November 9th, 2009, 01:59 AM  #1 
Newbie Joined: Nov 2009 Posts: 2 Thanks: 0  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? 
November 13th, 2009, 02:29 AM  #2 
Member Joined: Nov 2009 Posts: 72 Thanks: 0  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 usubstitution, but that'd be pretty messy. But if you can take initial velocity to be zero and , then so 
November 13th, 2009, 03:08 AM  #3 
Newbie Joined: Nov 2009 Posts: 2 Thanks: 0  Re: Drag force, analytical solution
amazing! Thanks so very much 

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