Question

A body is projected with the same speed at two different angles such that the horizontal range is same in both the cases. If the maximum height attained are $20m$ and $80m$ respectively in the above two cases, then the range is

• A. $120m$
• B. $20m$
• C. $160m$
• D. $40m$

Answer 1

Answer: (C)

$160m$

Range $R\propto \sin 2\theta$
$\Rightarrow R_1=R_2$ and thus $\sin 2{\theta }_1=\sin 2{\theta }_2$
$\Rightarrow 2{\theta }_1=180-2{\theta }_2$
$\Rightarrow {\theta }_1+{\theta }_2={90}^{\circ }$.......$\left(1\right)$

$H_1=\frac{u^2{\sin }^2{\theta }_1}{2g}=20$
$H_2=\frac{u^2{\sin }^2{\theta }_2}{2g}=80\Rightarrow \frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{1}{2}$

from $\left(1\right)$ $\sin {\theta }_2=\cos {\theta }_1\Rightarrow \tan {\theta }_1=\frac{1}{2}\Rightarrow \sin {\theta }_1=\frac{1}{\sqrt{5}}$

$\Rightarrow \frac{u^2}{2g}=100$
$R=\frac{u^2}{g}\sin 2\theta =200\sin 2{\theta }_1=400\sin {\theta }_1\cos {\theta }_1$

$=400\times \frac{1}{\sqrt{5}}\times \frac{2}{\sqrt{5}}=160m$