A projectile can have same range `R` for two angles of projection. It `t_1 and t_2` are the times of flight in the two cases, then what is the product of two times of flight ?

Options

A. `t_1 t_2 prop R^2`
B. `t_1 t_2 prop R`
C. `t_1 t_2 prop (1)/( R)`
D. `t_1 t_2 prop (1)/(R^2)`

Explanation

To find the product of the two times of flight \( T_1 \) and \( T_2 \) for a projectile that has the same range \( R \) for two different angles of projection, we can follow these steps:

### Step 1: Understand the relationship between angles and range
For a projectile launched at two angles \( \theta \) and \( \phi \), the range \( R \) can be expressed as:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
and
\[ R = \frac{u^2 \sin(2\phi)}{g} \]
Since the ranges are equal, we have:
\[ \sin(2\theta) = \sin(2\phi) \]

### Step 2: Establish the relationship between the angles
From the property of sine, we know that:
\[ \phi = 90^\circ - \theta \]
This means that \( \theta + \phi = 90^\circ \).

### Step 3: Calculate the time of flight for each angle
The time of flight \( T \) for a projectile is given by:
\[ T = \frac{2u \sin(\theta)}{g} \]
Thus, for angles \( \theta \) and \( \phi \), the times of flight are:
\[ T_1 = \frac{2u \sin(\theta)}{g} \]
\[ T_2 = \frac{2u \sin(\phi)}{g} \]

### Step 4: Find the product of the times of flight
Now, we can find the product \( T_1 \times T_2 \):
\[ T_1 \times T_2 = \left(\frac{2u \sin(\theta)}{g}\right) \times \left(\frac{2u \sin(\phi)}{g}\right) \]
\[ = \frac{4u^2 \sin(\theta) \sin(\phi)}{g^2} \]

### Step 5: Substitute \( \sin(\phi) \)
Since \( \phi = 90^\circ - \theta \), we have:
\[ \sin(\phi) = \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can rewrite the product:
\[ T_1 \times T_2 = \frac{4u^2 \sin(\theta) \cos(\theta)}{g^2} \]

### Step 6: Use the double angle identity
Using the double angle identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we can express the product as:
\[ T_1 \times T_2 = \frac{2u^2 \sin(2\theta)}{g^2} \]

### Step 7: Relate it to the range
Since we know that the range \( R \) is given by \( R = \frac{u^2 \sin(2\theta)}{g} \), we can substitute:
\[ T_1 \times T_2 = \frac{2R}{g} \]

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