Question

Two stones are projected with the same speed but making different angles with horizontal. Their ranges are equal. If the ranges of projection of one is $\pi /3$ and its maximum height is $h_1$ then the maximum height of the other will be

• A. $3h_1$
• B. $2h_1$
• C. $h_1/2$
• D. $h_1/3$

Answer 1

The correct option is D $h_1/3$

Speed=u is same for both stones.

Also the range is equal for both stones.

$R_1=R_2$

$\frac{u^2\sin 2{\theta }_1}{g}=\frac{u^2\sin 2{\theta }_2}{g}$

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

${\theta }_1=\frac{\pi }{3},{\theta }_2={30}^{\circ }=\frac{\pi }{6}$

Height $H=\frac{u^2{\sin }^2\theta }{2g}$

$H_1=\frac{u^2{\sin }^2{\theta }_1}{2g},H_2=\frac{u^2{\sin }^2{\theta }_2}{2g}$

$\frac{H_2}{H_1}=\frac{{\sin }^2{\theta }_2}{{\sin }^2{\theta }_1}$

$H_2=h_1\times \frac{{\sin }^2{30}^{\circ }}{{\sin }^2{60}^{\circ }}$

$=h_1\times \frac{1}{4}\times \frac{4}{3}$

$H_2=\frac{h_1}{3}$