Question

Two stones are projected with the same speed but making different angles with the horizontal. Their horizontal ranges are equal. The angle of projection of one is $\frac{\pi }{3}$ and the maximum height reached by it is $102m$. Then the ratio of maximum heights is

• A. $1:1$
• B. $1:3$
• C. $1:\sqrt{3}$
• D. $2:\sqrt{3}$

Answer 1

Answer: (B)

$1:3$

Range of a projectile $R\propto \sin 2\theta$

$\Rightarrow \sin 2{\theta }_1=\sin 2{\theta }_2$

$\Rightarrow 2{\theta }_1=180-2{\theta }_2$

$\Rightarrow [{\theta }_1+{\theta }_2={90}^0]$

${\theta }_2={60}^0,{\theta }_1={30}^0$

$H_{max}\propto {\sin }^2\theta$

$\Rightarrow \frac{(H_{max}{)}_1}{(H_{max}{)}_2}=\frac{{\sin }^2{30}^o}{{\sin }^2{60}^o}=\frac{(\frac{1}{2}{)}^2}{(\frac{\sqrt{3}}{2}{)}^2}=\frac{1}{3}$