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.

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=v₀/g

for b I had to use a 2nd order Diff Eq and I got t_max=(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

 

[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

 

[ehild]

for b I had to use a 2nd order Diff Eq and I got t_max=(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 dimension (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 don't 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.