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I'm probably being very dense here, but why isn't the x and y motion totally
separable. That is
Fx = -kvx^n
Fy = -kvy^n
Isn't it only the horizontal motion through the media that adds a horizontal
retarding force and the vertical motion that adds a vertical retarding
force?
This is how I set up my calculation..with the x and y forces,
accelerations, velocities, and displacements calculated separately--good old
introductory, algebra-based physics.
Even with your formulae, since vy << vx, there should be little difference
in the results.