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## Homework Statement

An object is ejected straight up into the air at an initial velocity v0.

(a) Determine the time for reaching the maximal elevation when the object is subject

to gravity alone.

(b) Determine the time for reaching the maximal elevation when the object is subject

to gravity combined with a retarding force of the form kmv.

(c) Carefully expand your result from 1b to determine that it agrees with 1a in the

limit of k approaching 0.

(d) On the basis of the expansion 1c decide whether the retarding force extends or

shortens the time to reach the maximal elevation.

## Homework Equations

F

_{z}=ma

_{z}

F

_{z}=-mg

F

_{z}=-mkv

## The Attempt at a Solution

I had no real trouble with part a and b

For a I got t

_{max}=v

_{0}/g

for b I had to use a 2nd order Diff Eq and I got t

_{max}=(2/k) ln([kv

_{0}/g]-1)

for part c I tried using a Maclaurin polynomial for my result t(k)...(t as a function of k) for part b around k=0. However I was unable to do the expansion as I could not find the value of t(0)... (this is the value for the time as calculated in part b as k approaches zero) I tried using a limit to evaluate the value, but couldn't figure it out. Any ideas?

Thanks