Deriving the speed of a falling ball

  • Thread starter burhan619
  • Start date
  • #1
7
0

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 vt= [itex]\sqrt{\frac{mg}{D}}[/itex]
  4. (d) Derive the equation for the speed at any time.

Homework Equations


[itex]\int[/itex]((a2-x2)-1)dx = [itex]\frac{1}{a}[/itex]arctanh([itex]\frac{x}{a}[/itex]), where tanh(x)=(ex-e-x)/(ex+e-x)= (e2x-1)/(e2x+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.
 

Attachments

Answers and Replies

  • #2
55
1
[itex]tanh^{-1}(\frac{v\sqrt{D}}{\sqrt{mg}}) = \sqrt{\frac{mg}{D}}*t[/itex]
...
Finally
[itex]v = \sqrt{\frac{mg}{D}}*tanh(\sqrt{\frac{mg}{D}}*t)[/itex]
 
  • Like
Likes hhhh
  • #3
7
0
Oh wow. That was pretty simple in retrospect... thanks.
 

Related Threads on Deriving the speed of a falling ball

Replies
2
Views
7K
Replies
2
Views
1K
Replies
5
Views
5K
Replies
22
Views
16K
Replies
6
Views
870
Replies
32
Views
2K
Replies
2
Views
5K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
9
Views
331
Replies
7
Views
599
Top