# Car turning without skidding

## Homework Statement F = ma

## The Attempt at a Solution

First off, to keep the car from skidding to the side we want it to be constrained to move in a circle about the center of the turn because if it skids then it has deviated / veered off tangentially at some point along a circular path. For the minimum velocity, we want friction to point in the direction away from the center so that the net centripetal force is less thus requiring a minimal velocity. The equations of motion for the car are $Nsin\theta - fcos\theta = M\frac{v_{min}^{2}}{R}, Ncos\theta + fsin\theta - Mg = 0$ which tells us that $v_{min}^{2} = gR\frac{tan\theta - \mu }{1 + \mu tan\theta }$. For the maximum case we just negate the direction of friction and this gives us, in a very similar way, $v_{max}^{2} = gR\frac{tan\theta + \mu }{1 - \mu tan\theta }$. Plugging in the parameters give in the ans. clue we see that $v_{min} = 0, v_{max} = \infty$ so any speed in the range $0 \leq v< \infty$ will keep it from skidding which is all possible speeds. Is that what the text means by "all speeds are possible"? Thanks!