Question

Two stones are projected with same velocity $v$ at an angle $\theta$ and $(90-\theta)$. If $H$ and $H_{1}$ are the 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. $R =4 HH _{1}$
• D. None of the above

Answer 1

Answer: (A)

$R =4 \sqrt{ HH _{1}}$

Answer 2

Range of projectile, $R=\frac{2 u^{2} \sin \theta \cos \theta}{g}$ ...(i) Height $H=\frac{u^{2} \sin ^{2} \theta}{2 g}$ ...(ii) $H_{1}=\frac{u^{2} \sin ^{2}\left(90^{\circ}-\theta\right)}{2 g}=\frac{u^{2} \cos ^{2} \theta}{2 g}$ ...(iii) Then, $H H_{1}=\frac{u^{2} \sin ^{2} \theta u^{2} \cos ^{2} \theta}{2 g 2 g}$ ...(iv) From E (i), we get $R^{2}=\frac{4 u^{2} \sin ^{2} \theta u^{2} \cos ^{2} \theta \times 4}{2 g 2 g}$ $R=\sqrt{16 H H_{1}}$ [from E (iv)] $=4 \sqrt{H H_{1}}$