JesseM said:And as I've said, the fact that the path of a tossed ball looks like a parabola has nothing to do with air resistance. Again, if we were to actually calculate the difference between an elliptical path and a parabolic path for a ball tossed above the surface (not continuing the paths past the surface) at ordinary velocities, I am confident that the difference will be microscopic.

Nonsense. Again, as far as I know there is no well-defined limit in which the curve of a ball tossed a small height would approach a circle or a hyperbola (again, only looking at the section of the curve above the surface, not extrapolating it further), but there is such a limit for a parabola. And if I come up with an actual numerical example, calculate the exact heights at different times using the exact elliptical solution, then calculate the heights using the parabolic approximation and find only a microscopic difference, you willRandallB said:And as I've said well within the microscopic range of a circle and a hyperbola as well

*not*be able to give me the equation of a circle or hyperbola that fits the elliptical curve so well that there is only a microscopic difference between them.

JesseM said:Are you willing to actually look at such an explicit numerical calculation and reconsider your position if it turns out I am right?

I didn't ask you whether you'd look at them, I asked you whether you wouldRandallB said:Sure if you can work out the numbers that translate these curve shapes to a flat surface I'll look at them.

**reconsider your position**if it turns out I'm correct that in a typical numerical example there is only a microscopic difference between the exact Newtonian prediction and the parabolic approximation (and if it turns out I'm also correct that it's not possible to find a circle or a hyperbola that fits the exact solution with such accuracy). Will you or won't you reconsider your position if this is the case? And are you in fact predicting that I am wrong that this is what will happen when we look at a numerical example, or are you unwilling to commit to a definite prediction?

I'm not sure I'll know how to solve the differential equations for the path when air resistance is included, but I can give it a try.RandallB said:Why not include what you expect the curve shape with air resistance to be as well, if it’s not a parabola it be worth knowing what it matches up best with, ellipse at perigee, circle, ellipse at apogee, hyperbola.