How Do You Calculate Stopping Distance with Velocity-Dependent Resistive Force?

[mgeoghe2]

dynamics problem!

this is a problem about a resistive force that is a function of v, not x.
a block with an initial velocity goes through a medium that resists motion with force proportional to the block's velocity, with proportionality constant K. so basically the resistive force = -Kx (i choose the pos x dir to be in the dir of initial velocity)
i am supposed to find the stopping distance by finding v as a function of x and then i am supposed to find it again by finding v as a function of t.
i cruised wikipedia because it helped me on an escape velocity problem, and it gave me some help, but i don't know how they got what they did and I am not entirely sure if its what i want.

they said:
-Kx = ma = m dv/dt which i have and understand
they say it integrates to
v = v(initial) * e^(-Kt/m)
can anyone explain how they got that? i am missing some math skillz here.
any other insights to the problem would be greatly appreciated.

 

[Astronuc]

The problem statement indicates that a resistive force a function of velocity, i.e. is proportional to velocity, then

force = -Kv, or -K\dot{x}, and not force = -Kx.

So the force balance becomes,

m dv/dt = -Kv which is a first order linear differential equation,

which gives the solution

v = v(initial) * e^(-Kt/m) for v=v(initial) at t=0.