Four bodies A,B,C and D are projected with equal velocities having angles of projection `15^(@),30^(@),45^(@) and 60^(@)` with the horizontal respectively. The body having the shortest range is

Options

A. A
B. B
C. C
D. D

Explanation

To determine which body among A, B, C, and D has the shortest range when projected with equal velocities at angles of 15°, 30°, 45°, and 60° respectively, we can use the formula for the range of a projectile:

### 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, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of projection.

2. **Identifying the Angles**:
- For body A: \( \theta_A = 15^\circ \)
- For body B: \( \theta_B = 30^\circ \)
- For body C: \( \theta_C = 45^\circ \)
- For body D: \( \theta_D = 60^\circ \)

3. **Calculating the Range for Each Body**:
- **For Body A**:
\[ R_A = \frac{u^2 \sin(2 \times 15^\circ)}{g} = \frac{u^2 \sin(30^\circ)}{g} = \frac{u^2 \cdot \frac{1}{2}}{g} = \frac{u^2}{2g} \]

- **For Body B**:
\[ R_B = \frac{u^2 \sin(2 \times 30^\circ)}{g} = \frac{u^2 \sin(60^\circ)}{g} = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

- **For Body C**:
\[ R_C = \frac{u^2 \sin(2 \times 45^\circ)}{g} = \frac{u^2 \sin(90^\circ)}{g} = \frac{u^2 \cdot 1}{g} = \frac{u^2}{g} \]

- **For Body D**:
\[ R_D = \frac{u^2 \sin(2 \times 60^\circ)}{g} = \frac{u^2 \sin(120^\circ)}{g} = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

4. **Comparing the Ranges**:
- We have:
- \( R_A = \frac{u^2}{2g} \)
- \( R_B = \frac{\sqrt{3} u^2}{2g} \)
- \( R_C = \frac{u^2}{g} \)
- \( R_D = \frac{\sqrt{3} u^2}{2g} \)

- Since \( \frac{u^2}{g} > \frac{\sqrt{3} u^2}{2g} > \frac{u^2}{2g} \), we can conclude:
- \( R_C > R_B = R_D > R_A \)

5. **Conclusion**:
The body with the shortest range is body A, which is projected at an angle of 15°.

### Final Answer:
The body having the shortest range is **Body A**.