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Q.
The horizontal range $R$ of two projectiles are same when their maximum heights are $H_{1}$ and $H_{2}$ . The value of $R$ is (assume equal speed of projection)
NTA AbhyasNTA Abhyas 2020
Solution:
Hint : For same range, sum of angle of projection is $90^\circ .$
$H_{1}=\frac{u^{2} \left(sin\right)^{2} \left(\left(\theta \right)_{1}\right)}{2 g}H_{2}=\frac{u^{2} \left(sin\right)^{2} \left(\theta \right)_{2}}{2 g}$
since $R$ is same
$\Rightarrow \theta _{1}+\theta _{2}=90^\circ $
$\Rightarrow H_{1}=\frac{u^{2} sin^{2} \theta _{1}}{2 g},H_{2}=\frac{u^{2} cos^{2} \theta _{1}}{2 g}$
Multiplying
$H_{1}H_{2}=\frac{u^{4} sin^{2} \theta _{1} cos^{2} \theta _{1}}{4 g}$
$\sqrt{H_{1} H_{2}}=\frac{u^{2} sin \theta _{1} cos \theta _{1}}{2 g}$
$\sqrt{H_{1} H_{2}}=\frac{u^{2} sin \left(2 \left(\theta \right)_{1}\right)}{4 g}$
$\Rightarrow R=4\sqrt{H_{1} H_{2}}$