Analytical Drag Models For Projectiles?

[Three1415]

Are there any for scenarios involving Newton (i.e, F = kv^2) drag? Despite my own searching I have not yet found any; all models of this sort seem to be utilizing either the fairly inaccurate Stokes drag (F=kv) or using Taylor series-like numerical approximations. I am asking because I appear to have derived a general analytical solution without extreme difficulty, and am simply surprised that no-one has done something like this before. Are approximations of this sort just generally disregarded in favor of numerical solutions to the Navier-Stokes equations, such that no-one thinks about Newton drag models? Or am I just missing published papers?

[arkie87]

Are there any for scenarios involving Newton (i.e, F = kv^2) drag? Despite my own searching I have not yet found any; all models of this sort seem to be utilizing either the fairly inaccurate Stokes drag (F=kv) or using Taylor series-like numerical approximations. I am asking because I appear to have derived a general analytical solution without extreme difficulty, and am simply surprised that no-one has done something like this before. Are approximations of this sort just generally disregarded in favor of numerical solutions to the Navier-Stokes equations, such that no-one thinks about Newton drag models? Or am I just missing published papers?

Perhaps, because k is piece-wise defined (since k is a function of mach number and geometry etc...), whereas for stokes flow, k is constant?

Also, if the projectile is moving in a gravitational field, it will fall so the attitude will change, and i'm sure k is a function of attitude...

[K^2]

There are two cases with simple analytical solutions. No gravity, or strictly along gravity. The general case for 2D motion with gravity is extremely complicated. There are analytical treatments, but you can never get a fully analytical general solution. If you understand PDEs, I can walk you through the key steps of derriving 2D motion. Otherwise, I'm afraid the only useful formulae I can give you are these for simple special cases.

No gravity: v(t) = v₀ / (1 + k v₀ t). v(0) = v₀.

Gravity, object is falling down: v(t) = - v_t tanh(g t / v_t). v(0) = 0. v_t = sqrt(g/k) is terminal velocity. tanh is hyperbolic tangent function. You can find it on most calculators, or just use Google Caclulator.

Gravity, object is launched upwards: v(t) = v[sib]t tan(g t / v_t - a). You need to solve for a, such that v(0) = v₀. This solution works right up until body reaches the apex. Then you need to switch to the above solution for falling back down.