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Scattered Notes on the Parabolic Trajectory Project

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).


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