The factor of kinetic energy over force in the focus equations got me thinking about work integrals. The work integral also led me to think about the vertex height as a projection back on the y axis. Why? Because work is only done in the direction tangent to the force, which in this case is in the y direction. The calculation of the projection angle was a bit messy, but wound up with a clean if rather obscure result.

The following is neither here nor there, and I suspect will waste more time than it's worth at the moment, so I'm just including it as extra notes to go back to later.

I already know there's a Gudermannian lurking in all of this. The function for the arc length of the parabola contains one. The projection angle consisted of a tanget half angle formula which also leads back to Gudermannians, (seehttps://en.wikipedia.org/wiki/Tangent_half-angle_formula#The_Gudermannian_function).

The following is neither here nor there, and I suspect will waste more time than it's worth at the moment, so I'm just including it as extra notes to go back to later.

I already know there's a Gudermannian lurking in all of this. The function for the arc length of the parabola contains one. The projection angle consisted of a tanget half angle formula which also leads back to Gudermannians, (seehttps://en.wikipedia.org/wiki/Tangent_half-angle_formula#The_Gudermannian_function).