The range of a projectile when launched at angle `theta` is same as when launched at angle `2theta`. What is the value of `theta` ?

Options

A. `15^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the problem, we need to find the angle \( \theta \) such that the range of a projectile launched at angle \( \theta \) is the same as when launched at angle \( 2\theta \).

### Step-by-Step Solution:

1. **Understand the Range Formula**:
The range \( R \) of a projectile launched with an initial velocity \( u \) at an angle \( \theta \) is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \( g \) is the acceleration due to gravity.

2. **Set Up the Equations**:
For the first launch at angle \( \theta \):
\[ R_1 = \frac{u^2 \sin(2\theta)}{g} \]
For the second launch at angle \( 2\theta \):
\[ R_2 = \frac{u^2 \sin(2 \cdot 2\theta)}{g} = \frac{u^2 \sin(4\theta)}{g} \]

3. **Equate the Ranges**:
Since it is given that the ranges are the same:
\[ R_1 = R_2 \]
This leads to:
\[ \frac{u^2 \sin(2\theta)}{g} = \frac{u^2 \sin(4\theta)}{g} \]
We can cancel \( \frac{u^2}{g} \) from both sides:
\[ \sin(2\theta) = \sin(4\theta) \]

4. **Use the Sine Identity**:
The sine function has the property that \( \sin A = \sin B \) implies:
\[ A = B + n \cdot 360^\circ \quad \text{or} \quad A = 180^\circ - B + n \cdot 360^\circ \]
For our case:
\[ 2\theta = 4\theta + n \cdot 360^\circ \quad \text{or} \quad 2\theta = 180^\circ - 4\theta + n \cdot 360^\circ \]

5. **Solving the First Case**:
From \( 2\theta = 4\theta + n \cdot 360^\circ \):
\[ -2\theta = n \cdot 360^\circ \quad \Rightarrow \quad \theta = -n \cdot 180^\circ \]
This does not yield a valid angle for projectile motion.

6. **Solving the Second Case**:
From \( 2\theta = 180^\circ - 4\theta + n \cdot 360^\circ \):
\[ 2\theta + 4\theta = 180^\circ + n \cdot 360^\circ \]
\[ 6\theta = 180^\circ + n \cdot 360^\circ \]
For \( n = 0 \):
\[ 6\theta = 180^\circ \quad \Rightarrow \quad \theta = 30^\circ \]

7. **Conclusion**:
Thus, the value of \( \theta \) is:
\[ \theta = 30^\circ \]