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 solve the problem, we need 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 \(u\).

### Step 1: Determine the range formula for projectile motion
The range \(R\) of a projectile launched at an angle \(\theta\) with initial speed \(u\) is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \(g\) is the acceleration due to gravity.

### Step 2: Calculate the range for the first projectile
For the first projectile launched at an angle \((\frac{\pi}{4} + \theta)\):
\[ R_1 = \frac{u^2 \sin(2(\frac{\pi}{4} + \theta))}{g} \]
Using the identity \(\sin(A + B) = \sin A \cos B + \cos A \sin B\):
\[ R_1 = \frac{u^2 \sin(\frac{\pi}{2} + 2\theta)}{g} \]
Since \(\sin(\frac{\pi}{2} + x) = \cos(x)\):
\[ R_1 = \frac{u^2 \cos(2\theta)}{g} \]

### Step 3: Calculate the range for the second projectile
For the second projectile launched at an angle \((\frac{\pi}{4} - \theta)\):
\[ R_2 = \frac{u^2 \sin(2(\frac{\pi}{4} - \theta))}{g} \]
Using the same sine identity:
\[ R_2 = \frac{u^2 \sin(\frac{\pi}{2} - 2\theta)}{g} \]
Since \(\sin(\frac{\pi}{2} - x) = \cos(x)\):
\[ R_2 = \frac{u^2 \cos(2\theta)}{g} \]

### Step 4: Calculate the ratio of the ranges
Now, we can find the ratio of the ranges \(R_1\) and \(R_2\):
\[ \frac{R_1}{R_2} = \frac{\frac{u^2 \cos(2\theta)}{g}}{\frac{u^2 \cos(2\theta)}{g}} = 1 \]

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