Question

Two particles are projected from the same point with the same speed $u$ such that they have the same range $R$, but different maximum heights, $h_1$ and $h_2$. Which of the following is correct ?

• A. $R^2=4h_1{h}_2$
• B. $R^2=16h_1{h}_2$
• C. $R^2=2h_1{h}_2$
• D. $R^2=h_1{h}_2$

Answer 1

The correct option is B $R^2=16h_1{h}_2$

For the same range, the angle of projection will be $\theta$ and ${90}^{\circ }-\theta$. So

$h_1=\frac{u^2{\sin }^2\theta }{2g}$ and

$h_2=\frac{u^2{\sin }^2({90}^{\circ }-\theta )}{2g}=\frac{u^2{\cos }^2\theta }{2g}$

$\text{Also, R}=\frac{u^2\sin 2\theta }{g}$

So, $h_1{h}_2=\frac{u^2{\sin }^2\theta }{2g}\times \frac{u^2{\cos }^2\theta }{2g}$

$h_1{h}_2=\frac{u^4}{16g^2}(2\sin \theta \cos \theta {)}^2=\frac{1}{16}{\left(\frac{u^2\sin 2\theta }{g}\right)}^2$

$h_1{h}_2=\frac{R^2}{16}$

$\therefore R^2=16h_1{h}_2$

Hence, option $\left(\text{B}\right)$ is correct.