles, we can follow these steps:

### Step 1: Understand the Complementary Angles
Let the angle of projection be \( \theta \). The complementary angle is \( 90^\circ - \theta \). The times of flight for these angles are given as:
- For angle \( \theta \):
\[ T_1 = \frac{2u \sin \theta}{g} \]
- For angle \( 90^\circ - \theta \):
\[ T_2 = \frac{2u \sin(90^\circ - \theta)}{g} = \frac{2u \cos \theta}{g} \]

### Step 2: Express \( \sin \theta \) and \( \cos \theta \) in terms of \( T_1 \) and \( T_2 \)
From the equations for \( T_1 \) and \( T_2 \), we can express \( \sin \theta \) and \( \cos \theta \):
- Rearranging for \( \sin \theta \):
\[ \sin \theta = \frac{g T_1}{2u} \]
- Rearranging for \( \cos \theta \):
\[ \cos \theta = \frac{g T_2}{2u} \]

### Step 3: Use the Range Formula
The range \( R \) of the projectile is given by:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
Using the double angle identity, we can express \( \sin 2\theta \) as:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Substituting the expressions for \( \sin \theta \) and \( \cos \theta \):
\[ R = \frac{u^2 \cdot 2 \left(\frac{g T_1}{2u}\right) \left(\frac{g T_2}{2u}\right)}{g} \]

### Step 4: Simplify the Expression
Now, simplify the expression for \( R \):
\[ R = \frac{u^2 \cdot 2 \cdot \frac{g T_1}{2u} \cdot \frac{g T_2}{2u}}{g} \]
Cancelling out \( u^2 \) and \( g \):
\[ R = \frac{g T_1 T_2}{2} \]

### Final Result
Thus, the range \( R \) of the projectile in terms of the times of flight \( T_1 \) and \( T_2 \) is:
\[ R = \frac{g T_1 T_2}{2} \]