Calculating Time of Freefalling Sphere w/Drag and Varying Air Density

[ARROW 3]

I am looking at the problem of a freefalling sphere and want to calculate time taken to fall a distance. I want to model drag as well as take account of the varying density in the atmosphere.
I am using Drag = 1/2*A*C_d*v²*ρ; A - cross-sectional area, C_d - drag coefficient, v² - velocity squared and ρ - air density.
I have been able to form a DE and solve it for v and s (distance) but this assumes a constant density of air. I am also having trouble with the hyperbolic trig functions so think i might be taking a 'large hammer to break a small nut'.
Any ideas on how to simplify this or is the DE the right way to approach?

 

[Simon Bridge]

The simplest model I've seen just adds a drag proportional to the cube of the instantanious speed.

If you have a problem with density not being contant just replace that part with a function that describes how the density varies. You thinking of something like free-fall from near-space?

 

[ARROW 3]

Thanks for the reply.
With the simple model you mention, does the drag proportional to v3 replace the drag proportional to v2?
With regards to the function for density, I think the calculus is getting too complicated.
I am looking at freefall near space and want to model time taken to return the Earth's surface.

 

[Simon Bridge]

Thanks for the reply.
With the simple model you mention, does the drag proportional to v3 replace the drag proportional to v2?

Yes, and it would be empirically fitted.
In general you can make any model you feel you can get away with.

With regards to the function for density, I think the calculus is getting too complicated.
I am looking at freefall near space and want to model time taken to return the Earth's surface.

Well that's what you are going to have to do I'm afraid.

drag has form kf(x)v^2: k = constant so you'll be solving

$$m\frac{d^2y}{dt^2} = mg-k\rho(y)\left ( \frac{dy}{dt}\right )^2$$
... you'll have to make an approximation for the air density function anyway ... but you don't need a general solution: you could do this numerically!

 

[Vadar2012]

If you're modelling a small spehere that means you're probably modelling some type of asteroid. It can't be a spacecraft because they wouldn't follow such a trajectory. If it is an asteroid, what about modelling the reduction in mass due to break up in the atmosphere?

 

[Enthalpy]

The hyperbolic function the correct one, there is no simpler solution.
Something like V(t) = Vmax*atanh(...*t) with Vmax limited by drag
and x(t) = log(acos(...*t)) if I remember properly.

That was at constant density. If density varies, you need to model it first, like exp(-height), but then the problem is probably too complicated for hand calculation.

One more difficulty: the sphere has no constant Cx. It varies a lot with Reynold's number. The best-known physicists were historically fooled with that. It's because the place where the stream rips off wanders a lot. The corrugated golf ball avoids this, and also decreases its drag.