Two projectiles A and B are projected with angle of projection `15^0` for the projectile A and `45^0` for the projectile B. If `R_A and R_B` be the horizontal range for the two projectile then.

Options

A. `R_AltR_B`
B. `R_A=R_B`
C. `R_AgtR_B`
D. the information is insufficient to decide the relation of `R_A with R_B`

Answer

To solve the problem of comparing the horizontal ranges \( R_A \) and \( R_B \) of two projectiles A and B projected at angles of \( 15^\circ \) and \( 45^\circ \) respectively, we can follow these steps:

### Step 1: Write the formula for horizontal range
The horizontal 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: Calculate the range for projectile A
For projectile A, with an angle of projection \( \theta_A = 15^\circ \):
\[ R_A = \frac{u_A^2 \sin(2 \times 15^\circ)}{g} = \frac{u_A^2 \sin(30^\circ)}{g} \]
Since \( \sin(30^\circ) = \frac{1}{2} \), we can simplify this to:
\[ R_A = \frac{u_A^2 \cdot \frac{1}{2}}{g} = \frac{u_A^2}{2g} \]

### Step 3: Calculate the range for projectile B
For projectile B, with an angle of projection \( \theta_B = 45^\circ \):
\[ R_B = \frac{u_B^2 \sin(2 \times 45^\circ)}{g} = \frac{u_B^2 \sin(90^\circ)}{g} \]
Since \( \sin(90^\circ) = 1 \), we can simplify this to:
\[ R_B = \frac{u_B^2}{g} \]

### Step 4: Compare the ranges \( R_A \) and \( R_B \)
To compare \( R_A \) and \( R_B \), we can set up the ratio:
\[ \frac{R_A}{R_B} = \frac{\frac{u_A^2}{2g}}{\frac{u_B^2}{g}} = \frac{u_A^2}{2u_B^2} \]
This simplifies to:
\[ \frac{R_A}{R_B} = \frac{u_A^2}{2u_B^2} \]

### Step 5: Analyze the result
From the ratio \( \frac{R_A}{R_B} = \frac{u_A^2}{2u_B^2} \), we see that the relationship between \( R_A \) and \( R_B \) depends on the values of \( u_A \) and \( u_B \). Since we do not have any information about the initial velocities \( u_A \) and \( u_B \), we cannot definitively conclude whether \( R_A \) is greater than, less than, or equal to \( R_B \).

### Conclusion
Thus, the information provided is insufficient to determine the relationship between \( R_A \) and \( R_B \).