Question

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)

• A. $\frac{H_1+H_2}{2}$
• B. $\sqrt{H_1{H}_2}$
• C. $4\sqrt{H_1{H}_2}$
• D. $H_1+H_2$

Answer 1

Answer: (C)

Hint : For same range, sum of angle of projection is $90^{\circ }.$
$H_1=\frac{u^2{\left(sin\right)}^2\left({\left(\theta \right)}_1\right)}{2g}{H}_2=\frac{u^2{\left(sin\right)}^2{\left(\theta \right)}_2}{2g}$
since $R$ is same
$\Rightarrow {\theta }_1+{\theta }_2=90^{\circ }$
$\Rightarrow H_1=\frac{u^{2}si{n}^2{\theta }_1}{2g},H_2=\frac{u^{2}co{s}^2{\theta }_1}{2g}$
Multiplying
$H_1{H}_2=\frac{u^{4}si{n}^2{\theta }_{1}co{s}^2{\theta }_1}{4g}$
$\sqrt{H_1{H}_2}=\frac{u^{2}sin{\theta }_{1}cos{\theta }_1}{2g}$
$\sqrt{H_1{H}_2}=\frac{u^{2}sin\left(2{\left(\theta \right)}_1\right)}{4g}$
$\Rightarrow R=4\sqrt{H_1{H}_2}$