# Falling object in a gravitational field with v^2 drag force

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1. Sep 24, 2016

### Elvis 123456789

1. The problem statement, all variables and given/known data
Consider a particle in a gravitational field that is also subject to a resisting force proportional to the velocity squared ( Fdrag = + or - cv2).
a) Find the terminal velocity, vT, for the object as it falls.

b) Show that for an object dropped from rest that the velocity is given by v = vTtanh-1(-t/β)
where β = sqrt( m/gc )

2. Relevant equations

3. The attempt at a solution
I did part a and got vT = sqrt( mg/c )
but for part b I am not getting the same result that is shown.
My work is shown in the attachment (the tau in the attachment is the β. I wrote it as β here because its hard to distinguish the t, T, and the tau on here)

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Last edited: Sep 24, 2016
2. Sep 25, 2016

### TSny

Your work looks correct to me. Note that your solution has the correct asymptotic behavior for t → ∞; but the answer stated in the problem does not.

3. Sep 25, 2016

### ehild

Your integral with respect to v is not correct,

Last edited: Sep 25, 2016
4. Sep 25, 2016

### Elvis 123456789

I cannot spot the mistake. Do you mind pointing out where I went wrong?

5. Sep 25, 2016

### Elvis 123456789

I used an integral table for this which says that
int [ dx/(a2-x2)] = (1/a) * tanh-1 (x/a)

where a = sqrt (mg/c) and this is for x2 < a2 which is the case here

6. Sep 25, 2016

### TSny

To obtain this result, you can change integration variable from x to z, where tanh(z) = x/a.

If you do the integral by partial fractions using 1/(a2 - x2) = 1/(2a) * [ 1/(a + x) + 1/(a - x) ] , then you get the integral expressed in terms of a logarithm instead of tanh-1.

7. Sep 25, 2016

### Elvis 123456789

Right, I was just trying to express it as was requested in the question details. I'm still wondering if I made an error though as ehild suggests?

8. Sep 25, 2016

### TSny

I believe your solution is correct. I thought maybe ehild was thinking of the logarithmic form of the integral.

9. Sep 25, 2016

### Elvis 123456789

Thanks for the help. The result shown in the problem description is definitely wrong for the incorrect asymptotic behavior that you pointed out as t -> infinity. I emailed the professor about this so he can let the class know just in case anybody else was scratching their heads.

10. Sep 25, 2016