Three projectiles A, B and Care projected at an angle of `30^@, 45^@, 60^@` respectively. If R_A, R_B and R_C` are ranges of A, B and C respectively then (velocity of projection is same for A, B and C) 

Options

A. `R_(A) = R_(B) =R_(C)`
B. `R_(A) = R_(C) gt R_(B)`
C. `R_(A) lt R_(B) lt R_(C)`
D. `R_(A) = R_(C) lt R_(B)`

Explanation

To solve the problem of comparing the ranges of three projectiles A, B, and C projected at angles of \(30^\circ\), \(45^\circ\), and \(60^\circ\) respectively, we can use the formula for the range of a projectile:

\[ R = \frac{u^2 \sin(2\theta)}{g} \]

where:
- \( R \) is the range,
- \( u \) is the initial velocity of projection,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity.

Since the initial velocity \( u \) and \( g \) are the same for all three projectiles, we can focus on the term \( \sin(2\theta) \).

### Step 1: Calculate the range for each projectile.

1. **For Projectile A (angle = \(30^\circ\))**:
\[ R_A = \frac{u^2 \sin(2 \times 30^\circ)}{g} = \frac{u^2 \sin(60^\circ)}{g} \]
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Therefore,
\[ R_A = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} \]

2. **For Projectile B (angle = \(45^\circ\))**:
\[ R_B = \frac{u^2 \sin(2 \times 45^\circ)}{g} = \frac{u^2 \sin(90^\circ)}{g} \]
\[ \sin(90^\circ) = 1 \]
Therefore,
\[ R_B = \frac{u^2}{g} \]

3. **For Projectile C (angle = \(60^\circ\))**:
\[ R_C = \frac{u^2 \sin(2 \times 60^\circ)}{g} = \frac{u^2 \sin(120^\circ)}{g} \]
\[ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Therefore,
\[ R_C = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} \]

### Step 2: Compare the ranges.

Now we have:
- \( R_A = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} \)
- \( R_B = \frac{u^2}{g} \)
- \( R_C = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} \)

From this, we can see that:
- \( R_A = R_C \)
- \( R_B > R_A \) and \( R_B > R_C \)

### Conclusion:

Thus, the ranges can be summarized as:
\[ R_A = R_C < R_B \]

So, the projectile B, which is launched at \(45^\circ\), has the maximum range, while projectiles A and C, launched at \(30^\circ\) and \(60^\circ\) respectively, have equal ranges.

### Final Result:
- \( R_A = R_C \)
- \( R_B > R_A \) and \( R_B > R_C \)