Two projectiles are projected angle `((pi)/(4) + theta)` and `((pi)/(4) - theta)` with the horizontal , where `theta lt (pi)/(4)`, with same speed. The ratio of horizontal ranges described by them is

Options

A. tan `theta` : 1
B. 1 : `tan^(2) theta `
C. ` 1 : 1 `
D. `1 : sqrt(3)`

Answer

To find the ratio of the horizontal ranges of two projectiles projected at angles \((\frac{\pi}{4} + \theta)\) and \((\frac{\pi}{4} - \theta)\) with the same speed, we can follow these steps:

### Step 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 speed, \(\theta\) is the angle of projection, and \(g\) is the acceleration due to gravity.

### Step 2: Identify the Angles
For the two projectiles:
- The first projectile is launched at an angle \(\theta_1 = \frac{\pi}{4} + \theta\).
- The second projectile is launched at an angle \(\theta_2 = \frac{\pi}{4} - \theta\).

### Step 3: Calculate the Ranges
Using the range formula, we can express the ranges \(R_1\) and \(R_2\) for the two projectiles:
\[ R_1 = \frac{u^2 \sin(2(\frac{\pi}{4} + \theta))}{g} \]
\[ R_2 = \frac{u^2 \sin(2(\frac{\pi}{4} - \theta))}{g} \]

### Step 4: Simplify the Sine Functions
Now, we simplify the sine functions:
- For \(R_1\):
\[ R_1 = \frac{u^2 \sin(\frac{\pi}{2} + 2\theta)}{g} = \frac{u^2 \cos(2\theta)}{g} \]
- For \(R_2\):
\[ R_2 = \frac{u^2 \sin(\frac{\pi}{2} - 2\theta)}{g} = \frac{u^2 \cos(2\theta)}{g} \]

### Step 5: Find the Ratio of Ranges
Now, we can find the ratio of the ranges:
\[ \frac{R_1}{R_2} = \frac{\frac{u^2 \cos(2\theta)}{g}}{\frac{u^2 \cos(2\theta)}{g}} = 1 \]

### Conclusion
Thus, the ratio of the horizontal ranges described by the two projectiles is:
\[ \frac{R_1}{R_2} = 1 \]

### Final Answer
The ratio of the horizontal ranges is \(1:1\).

---