An object is projected at an angle of `45^(@)` with the horizontal. The horizontal range and the maximum height reached will be in the ratio.

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Explanation

To find the ratio of the horizontal range (R) to the maximum height (H) for an object projected at an angle of 45 degrees with the horizontal, we can follow these steps:

### Step 1: Write the formulas for horizontal range and maximum height
The formulas for horizontal range (R) and maximum height (H) when an object is projected at an angle θ are:
- Horizontal Range, \( R = \frac{u^2 \sin(2\theta)}{g} \)
- Maximum Height, \( H = \frac{u^2 \sin^2(\theta)}{2g} \)

### Step 2: Substitute θ = 45 degrees
Since the angle of projection θ is given as 45 degrees, we can substitute this value into the formulas:
- \( \sin(2\theta) = \sin(90^\circ) = 1 \)
- \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \)

### Step 3: Calculate R and H
Substituting θ into the formulas:
- For the horizontal range:
\[ R = \frac{u^2 \cdot 1}{g} = \frac{u^2}{g} \]
- For the maximum height:
\[ H = \frac{u^2 \left(\frac{1}{\sqrt{2}}\right)^2}{2g} = \frac{u^2 \cdot \frac{1}{2}}{2g} = \frac{u^2}{4g} \]

### Step 4: Find the ratio R to H
Now we can find the ratio \( \frac{R}{H} \):
\[ \frac{R}{H} = \frac{\frac{u^2}{g}}{\frac{u^2}{4g}} = \frac{u^2}{g} \cdot \frac{4g}{u^2} = 4 \]

### Step 5: Express the ratio in the form R:H
Thus, the ratio of the horizontal range to the maximum height is:
\[ R : H = 4 : 1 \]

### Conclusion
The ratio of the horizontal range to the maximum height when the object is projected at an angle of 45 degrees is \( 4 : 1 \).