Question

A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range R.In terms of its original range R, what is the range R_max the rock can attain if it is launched at the same speed but at the optimal angle for maximum range?

Answer 1

$R=\dfrac{u^2\sin{2\theta}}{g}\\ H=\dfrac{u^2\sin^2{\theta}}{2g}\\ \dfrac{u^2\sin{2\theta}}{g}=\dfrac{u^2\sin^2{\theta}}{2g}\\ 2\sin{2\theta}=\sin^2{\theta}\\ 4\sin{\theta}\cos{\theta}=\sin^2{\theta}\\ \tan{\theta}=4\\ \theta=\arctan{4}=76\degree\\ R=\dfrac{u^2\sin{2\theta}}{g}\\ At\ same\ speed\ and\ optimal\ angle\ for\\ maximum\ range\\ \theta=45\degree\\ R_{max}=\dfrac{u^2}{g}\\ R=R_{max}\sin{2\theta}\\ R_{max}=\dfrac{R}{\sin{2\theta}}$