Range of a projectile is R, when the angle of projection is `30^(@)`. Then, the value of the other angle of projection for the same range is

Options

A. `45^(@)`
B. `60^(@)`
C. `50^(@)`
D. `40^(@)`

Answer

To find the other angle of projection for the same range when the initial angle is \(30^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand 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 Range for \(30^\circ\)**:
For the angle of projection \( \theta = 30^\circ \):
\[ R_1 = \frac{u^2 \sin(2 \times 30^\circ)}{g} = \frac{u^2 \sin(60^\circ)}{g} \]
Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), we can write:
\[ R_1 = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} \]

3. **Set Up the Equation for the Other Angle**:
Let the other angle of projection be \( \theta \). The range for this angle is:
\[ R_2 = \frac{u^2 \sin(2\theta)}{g} \]
According to the problem, \( R_1 = R_2 \). Therefore:
\[ \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{u^2 \sin(2\theta)}{g} \]

4. **Cancel Common Terms**:
We can cancel \( \frac{u^2}{g} \) from both sides (assuming \( u \) and \( g \) are not zero):
\[ \frac{\sqrt{3}}{2} = \sin(2\theta) \]

5. **Find \(2\theta\)**:
To find \(2\theta\), we take the inverse sine:
\[ 2\theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), so:
\[ 2\theta = 60^\circ \]

6. **Calculate \( \theta \)**:
Dividing both sides by 2 gives:
\[ \theta = \frac{60^\circ}{2} = 30^\circ \]

7. **Find the Other Angle**:
However, we also know that the sine function has a property where:
\[ \sin(180^\circ - x) = \sin(x) \]
Therefore, the other angle that gives the same sine value is:
\[ 2\theta = 180^\circ - 60^\circ = 120^\circ \]
Thus:
\[ \theta = \frac{120^\circ}{2} = 60^\circ \]

### Conclusion:
The other angle of projection for the same range \( R \) is \( \theta = 60^\circ \).