Ratio of the ranges of the bullets fired from a gun `(` of constant muzzle speed `)` at angle `theta, 2 theta &4 theta` is found in the ratio `x:2:2`, then the value of `x` will be `(` Assume same muzzle speed of bullets `)`

Options

A. 1
B. 2
C. `sqrt(3)`
D. none of these

Explanation

To solve the problem, we need to find the value of \( x \) in the ratio of the ranges of bullets fired at angles \( \theta \), \( 2\theta \), and \( 4\theta \) given that the ranges are in the ratio \( x : 2 : 2 \).

### Step-by-Step Solution:

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

2. **Calculate Ranges for Each Angle**:
- For angle \( \theta \):
\[ R_1 = \frac{u^2 \sin(2\theta)}{g} \]
- For angle \( 2\theta \):
\[ R_2 = \frac{u^2 \sin(4\theta)}{g} \]
- For angle \( 4\theta \):
\[ R_3 = \frac{u^2 \sin(8\theta)}{g} \]

3. **Set Up the Ratio**:
According to the problem, the ranges are in the ratio:
\[ R_1 : R_2 : R_3 = x : 2 : 2 \]
This implies:
\[ \frac{R_1}{R_2} = \frac{x}{2} \quad \text{and} \quad \frac{R_2}{R_3} = \frac{2}{2} = 1 \]

4. **Equate Ranges**:
From the second ratio \( R_2 = R_3 \):
\[ \frac{u^2 \sin(4\theta)}{g} = \frac{u^2 \sin(8\theta)}{g} \]
This simplifies to:
\[ \sin(4\theta) = \sin(8\theta) \]

5. **Use the Sine Identity**:
The sine function has the property that \( \sin A = \sin B \) implies \( A = B + n \cdot 180^\circ \) or \( A = 180^\circ - B + n \cdot 180^\circ \) for some integer \( n \). Thus, we can write:
\[ 4\theta = 8\theta - n \cdot 180^\circ \]
Rearranging gives:
\[ 4\theta = n \cdot 180^\circ \implies \theta = \frac{n \cdot 180^\circ}{4} = n \cdot 45^\circ \]

6. **Calculate the Ranges**:
Now, substituting \( \theta = 45^\circ \) into the range equations:
- \( R_1 = \frac{u^2 \sin(90^\circ)}{g} = \frac{u^2}{g} \)
- \( R_2 = \frac{u^2 \sin(180^\circ)}{g} = 0 \) (not applicable)
- \( R_3 = \frac{u^2 \sin(360^\circ)}{g} = 0 \) (not applicable)

Therefore, we need to find the correct angles that satisfy the ratio \( x : 2 : 2 \).

7. **Final Ratio Calculation**:
Now, we can find the ratio of \( R_1 \) and \( R_2 \):
\[ R_1 = \frac{u^2 \sin(2\theta)}{g}, \quad R_2 = \frac{u^2 \sin(4\theta)}{g} \]
Since \( R_2 = R_3 \), we can equate:
\[ \frac{R_1}{R_2} = \frac{\sin(2\theta)}{\sin(4\theta)} = \frac{x}{2} \]
Using the double angle identity, we can find \( x \).

8. **Conclusion**:
After calculating the values, we find that:
\[ x = 2 \cdot \frac{\sin(2\theta)}{\sin(4\theta)} \]
By substituting the angles, we can find the value of \( x \).