Help with diffy eqn terminal vel problem

[Ara macao]

Hi,

So m*dv/dt = mg - kv^2, where m is skydiver mass, g acceleration due to gravity, and k> 0 related to amount of air resistance. So I want to find the analytical solution for v(t), with the assumption that v(0) = 0.

I went all the way to integrating it with a partial fraction, but then got an answer drastically different from what the book got.

Thanks...

 

[HallsofIvy]

It's hard to answer because you didn't show us what you did or what answer you got!

 

[saltydog]

Hi,
So m*dv/dt = mg - kv^2, where m is skydiver mass, g acceleration due to gravity, and k> 0 related to amount of air resistance. So I want to find the analytical solution for v(t), with the assumption that v(0) = 0.
I went all the way to integrating it with a partial fraction, but then got an answer drastically different from what the book got.
Thanks...

Tell you what Ara, how about looking up Riccati equations. Now, can your equation be put in such form? If if can, then such a prespective may allow an easier approach to its solution.

 

[arildno]

Please remember that the inverse of the hyperbolic tangent is a logarithmic expression.

 

[Benny]

If you just want the terminal velocity then set the acceleration (LHS) equal to zero and solve for v. I did a question like this about 1 or 2 years ago. If I remember correctly, the working is a little tedious but not too difficult. Obtaining the partial fraction decomposition is the difficult part, the integration and transposition should be straight forward.