# Deriving the speed of a falling ball

1. Oct 1, 2011

### burhan619

1. The problem statement, all variables and given/known data
You drop a baseball from the roof of a tall building. As the ball falls, the air exerts a drag force proportional to the square of the ball's speed (f=Dv2).
1. (a) In a diagram, show the direction of motion and indicate, with the aid of vectors, all the forces acting on the ball.
2. (b) Apply Newton's second laws second law and infer from the resulting equation the general properties of the motion.
3. (c) Show that the ball acquires a terminal speed that is described by vt= $\sqrt{\frac{mg}{D}}$
4. (d) Derive the equation for the speed at any time.

2. Relevant equations
$\int$((a2-x2)-1)dx = $\frac{1}{a}$arctanh($\frac{x}{a}$), where tanh(x)=(ex-e-x)/(ex+e-x)= (e2x-1)/(e2x+1)

3. The attempt at a solution
In the attachment.

I'm at a block with the last step of part d. How do I isolate v from that? Any help is appreciated.

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2. Oct 1, 2011

### Uniquebum

$tanh^{-1}(\frac{v\sqrt{D}}{\sqrt{mg}}) = \sqrt{\frac{mg}{D}}*t$
...
Finally
$v = \sqrt{\frac{mg}{D}}*tanh(\sqrt{\frac{mg}{D}}*t)$

3. Oct 1, 2011

### burhan619

Oh wow. That was pretty simple in retrospect... thanks.