Two projectiles A and B are thrown with initial velocities of 40m s...

The range $\left({R}_{1}\right)$ of the first projectile is given by

${R}_{1} = \dfrac{{{u}_{1}}^{2}\sin 2{\theta}_{1}}{g} . . . ( 1 )$

The range $\left({R}_{2}\right)$ of the second projectile is given by

${R}_{2} = \dfrac{{{u}_{2}}^{2}\sin 2{\theta}_{2}}{g} . . . ( 2 )$

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

$\begin{cases}\dfrac{{R}_{1}}{{R}_{2}} & = & \dfrac{\dfrac{{{u}_{1}}^{2}\sin 2{\theta}_{1}}{g}}{\dfrac{{{u}_{2}}^{2}\sin 2{\theta}_{2}}{g}} \\ & = & \dfrac{{{u}_{1}}^{2}\sin 2{\theta}_{1}}{{{u}_{2}}^{2}\sin 2{\theta}_{2}} . . . \left(3\right)\end{cases}$

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

$\begin{cases}\dfrac{{R}_{1}}{{R}_{2}} & = & \dfrac{{40}^{2} \times sin{60}^{°}}{{60}^{2} \times sin{120}^{°}} \\ & = & \dfrac{4}{9}\end{cases}$