Obtain an expression for the maximum speed of a vehicle along horizontal circular curved road.

Answer: \(v_{max}=\sqrt{\mu_s r g}\)

Consider a vehicle of mass \(m\) moving with speed \(v\) on a flat (unbanked) horizontal curve of radius \(r\). [Figure: car on a level road with weight \(mg\) downward, normal reaction \(N\) upward and friction \(f\) along the road towards the centre of the curve.]

Vertically there is no acceleration, so \(N=mg\).

The only horizontal force towards the centre is the static friction between the tyres and the road, which supplies the centripetal force: \(f=\dfrac{mv^2}{r}\).

Static friction has a maximum value \(f_{max}=\mu_s N=\mu_s mg\). For safe motion, \(\dfrac{mv^2}{r}\le\mu_s mg\), i.e. \(v^2\le\mu_s r g\).

Hence the maximum safe speed is \(v_{max}=\sqrt{\mu_s r g}\). It is independent of the mass of the vehicle. If the speed exceeds this, the vehicle skids outward.