Understanding Simple Kinematics: Solving for Velocity in Resisted Motion

[bcjochim07]

Homework Statement

I should know this, but it's been awhile since I've dealt w/ kinematics.

As the simplest example of resisted motion of a particle, find the velocity of horizontal motion in a medium in which the retarding force is proportional to velocity.

So Fr is something like -kmv, where k is a constant.

I'm tempted to use v=vo+at in this manner:

ma=-kmv, so a = -kv
then v=vo-kvt
v(1+kt)=vo
v=vo/(1+kt)

But my book uses integrals:
mdv/dt=-kmv
int(dv/v)=-k*int(dt)
lnv=-kt+C , v= c1e^-kt where (c1=e^C) and applying initial conditions you get
v=vo*e^-kt
and this makes a lot of sense to me.

So could somebody please refresh me on why I cannot solve for a and substitute into v=vo+at? I'm thinking it has to do with the constantly changing force, but I'm looking for a good explanation.

 

[rl.bhat]

Kinematic equation v = vo + at is applicable to a motion having constant acceleration. But in the given problem acceleration is not constant.