Two projectiles $A$ and $B$ are thrown with initial velocities of $40 \mathrm{m} {\mathrm{s}}^{-1}$ and $60 \mathrm{m} {\mathrm{s}}^{-1}$ at angles $30°$ and $60°$ with the horizontal respectively. The ratio of their ranges respectively is $(\mathrm{g} = 10 \mathrm{m} {\mathrm{s}}^{-2})$

The range $\left(R_1\right)$ of the first projectile is given by

$R_1=\frac{{u_1}^2\sin 2{\theta }_1}{g} ...\left(1\right)$

The range $\left(R_2\right)$ of the second projectile is given by

$R_2=\frac{{u_2}^2\sin 2{\theta }_2}{g} ...\left(2\right)$

Divide equation (1) by equation (2) and simplify to obtain the ratio of the ranges.

$\begin{array}{rcl}\frac{R_1}{R_2}& =& \frac{\frac{{u_1}^2\sin 2{\theta }_1}{g}}{\frac{{u_2}^2\sin 2{\theta }_2}{g}}\\ & =& \frac{{u_1}^2\sin 2{\theta }_1}{{u_2}^2\sin 2{\theta }_2} ...\left(3\right)\end{array}$

Substitute the values of the known parameters into equation (3) to calculate the required ratio.

 $\begin{array}{rcl}\frac{R_1}{R_2}& =& \frac{{40}^2\times \sin {60}^{°}}{{60}^2\times \sin {120}^{°}}\\ & =& \frac{4}{9}\end{array}$