Question

A projectile thrown with an initial velocity $u$ has the same range R when the maximum height attained by it is either $h_1$ or $h_2.$ Then $R,h_1$ and $h_2$ are related as

• A. $R=\sqrt{h_1{h}_2}$
• B. $R=2\sqrt{h_1{h}_2}$
• C. $R=3\sqrt{h_1{h}_2}$
• D. $R=4\sqrt{h_1{h}_2}$

Answer 1

The correct option is D

$R=4\sqrt{h_1{h}_2}$

The range of a projectile is the same for two angles of projection $\theta$ and ${90}^{\circ }-\theta .$ For these two angles of projection, the maximum heights are
$h_1=\frac{v_0^{2}si{n}^2\theta }{2g}and{h}_2=\frac{v_0^{2}si{n}^2({90}^{\circ }-\theta )}{2g}=\frac{v_0^{2}co{s}^2\theta }{2g}\therefore h_1{h}_2=\frac{v_0^{4}si{n}^2\theta co{s}^2\theta }{4g^2}$
Also, $R=\frac{2v_0^{2}sin\theta cos\theta }{g}\Rightarrow R^2=\frac{4v_0^{4}si{n}^2\theta co{s}^2\theta }{g^2}=16\times \frac{v_0^{4}si{n}^2\theta co{s}^2\theta }{4g^2}$
$\Rightarrow R^2=16h_1{h}_2$
$\Rightarrow R=4\sqrt{h_1{h}_2}$
Hence, the correct choice is (d).