# Differential equation involving falling object

1. Sep 14, 2008

### MissMCHP

1. The problem statement, all variables and given/known data

A body of mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k|v|, where k is a constant. Assume that the gravitational attraction of the earth is constant.
a) Find the velocity v(t) of the body at any time.
b) Using the result of part a) to calculate the limit of v(t) as k->0

2. Relevant equations

3. The attempt at a solution
So I managed part a) with a solution of

v(t) = -mg/k + (v0+mg/k)e-kt/m,

which after checking and re-checking, I am fairly confident in. The problem is that when I try to find the limit as k->0, all I could come up with is v(t)=v0. This is obviously wrong since the as k->0, air resistance become negligible, and the answer should be the all too familiar v(t) = v0-gt. I tried to graph the function, and I found that after picking out all the useless part, it boils down to

lim k->t m/k(1-e-kt/m) = t!

Well. It does make sense, but I could not for the life of me figure out how to derive it mathematically. I am stuck, and any help would be greatly appreciated!

2. Sep 14, 2008

### Dick

Good deductive work on figuring out what the limit MUST be, you meant k->0, right? Now use l'Hopital's rule. It's a limit of the form 0/0. The limit is t!

Last edited: Sep 14, 2008
3. Sep 15, 2008

### MissMCHP

Yes! I did mean k->0. Thank you! I never would have thought of L'Hopital's rule (even though I should)