Two particle A and B are projectied from the same point at angles `37^(@)` and `45^(@)` respectile. If `R_(A)` and `R_(B)` are the range of two projectile ,then

Options

A. `R_(A)=R_(B)`
B. `R gt R_(B)`
C. Information is insuffcient to compare `R_(A) and R_(B)`.
D. `R_(A) lt R_(B)`

Explanation

To solve the problem, we need to find the relationship between the ranges \( R_A \) and \( R_B \) of two projectiles A and B projected at angles \( 37^\circ \) and \( 45^\circ \) respectively.

### Step-by-Step Solution:

1. **Identify 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. **Calculate the Range for Particle A**:
For particle A projected at an angle of \( 37^\circ \):
\[ R_A = \frac{U_A^2 \sin(2 \times 37^\circ)}{g} \]
We can compute \( \sin(74^\circ) \) using a calculator or trigonometric tables.

3. **Calculate the Range for Particle B**:
For particle B projected at an angle of \( 45^\circ \):
\[ R_B = \frac{U_B^2 \sin(2 \times 45^\circ)}{g} \]
Since \( \sin(90^\circ) = 1 \), this simplifies to:
\[ R_B = \frac{U_B^2}{g} \]

4. **Compare the Ranges**:
Now we have:
\[ R_A = \frac{U_A^2 \sin(74^\circ)}{g} \]
\[ R_B = \frac{U_B^2}{g} \]
To compare \( R_A \) and \( R_B \), we need to know the relationship between \( U_A \) and \( U_B \).

5. **Conclusion**:
Since we do not have any information about the initial velocities \( U_A \) and \( U_B \), we cannot definitively compare \( R_A \) and \( R_B \). Therefore, the information is insufficient to determine the relationship between the ranges.

### Final Answer:
The correct option is that the information is insufficient to compare \( R_A \) and \( R_B \).