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Physics Question on laws of motion

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

A

gRμs+tan⁡θ1−μstan⁡θ\sqrt{g R \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}

B

gRμs+tan⁡θ1−μstan⁡θ\sqrt{\frac{g}{R} \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}

C

gR2μs+tan⁡θ1−μstan⁡θ\sqrt{\frac{g}{R^2} \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}

D

gR2μs+tan⁡θ1−μstan⁡θ\sqrt{g R^2 \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}

Answer

gRμs+tan⁡θ1−μstan⁡θ\sqrt{g R \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}

Explanation

Solution

v2Rg=(μs+tan⁡θ1−μstan⁡θ)\frac{v^{2}}{Rg} = \left(\frac{\mu_{s} + \tan \theta}{1 - \mu_{s} \tan\theta}\right)
⇒v=Rg[μs+tan⁡θ1−μstan⁡θ]\Rightarrow v = \sqrt{Rg \left[\frac{\mu_{s} + \tan\theta}{1- \mu_{s} \tan\theta}\right]}