How can I solve initial velocity for a projectile given the max height and distance

Question

I'm trying to find the initial velocity of a tennis ball (magnitude and angle) given the initial height, max height and max range.

I believe this is possible with the equations below but I'm struggling with the algebra and trig to make it work!

Any help would be appreciated.

Answer 1

Compute the time to fall from $y_{max}$ to $0$ and from $y_{max}$ to $y_0$. This gives you the time of flight. From $x_f$ you can get $V_0\cos \theta$

Compute the vertical speed after the fall from $y_{max}$ to $y_0$. This is the initial upward speed $V_0 \sin \theta$

Take the $\arctan$ of the ratio of initial velocities to get $\theta$. Use that to get $V_0$.