(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A body of mass m is projected vertically upward with an initial velocity v_{0}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 + (v_{0}+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)=v_{0}. This is obviously wrong since the as k->0, air resistance become negligible, and the answer should be the all too familiar v(t) = v_{0}-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!

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# Differential equation involving falling object

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