Deriving the speed of a falling ball

[burhan619]

Homework Statement

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 v_t= $\sqrt{\frac{mg}{D}}$
4. (d) Derive the equation for the speed at any time.

Homework Equations

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

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.

 

[Uniquebum]

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

 

[burhan619]

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