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 $h_1\text{and}{h}_2$. Which of the following is correct ?

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

Answer 1

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

For the same range, angles of projection are, $\theta \text{and}{90}^{\circ }-\theta$

So,

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

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

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

Now,

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

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

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

Hence, $\left(B\right)$ is the correct answer.