Two balls are projected from the same point in directions 30°and 60° with respect to the horizontal. What is the ratio of their initial velocities if they (a) attain the same height? (b) have the same range?

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Two balls are projected from the same point in directions 30°and 60° with respect to the horizontal. What is the ratio of their initial velocities if they (a) attain the same height? (b) have the same range?

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Detailed Solution

$θ_{1} = 30^{\circ}; θ_{2} = 60^{\circ}$
$a) Height = \frac{u^{2} \sin^{2} θ}{2 g}$
Same height $usin θ = constant$
$\frac{u_{1}}{u_{2}} = \frac{\sin θ_{2}}{\sin θ_{1}} = \frac{\sqrt{3}}{1}$
Horizontal range $R = \frac{u^{2} \sin 2 θ}{g}$
For same range $u^{2} \Rightarrow \sin 2 θ = constant$
$\frac{u_{1}}{u_{2}} = \sqrt{\frac{\sin 2 θ_{2}}{\sin 2 θ_{1}}} = \frac{1}{1}$