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November 9th, 2009, 01:59 AM   #1
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Drag force, analytical solution

Hi

It's been a very long time since i did any advanced math. I am currently studying bodies under water. The forces acting on them are
* Gravity F=m*g
* Buoyancy F=roh*Vol*g
* Drag F=0.5*roh*vel^2 * Area * Cd

Simplifying I get

a = accelleration
v = velocity
k1 = constant 1
k2 = constant 2

a = k1 - k2 *v^2

So is there a analytical solution for displacement for this equation?

I have solved it numerically but would like to have an analytical solution if possible.
All the problems with drag (parachuting, golf balls, baseballs, free fall etc etc) either solve this numerically or simplify drag to be F=k*v

Any ideas?
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November 13th, 2009, 02:29 AM   #2
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Re: Drag force, analytical solution

There is. Let be the displacement. Then your equation is which, by defining , can be written as

Integrate by parts:

To simplify the equations define and , so that the above can be written as



And so then



If you want a nicer formula than that, you can do a u-substitution, but that'd be pretty messy.






But if you can take initial velocity to be zero and , then



so

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November 13th, 2009, 03:08 AM   #3
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Re: Drag force, analytical solution

amazing!

Thanks so very much
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