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).