A projectile has the same range R for angles of projections. If `T_(1) and T_(2)` be the times of fight in the two cases, then ( using ` theta` as the angle of projection corresponding to ` T_(1)` )

Options

A. `T_(1) T_(2) = 2R/g `
B. ` T_(1) T_(2) = R^(2)`
C. `T_(1)/T_(2) = tan theta`
D. `T_(1)/T_(2) = 1`

Answer

To solve the problem step by step, we will derive the relationship between the times of flight \( T_1 \) and \( T_2 \) for a projectile launched at two complementary angles that yield the same range.

### Step-by-Step Solution:

1. **Understanding the Time of Flight**:
The time of flight \( T \) for a projectile launched at an angle \( \theta \) with an initial speed \( u \) is given by:
\[ T = \frac{2u \sin \theta}{g} \]
For our two angles, let \( T_1 \) be the time of flight for angle \( \theta \) and \( T_2 \) for angle \( 90^\circ - \theta \).

2. **Finding \( T_2 \)**:
The time of flight for the angle \( 90^\circ - \theta \) can be expressed as:
\[ T_2 = \frac{2u \sin(90^\circ - \theta)}{g} = \frac{2u \cos \theta}{g} \]

3. **Expressing the Range**:
The range \( R \) of a projectile is given by:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can rewrite the range as:
\[ R = \frac{2u^2 \sin \theta \cos \theta}{g} \]

4. **Relating \( \sin \theta \) and \( \cos \theta \) to Time of Flight**:
From the expressions for \( T_1 \) and \( T_2 \):
- From \( T_1 \):
\[ \sin \theta = \frac{g T_1}{2u} \]
- From \( T_2 \):
\[ \cos \theta = \frac{g T_2}{2u} \]

5. **Substituting into the Range Equation**:
Substitute \( \sin \theta \) and \( \cos \theta \) into the range equation:
\[ R = \frac{2u^2 \left(\frac{g T_1}{2u}\right) \left(\frac{g T_2}{2u}\right)}{g} \]
Simplifying this gives:
\[ R = \frac{2u^2 \cdot g T_1 T_2}{4u^2 g} = \frac{T_1 T_2}{2} \]

6. **Final Relationship**:
Rearranging the equation gives:
\[ T_1 T_2 = 2R \]

### Conclusion:
Thus, we find that the relationship between the times of flight \( T_1 \) and \( T_2 \) for the two angles of projection that yield the same range \( R \) is:
\[ T_1 T_2 = 2R \]