Q. Two stones are projected with same velocity $v$ at an angle $\theta$ and $(90{^\circ} - \theta)$. If $H$ and $H_{1}$ are greatest heights in the two paths, what is the relation between $R,\ H$ and $H_{1}$?
A) $R = 4\sqrt{HH_{1}}$
B) $R = \sqrt{HH_{1}}$
C) $R = 4HH_{1}$
D) None of these
Explanation
(a) Range of projectile,
$R = \frac{2u^{2}\sin^{2}\theta}{g}$ …(i)
Height, $H = \frac{u^{2}\sin^{2}\theta}{2g}$ …(ii)
$H_{1} = \frac{u^{2}{\sin^{2}(90{^\circ} -}{\theta)}}{2g} = \frac{u^{2}\cos^{2}\theta}{g}$ …(iii)
Then $HH_{1} = \frac{u^{2}\sin^{2}\theta \times u^{2}\cos^{2}\theta}{2g \times 2g}$ …(iv)
From Eq. (i) we get,
$R^{2} = \frac{4u^{2}\sin^{2}\theta \times u^{2}\cos^{2}\theta \times 4}{2g \times 2g}$
$R = \sqrt{16HH_{1}}$ [from Eq. (iv)]
$\therefore\ R = 4\sqrt{HH_{1}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.