Question

Ratio of minimum kinetic energies of two projectiles of same mass is $4:1$ . The ratio of the maximum height attained by them is also $4:1$ . The ratio of their ranges would be

• A. $16:1$
• B. $4:1$
• C. $8:1$
• D. $2:1$

Answer 1

Answer: (B)

Minimum kinetic energies are at highest point
$\frac{\frac{1}{2} m u_{1}^{2} cos^{2} \theta _{1}}{\frac{1}{2} m u_{2}^{2} cos^{2} ⁡ \theta _{2}}=\frac{4}{1}$
$\Rightarrow \, \, \frac{u_{1} cos \theta _{1}}{u_{2} cos ⁡ \theta _{2}}=2$ ......(1)
and ratio of the maximum height
$\frac{u_{1}^{2} sin^{2} \theta _{1}}{u_{2}^{2} sin^{2} ⁡ \theta _{2}}=\frac{4}{1}$
or $\frac{u_{1} sin \theta _{1}}{u_{2} sin ⁡ \theta _{2}}=\frac{2}{1}$ ...(2)
from equation no. (1) and (2)
$\frac{u_{1} sin \theta _{1} . u_{1} cos ⁡ \theta _{1}}{u_{2} sin ⁡ \theta _{2} . u_{2} cos ⁡ \theta _{2}}=\frac{4}{1}$
$\frac{\frac{g R_{1}}{2}}{\frac{g R_{2}}{2}}=\frac{4}{1}$
$\Rightarrow \, \, \frac{R_{1}}{R_{2}}=\frac{4}{1}$