Parabolas and projectiles

Question

Given $2$ points, $A$ and $B$, if I am in $A$ and I have an inclination angle $c$, with how many velocity do I need to shoot a projectile to hit $B$ ? My problem is, how do I setup this data in an equation so that I can solve it with a program?

Answer 1

Let $A$ be the origin and $B$ be $(x,y)$. The angle of projecting is $c$. The parabolic trajectory is given by the formula below $$y = x \tan(c) - \dfrac{g}{2u^2 \cos^2(c)}x^2$$ This gives us $$u^2 = \dfrac{gx^2}{2 \cos^2(c) (y-x \tan(c))}$$ Hence the unique velocity is $$u = \left \vert \dfrac{x}{\cos(c)} \right \vert \sqrt{\dfrac{g}{2(y-x \tan(c))}}$$