I need an equation to relate drag, mass, initial velocity, launching angle, and range.?

Question

I'm building an egg catapult (well, giant slingshot, actually) and given the egg's drag coefficient, mass, launching angle, and initial velocity I'd like to be able to predict the range. More specifically, I'd like to be able to target specific ranges by adjusting the initial velocity. Been too long since I've had physics, so i need some help on this one! I know how to do it ignoring drag, but that is not an appropriate assumption in this case.....

Thanks!

The equation needn't be closed-form; I have a working knowledge of calculus.

Answer 1

There's no way to incorporate the nonlinear drag force into a closed-form equation. You'll need a (relatively simple) time-stepped simulation in which the acceleration is computed at each step based on drag and gravity. It'll be something like the old gunnery sims that came out in the early days of computing. This is the general idea:
do
t = t+deltat
drag = f(previous v)
a = f(drag,g,etc.)
v = previous v + a*deltat
s = previous s + v*deltat
loop
Of course there are more sophisticated integration schemes than this which reduce errors considerably.