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 μ s . The maximum safe velocity on this road is:

Step-by-step solution

N = m g cos ⁡ θ + m v 2 R sin ⁡ θ f max = μ N ⇒ f m a x = μ s m g cos ⁡ θ + μ s m v 2 R sin ⁡ θ mg ⁡ sin θ + f max = m v 2 R ⁡ cos θ m g sin ⁡ θ + μ s m g cos ⁡ θ + μ s m v 2 R sin ⁡ θ = m v 2 R cos ⁡ θ g sin ⁡ θ + μ s g cos ⁡ θ = v 2 R cos ⁡ θ - μ s sin ⁡ θ g R tan ⁡ θ + μ s 1 - μ s tan ⁡ θ = v 2