The maximum horizontal range of a projectile is 400 m. The maximum height attained by it will be

The correct answer is 100 m.

Step-by-Step Explanation:

To find the maximum height attained by the projectile, let us look at the mathematical formulas governing projectile motion.

1. Maximum Horizontal Range ($R_{\text{max}}$):
The horizontal range $R$ of a projectile launched with an initial velocity $u$ at an angle $\theta$ to the horizontal is given by the formula:
$R=\frac{u^{2}sin\left(2\theta \right)}{g}$
where $g$ is the acceleration due to gravity.

The horizontal range is maximum when the term $sin\left(2\theta \right)$ is equal to its maximum value, which is 1. This happens when:
$2\theta =90°\Rightarrow \theta =45°$

Thus, the formula for maximum horizontal range is:
$R_{\text{max}}=\frac{u^2}{g}$

We are given that $R_{\text{max}}=400\text{m}$. Therefore:
$\frac{u^2}{g}=400\text{m}$

2. Maximum Height ($H$):
The formula for the maximum height attained by a projectile launched at an angle $\theta$ is:
$H=\frac{u^2{sin}^2\theta }{2g}$

Since the projectile is launched at the angle that maximizes the range ($\theta =45°$), we substitute $\theta =45°$ into the height formula:
$sin(45°)=\frac{1}{\sqrt{2}}$
${sin}^2(45°)={\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{1}{2}$

Now, substituting this value back into the height equation:
$H=\frac{u^2}{2g}·\frac{1}{2}=\frac{u^2}{4g}$

3. Relating Maximum Height to Maximum Range:
We can express the maximum height in terms of the maximum horizontal range:
$H=\frac{1}{4}\left(\frac{u^2}{g}\right)=\frac{R_{\text{max}}}{4}$

Given $R_{\text{max}}=400\text{m}$:
$H=\frac{400}{4}=100\text{m}$

Therefore, the maximum height attained by the projectile is 100 m.