Calculating ballistic trajectory with changing conditions during flight

Question

There is a good compilation of trajectory math in wikipedia.

But I need to calculate a trajectory that has non uniform conditions. E.g. the wind speed changes above certain altitude. (Cannot be modeled easily.)

• Should I calculate projectile's velocity vector e.g. every second and then for the next second based on that (having small enough t_delta)

• Or should I try to split the trajectory into pieces - based on the parameters (e.g. wind is v_{wind 1} between y₁ and y₂ so I calculate for y<y₁, y₁≤y<y₂ and y₂≤y separately).

• Try to build and solve a symbolic equation - run time - with all the parameters modeled. (Is this completely utopistic? Traditional programmin languages aren't too good solving symbols.)

• Something completely different... ?

Are there good languages / frameworks for handling symbolic math?

Answer 1

I'd suggest an "improved" first approach: solve the differential equations of motion numerically with e.g. the classic Runge-Kutta method.

The nice part is that with these algorithms, once you correctly set up your framework, you just have to write an "evaluate" function for the motion law (which can be almost anything - you don't need to restrict to particular forces), and everything should work fine (as far as the integration step is adequate).

Answer 2

If the conditions really are cleanly divided into two domains like that, then the second approach is probably best. The first approach is both imprecise and overkill, and the third, if done right, will wind up being equivalent to the second.