A car is negotiating a curved road of radius R. The road is banked at an angle θ. The coefficient of friction between the tyres of the car and the road is μs. The maximum safe velocity on this road is :

\(\sqrt{gR \frac{\mu_s + tan \theta}{1-\mu_s tan \theta }}\)

For Vertical equilibrium : \(N cos \theta = mg + f_1 sin \theta\)   .....(1)

For Horizontal equilibrium : \( N sin \theta = \frac{mv^2}{r} - f_1 cos \theta \)

Now solving these two equatiopns we get: v = \(\sqrt{gR \frac{\mu_s + tan \theta}{1-\mu_s tan \theta }}\)