A basic question about movement in a circle There are two identical cars(same friction) and two different circles with one radius bigger than the other. Each car drives at the max speed it can drive while maintaining the circular motion. Which car will complete a circle first, the one in the smaller radius or will they both finish at the same time?
I may be wrong, but logically it looks to me that they will finish at the same time but I could not find a way to prove it.
 A: 
Each car drives at the max speed it can drive while maintaining the circular motion.

I am assuming that there is a certain maximum centripetal force that the cars can handle before starting to slip. I think that is what you're alluding to in the above quote.
Thus, this maximum centripetal force $\frac{m v_{max}^2}{R}$ is constant for both cars. Here $m$ is the car mass, $v_{max}$ is the maximum car speed without slipping and $R$ is the trajectory radius.
From this equation, you can see that if you double the trajectory radius, the maximum non-slipping velocity increases by a factor $\sqrt{2}$ (because the maximum centripetal force is constant).
Since doubling the radius of the trajectory also doubles the length of the trajectory, this means that the car in the larger circular trajectory will actually take longer to complete a lap, by a factor of $\sqrt{2}$.
So indeed, the car in the outer trajectory will be able to go at a faster speed, but this is not enough to compensate for the longer path length.
