A car is negotiating a curved road of radius R. The road is banked at an angle $\theta$ . The coefficient of friction between the types of the car and the road is ${\mu }_s$ . The maximum safe velocity on this road is

For vertical equilibrium on the road, 

$N\cos \theta =mg+f\sin \theta$

$mg=N\cos \theta -f\sin \theta$……….(1)

Centripetal force for safe turning, 

$N\sin \theta +f\cos \theta =\frac{m{v}^2}{R}$…….(2)

From eqns. (i) and (ii), we get 

$\begin{array}{r}\frac{v^2}{Rg}=\frac{N\sin \theta +f\cos \theta }{N\cos \theta -f\sin \theta }\\ \Rightarrow \frac{v_{\max }^2}{Rg}=\frac{N\sin \theta +{\mu }_{s}N\cos \theta }{N\cos \theta -{\mu }_{s}N\sin \theta }\\ v_{\max }=\sqrt{Rg\left(\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }\right)}\end{array}$