Two projectiles of same mass have their maximum kinetic energies in...

The kinetic energy of the projectiles is maximum at the point of release. The ratio of K.E. of projectiles $$\begin{aligned}& \frac{\frac{1}{2} m u_1^2}{\frac{1}{2} m u_2^2}=\frac{4}{1}\\ & \frac{u_1^2}{u_2^2}=\frac{4}{1} \end{aligned}$$The ratio of maximum heights of projectiles $\frac{H_1}{H_2}=\frac{4}{1}$$\begin{aligned}& \frac{u_1^2 \sin ^2 \theta_1}{2 g}\\ & \frac{u_2^2 \sin ^2 \theta_2}{2 g}=\frac{4}{1}\\ & \frac{u_1^2}{u_2^2} \cdot \frac{\sin ^2 \theta_1}{\sin ^2 \theta_2}=\frac{4}{1}\\ & \frac{4}{1} \cdot \frac{\sin ^2 \theta_1}{\sin ^2 \theta_2}=\frac{4}{1} \end{aligned}$$($from equation (i)) $$\begin{aligned}& \sin ^2 \theta_1=\sin ^2 \theta_2 \\ & \theta_1=\theta_2 \end{aligned}$$Now, ratio$of ranges of projectiles$$\begin{aligned}& \frac{R_1}{R_2}=\frac{u_1^2 \sin 2 \theta_1}{u_2^2 \sin 2 \theta_2}\\ & \frac{R_1}{R_2}=\frac{u_1^2 \sin 2 \theta_1}{u_2^2 \sin 2 \theta_1} \quad\left(\because \theta_1=\theta_2\right) \\ & \frac{R_1}{R_2}=\frac{u_1^2}{u_2^2}, \frac{R_1}{R_2}=\frac{4}{1} \end{aligned}$$