Force dependent on velocity of particle

[Bacat]

Homework Statement

A particle of mass m moves through a medium that resists its motions with a force of magnitude

-mk(v^2+av)

where k and a are positive constants. If no other force acts, and the particle has an initial velocity v0, find the distance traveled after a time t.

Show that the particle comes to rest as t \to \infty

Homework Equations

F=m\frac{dv}{dt}

The Attempt at a Solution

EOM: -k(v^2 + av) = \frac{dv}{dt}

dt=\frac{dv}{-k(v^2+av)}

\int \!dt=-\frac{1}{k} \int \! \frac{dv}{(v^2+av)}...Integrate in Mathematica...

t-t_0 = \frac{Ln(a+v)-Ln(v)}{ak}

Exp(atk)=\frac{a+v}{v}

v(Exp(atk)-1)=a

v(t)=\frac{a}{Exp(atk)-1}

Set v = v0 at time t=0...

v(0) = v_0 = \frac{a}{Exp(0)-1} = \frac{a}{0}

But this is not defined!

Did I make a mistake? How do I set v = v0 if I get infinity?

Thank you for your time and help.

 

[kuruman]

The problem is with your limits of integration. The left side (time) goes from 0 to t. That's fine. The right side must go from v₀ (which is the velocity that matches time t = 0) to v (which is the velocity that matches time t).