# Air Resistance and Projectile Motion

AlexChandler

## 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.

Fz=maz
Fz=-mg
Fz=-mkv

## The Attempt at a Solution

I had no real trouble with part a and b

For a I got tmax=v0/g

for b I had to use a 2nd order Diff Eq and I got tmax=(2/k) ln([kv0/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

## Answers and Replies

Mandeep Deka
If you put k=0, you will get a 0/0 form,
So, you can apply L' Hospital Rule!

AlexChandler
Yes, I thought of that at first, but actually you get ln(-1)/0
and natural log(-1) = 3.14159265 i

Homework Helper
for b I had to use a 2nd order Diff Eq and I got tmax=(2/k) ln([kv0/g]-1)

Show your work. You have to solve a first-order equation to get the time of maximal elevation.

ehild

AlexChandler
F=ma. In vertical dimention (z) with up as positive
-kmv-mg=ma
-kmz'-mg=mz"
z"+kz'= -g
this is the second order diff eq I solved to find a function z(t)
I then set z'(t)=0 and solved for the time t at which z is max
this led me to the solution I posted.

AlexChandler
Haha I suppose it would be first order.
v'+kv=-g
Since I dont even really need the position function.
Is this the first order you were talking about?

AlexChandler
Ahh yes have solved the problem. Made a small algebraic mistake a few steps ago. The formula for Tmax turned out to be much simpler to expand. Actually, I was able to use l'Hôpital's rule! (that is... Bernoulli's rule). Thanks for your help.