A circular racetrack of radius 300 m is banked at an angle of `15^@` . If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the (a) optimum speed of the race- car to avoid wear and tear on its tyres, and (b) maximum permissible speed to avoid slipping ?

Explanation

On a banked road, the horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping. At the optimum speed, the normal reaction’s component is enough to provide the needed centripetal force, and the frictional force is not needed. The optimum speed `v_0` is given by Eq.
`v_0 = (Rg tan theta)^(1//2)`
Here `R = 300 m , theta = 15^@ , g =9.8 ms^(-2) ` , we have
`v_0 = 28.1 ms^(-1)`
The maximum permissible speed `v_(max) ` is given by eq.
`v_(max) = (Rg (mu_s + tan theta)/(1 -mu_s tan theta))^(1//2) = 38.1 ms^(-1)`