The horizontal range and miximum height attained by a projectile are `R and H`, respectively. If a constant horizontal acceleration `a = g//4` is imparted to the projectile due to wind, then its horizontal range and maximum height will be

Options

A. `(R + H),(H)/(2)`
B. `(R + (H)/(2)), 2H`
C. `(R + 2H),H`
D. `(R + H), H`

Explanation

To solve the problem, we need to analyze the effects of the horizontal acceleration on the projectile's motion. We will derive the new horizontal range and maximum height step by step.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- The initial horizontal range \( R \) and maximum height \( H \) of the projectile are given.
- The formulas for the horizontal range and maximum height are:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
\[ H = \frac{u^2 \sin^2(\theta)}{2g} \]
- Here, \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

2. **Identify the Effect of Horizontal Acceleration**:
- A constant horizontal acceleration \( a = \frac{g}{4} \) is imparted to the projectile due to wind.
- This acceleration affects the horizontal motion but does not affect the vertical motion, hence the maximum height \( H \) remains unchanged.

3. **Calculate the New Horizontal Range**:
- The new horizontal range \( R' \) can be calculated using the formula for motion with constant acceleration:
\[ R' = R + \frac{1}{2} a t^2 \]
- The time of flight \( t \) for the projectile is given by:
\[ t = \frac{2u \sin(\theta)}{g} \]
- Substitute \( a \) and \( t \) into the range formula:
\[ R' = R + \frac{1}{2} \left(\frac{g}{4}\right) \left(\frac{2u \sin(\theta)}{g}\right)^2 \]

4. **Simplify the Expression**:
- Calculate \( t^2 \):
\[ t^2 = \left(\frac{2u \sin(\theta)}{g}\right)^2 = \frac{4u^2 \sin^2(\theta)}{g^2} \]
- Substitute \( t^2 \) back into the equation for \( R' \):
\[ R' = R + \frac{1}{2} \cdot \frac{g}{4} \cdot \frac{4u^2 \sin^2(\theta)}{g^2} \]
- Simplifying gives:
\[ R' = R + \frac{u^2 \sin^2(\theta)}{2g} \]

5. **Relate the New Range to Maximum Height**:
- From the formula for maximum height \( H \):
\[ H = \frac{u^2 \sin^2(\theta)}{2g} \]
- Therefore, we can substitute \( H \) into the equation for \( R' \):
\[ R' = R + H \]

6. **Final Results**:
- The new horizontal range \( R' \) is:
\[ R' = R + H \]
- The maximum height \( H' \) remains:
\[ H' = H \]

### Conclusion:
- The new horizontal range is \( R + H \) and the maximum height remains \( H \).