A car starts from rest to cover a distance s. The coefficient of friction between the road and the tires is mu. The minimum time in which the car...

Hello!

To change car speed, a force (horizontal in this case) must be applied to it. The force with the same magnitude will be applied to the ground because of Newton's Third law. If one would try to apply a force greater than the friction force `mu mg,` a car would slip on the ground.

By Newton's Second law, the maximum acceleration will be `a=(mu mg)/m=mu g,` where `m` is the mass or a car and `g` is the gravity acceleration. The distance as a function of time will be:

`s(t)=(a t^2)/2=(mu g t^2)/2`

So, for a fixed distance `s` the minimum time will be:

`t=sqrt((2s)/(mu g)).`

This is proportional to  "1/square root of `mu` " (the fourth option). 

The answer is: the minimum time is proportional to 1/square root of mu.