an object is fired up in constant gravity. what is the time to max height with and without propotional air res

Question

Consider a projectile fired vertically in a constant gravitational field. For the same inital velocities, compare the times required for the projectile to reach its maximum height (a) for zero resisting force and (b) for a resisting force proportional to the instantaneous velocity of the projectile.

Answer 1

Well, I know that part a you solve as follows:

Vf = Vo - gt

0 = Vo - gt
t = Vo / g

Part b I am a little unsure if my method is correct, but here is what I have.


The net force on the object is given by mg + kv(t)

Thus ma = mg + kmv(t)

m [dv/dt] = m[g+kv(t)]

dv/dt = g + kv(t)

Separate Variables:

dv
--------- = dt
g+kv

Natural Log (g + kv) = kt + a

g + kv = e^(kt+a)

v(t) = Ce^(kt) - (g/k)

Set V(t) = 0

0 = Ce^(kt) - (g/k)

Ce^(kt) = g/k

ln [g / (kC)]
--------------- = t
k