Three balls of same masses are projected with equal speeds at angle `15^(@), 45^(@), 75^(@)`, and their ranges are respectively `R_(1),R_(2)` and `R_(3)`, then

Options

A. `R_(1) gt R_(2) gt R_(3)`
B. `R_(1) lt R_(2) lt R_(3)`
C. `R_(1) = R_(2) = R_(3)`
D. `R_(1) = R_(3) lt R_(2)`

Explanation

To solve the problem, we need to find the relationship between the ranges \( R_1, R_2, R_3 \) of three balls projected at angles \( 15^\circ, 45^\circ, \) and \( 75^\circ \) respectively. We will 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 speed,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity.

### Step 1: Calculate \( R_1 \) for \( \theta_1 = 15^\circ \)

Using the range formula:

\[ R_1 = \frac{u^2 \sin 2(15^\circ)}{g} \]

Calculating \( \sin 2(15^\circ) \):

\[ \sin 2(15^\circ) = \sin 30^\circ = \frac{1}{2} \]

Thus,

\[ R_1 = \frac{u^2 \cdot \frac{1}{2}}{g} = \frac{u^2}{2g} \]

### Step 2: Calculate \( R_2 \) for \( \theta_2 = 45^\circ \)

Using the range formula:

\[ R_2 = \frac{u^2 \sin 2(45^\circ)}{g} \]

Calculating \( \sin 2(45^\circ) \):

\[ \sin 2(45^\circ) = \sin 90^\circ = 1 \]

Thus,

\[ R_2 = \frac{u^2 \cdot 1}{g} = \frac{u^2}{g} \]

### Step 3: Calculate \( R_3 \) for \( \theta_3 = 75^\circ \)

Using the range formula:

\[ R_3 = \frac{u^2 \sin 2(75^\circ)}{g} \]

Calculating \( \sin 2(75^\circ) \):

\[ \sin 2(75^\circ) = \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2} \]

Thus,

\[ R_3 = \frac{u^2 \cdot \frac{1}{2}}{g} = \frac{u^2}{2g} \]

### Step 4: Compare the ranges

From our calculations, we have:

- \( R_1 = \frac{u^2}{2g} \)
- \( R_2 = \frac{u^2}{g} \)
- \( R_3 = \frac{u^2}{2g} \)

Now, we can see that:

\[ R_1 = R_3 \]

And since \( R_2 \) is greater than both \( R_1 \) and \( R_3 \):

\[ R_2 > R_1 \quad \text{and} \quad R_2 > R_3 \]

### Conclusion

The relationship between the ranges is:

\[ R_1 = R_3 < R_2 \]