Revisiting Ballistic Motion in the Presence of Air Resistance and Fluid Dynamics

[PlanckShift]

I'm thinking about the ballistic projectile again When you start out in first year phyz the first thing they teach is the ballistic projectile without air resistance Maybe at some time later they might go to a simple linear resistance term. Now I want to throw spin and oscillation into the mix. I'm convinced that this treatment requires the presence of air resistance but I'd ike to hear from someone who has actually studied this already maybe someone who was in the infantry or something?

 

[Simon Bridge]

Yes it does require air resistance.
You can get a good practical overview by looking up the physics of ball sports - where there is a great deal of money invested in this sort of research. Just because ballistics is usually taught in terms of cannons, does not mean military applications are the focus.

You hit a golf ball you do not get a parabola. You get loft, hook and slice depending on the spin. With millions in golf you can bet you this is well studied.

 

[Jolb]

I don't know what you mean by "oscillation..." do you mean the projectile itself is deforming in some oscillatory way?

Anyway, the linear resistance term itself is just an idealization of the viscosity of air (and the finite speed of sound). One huge field that you'd need to study to understand these projectile phenomena in the air (including why a ball spinning in the air might hook) is fluid dynamics.

The basic equation of fluid dynamics/hydrodynamics is the Navier-Stokes equation, which to this day is not well understood. (It was first studied by Euler.) It's nonlinear, it leads to chaos and turbulence, it's extremely difficult to simulate, etc. In fact there is still a million-dollar Millenium Prize still standing--they'll pay you a million dollars if you can simply show that for arbitrary initial conditions subject to the Navier-Stokes equation, a solution exists.

[As a joke, here's my "mathematical proof by physics" that solutions to the N-S equation exist:
1)Real fluids obey the Navier-Stokes equation. 2) Given an initial velocity field, I can set up a real fluid that's moving in that way. 3) The real fluid does its thing without the universe caving in on itself or screeching to a halt. QED]

Fluid dynamics has even been controversial in physics history. Bernoulli showed in the late 1700's what's now known as Bernoulli's principle, which showed the possibility of wing-based flying machines, but about a hundred years later Lord Kelvin proved Kelvin's Circulation Theorem, which led him to conclude that heavier-than-air flying machines are impossible. It took "experimental work" by the Wright brothers to prove him wrong. I've heard that to this day, aeronautical engineers will get in arguments over what really causes airplane wings to generate lift.

Also, there was an interesting debate because some entomologist claimed that it's physically impossible for bumblebees to fly, given their wingspan. Turned out the solution to the problem involved the turbulent phenomena that come out of the NS equation.