Can't work out integral in polar coordinates

[etotheipi]

Homework Statement

I made up a question where a car is moving around a quarter circle (from angle pi/2 to 0), where a tangential frictional force of magnitude kv acts in a direction opposite to the car's motion. It begins at the top of the quarter circle with speed u, and I want to work out its final speed. I chose to go about this in terms of energy considerations.

Relevant Equations

$$v = r \frac{d\theta}{dt}$$$$W = \int F dx$$

I considered the work done by the frictional force in an infinitesimal angular displacement:
$$dW = Frd\theta = (kr\omega) rd\theta = kr^{2} \frac{d\theta}{dt} d\theta$$I now tried to integrate this quantity from pi/2 to 0, however couldn't figure out how to do this$$W = kr^{2}\int_{\frac{\pi}{2}}^{0} \frac{d\theta}{dt} d\theta$$I was wondering if anyone could give me any tips! Thank you!Edit

I've made a little progress, I've changed the differential to$$d(\frac{1}{2}mv^{2}) = kr^{2}v d\theta$$so$$m dv = kr^{2} d\theta$$and consequently$$v = \frac{kr^{2}\theta}{m} + C$$Does this look right to anyone?

 

[Delta2]

I think in your final result for $v$, the square exponent in $r$ need not be there, should be just plain $r$ (this should propagate all the way back to dK=dW equation at the top of the edit.)

 

[etotheipi]

I think in your final result for $v$, the square exponent in $r$ need not be there, should be just plain $r$ (this should propagate all the way back to dK=dW equation at the top of the edit.)

Ahh yes you're absolutely right. Thanks a bunch!

 

[Delta2]

I also think there is a sign error because the force is supposed to be frictional i.e it opposes particle speed, it should be $F=-kv$.