# Projectile initial speed with maximum range and height.

A projectile has range R and maximum height H. Prove that the initial speed is $\sqrt{\frac{g(R^2+16H^2)}{8H}}$ I got the following equations:

Vertically: $H=utsin \theta -gt^2/2$,$usin \theta =gt$, $u^2sin^2\theta =2gh$

Horizontally: $R=2utcos\theta$

I then proceeded by eliminating $t$ to get $H=\frac{R}{2}tan\theta-\frac{gR^2}{8u^2cos^2\theta}$. Didn't get any closer to the answer after this point

$$R=\frac {u^2 \sin 2 \theta}{g} \\H=\frac {u^2 \sin ^2 \theta}{2g}$$
Now all you have to do is eliminate $\theta$ and solve for $u$ which is the initial velocity.