Three particles A, B and C are projected from the same point with the same initial speeds making angles `30^(@), 45^(@) and 60^(@)` respectively with the horizontal. Which of the following statement is correct?

Options

A. A,B and C have unequal ranges
B. Ranges of A and C are equal and less than that of B
C. Ranges of A and C are equal and greater then that of B
D. A,B and C have equal ranges

Explanation

To solve the problem, we need to calculate the range of each of the three particles A, B, and C, which are projected at angles of 30°, 45°, and 60° respectively, using the formula for the range of projectile motion.

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

2. **Calculate the Range for Particle A (30°)**:
- For particle A:
\[ R_A = \frac{u^2 \sin(2 \times 30^\circ)}{g} = \frac{u^2 \sin(60^\circ)}{g} \]
- Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ R_A = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

3. **Calculate the Range for Particle B (45°)**:
- For particle B:
\[ R_B = \frac{u^2 \sin(2 \times 45^\circ)}{g} = \frac{u^2 \sin(90^\circ)}{g} \]
- Since \( \sin(90^\circ) = 1 \):
\[ R_B = \frac{u^2}{g} \]

4. **Calculate the Range for Particle C (60°)**:
- For particle C:
\[ R_C = \frac{u^2 \sin(2 \times 60^\circ)}{g} = \frac{u^2 \sin(120^\circ)}{g} \]
- Since \( \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ R_C = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

5. **Compare the Ranges**:
- From the calculations:
\[ R_A = R_C = \frac{\sqrt{3} u^2}{2g} \]
\[ R_B = \frac{u^2}{g} \]
- Since \( \frac{\sqrt{3}}{2} \approx 0.866 \) and \( 1 > \frac{\sqrt{3}}{2} \), we find that:
\[ R_A = R_C < R_B \]

6. **Conclusion**:
- The ranges of particles A and C are equal and less than the range of particle B. Therefore, the correct statement is that the ranges of A and C are equal, and both are less than the range of B.

### Final Answer:
The correct statement is: "The ranges of A and C are equal and less than that of B."