A projectile can have the same range 'R' for two angles of projection . If `'T_(1)'` and `'T_(2)'` to be times of flights in the two cases, then the product of the two times of flights is directly proportional to .

Options

A. `1//R^(2)`
B. `1//R`
C. `R`
D. `R^(2)`

Explanation

To solve the problem, we need to find the relationship between the product of the times of flight \( T_1 \) and \( T_2 \) for a projectile launched at two different angles that yield the same range \( R \).

### Step-by-Step Solution:

1. **Understanding 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.

2. **Complementary Angles**:
For two angles \( \theta \) and \( 90^\circ - \theta \), the range remains the same:
\[ R = \frac{u^2 \sin 2\theta}{g} = \frac{u^2 \sin(90^\circ - 2\theta)}{g} \]
This means that the angles \( \theta \) and \( 90^\circ - \theta \) will give the same range.

3. **Time of Flight Formula**:
The time of flight \( T \) for a projectile is given by:
\[ T = \frac{2u \sin \theta}{g} \]
Therefore, for the two angles, we have:
\[ T_1 = \frac{2u \sin \theta}{g} \]
\[ T_2 = \frac{2u \sin(90^\circ - \theta)}{g} = \frac{2u \cos \theta}{g} \]

4. **Calculating the Product of Times of Flight**:
Now, we can calculate the product \( T_1 \times T_2 \):
\[ T_1 \times T_2 = \left(\frac{2u \sin \theta}{g}\right) \left(\frac{2u \cos \theta}{g}\right) \]
\[ = \frac{4u^2 \sin \theta \cos \theta}{g^2} \]

5. **Using the Trigonometric Identity**:
We know that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we can rewrite the product as:
\[ T_1 \times T_2 = \frac{2u^2 \sin 2\theta}{g^2} \]

6. **Relating to the Range**:
From the range formula, we have:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
Therefore, we can express \( \sin 2\theta \) in terms of \( R \):
\[ \sin 2\theta = \frac{Rg}{u^2} \]
Substituting this back into the product of times of flight:
\[ T_1 \times T_2 = \frac{2u^2}{g^2} \cdot \frac{Rg}{u^2} = \frac{2R}{g} \]

7. **Conclusion**:
Thus, we find that the product of the times of flight \( T_1 \) and \( T_2 \) is directly proportional to the range \( R \):
\[ T_1 \times T_2 \propto R \]

### Final Answer:
The product of the two times of flights \( T_1 \) and \( T_2 \) is directly proportional to the range \( R \).