Question

Two particles A and B are projected with same speed so that the ratio of their maximum height reached is 3: 1 . If the speed of A is doubled without altering other parameters, the ratio of the horizontal ranges attained by A and B is

• A. 1: 1
• B. 2: 1
• C. 4: 1
• D. 3: 2

Answer 1

Answer: (C)

Given, $\frac{H_{1}}{H_{2}}=\frac{3}{1}$
$ \therefore H_{1}=3 H_{2}$
or $\frac{u^{2} \sin ^{2} \theta_{1}}{2 g}=\frac{3 u^{2} \sin ^{2} \theta_{2}}{2 g}$
or $\frac{\sin ^{2} \theta_{1}}{\sin ^{2} \theta_{2}}=\frac{3}{1}$
It is true if $\theta_{1}=60^{\circ}$ and $\theta_{2}=30^{\circ}$
When speed of $A$ is doubled, then horizontal range
$R_{1}=\frac{(2 u)^{2} \sin \left(2 \times 60^{\circ}\right)}{g}$
$=\frac{4 u^{2} \sin 120^{\circ}}{g}$
$=\frac{4 u^{2} \sin 60^{\circ}}{g}$
and $R_{2}=\frac{u^{2} \sin \left(2 \times 30^{\circ}\right)}{g}$
$=\frac{u^{2} \sin 60^{\circ}}{g}$
$\therefore \frac{R_{1}}{R_{2}}=\frac{4}{1}$