Question

Two particles $A$ & $B$ are projected along different directions from the same point $P$ on the ground with same speed of $70\text{m/s}$ in the same vertical plane. They hit the ground at the same point $Q$ such that $PQ=480\text{m}$. Then :
(Use $g=9.8{\text{m/s}}^2,{\sin }^{-1}0.96={74}^{\circ },{\sin }^{-1}0.6={37}^{\circ }$)

• A. Ratio of their times of flights is $4:5$
• B. Ratio of their maximum hights is $9:25$
• C. Ratio of their minimum speeds during flight is $4:3$
• D. The bisector of the angle between their directions of projection makes 37º with horizontal.

Answer 1

The correct option is C Ratio of their minimum speeds during flight is $4:3$

As we know, that range is given by $R=\frac{u^2\sin 2\theta }{g}\Rightarrow 480=\frac{70\times 70\sin 2\theta }{10}$
${\theta }_1={37}^{\circ }$
${\theta }_2=90-37={53}^{\circ }$
(A) Ratio of time of flights $\frac{T_1}{T_2}=\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{\sin {37}^{\circ }}{\sin {53}^{\circ }}=\frac{3/5}{4/5}=\frac{3}{4}$
(B) Ratio of maximum heights $\frac{H_1}{H_2}=\frac{{\sin }^2{\theta }_1}{{\sin }^2{\theta }_2}=\frac{(\sin {37}^{\circ }{)}^2}{(\sin {53}^{\circ }{)}^2}=\frac{9/25}{16/25}=\frac{9}{16}$
(C) Ratio of minimum speed $\frac{V_{{\text{min}}_1}}{V_{{\text{min}}_2}}=\frac{\cos {\theta }_1}{\cos {\theta }_2}=\frac{\cos {37}^{\circ }}{\cos {53}^{\circ }}=\frac{4}{3}$