A circular racetrack of radius 300 m is banked at an angle of `15^(@)` The coefficient of friction between the wheels of a race car and the road is 0.2. The optimum speed of the race car to avoid wear and tear on its tyres is

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 `upsilon_(0)` is given by `upsilon_(0)=(Rg tan theta)^(1//2)`
Here `R = 300 m, theta = 15^(@), g = 9.8 ms^(-2)`
`upsilon_(0)=(R g tan theta)^(1//2)=(300xx9.8xx tan 15^(@))^(1//2)`
On simplification, `upsilon_(0)=28.1 ms^(-1)`
The maximum permissible speed `upsilon_(max)` is given by
`v_(max)=sqrt((Rg(tan theta + mu))/((1-mu tan theta)))`
`= sqrt((300xx9.8xx(tan 15^(@)+0.2))/((1-0.2 tan 15^(@))))`
On simplification `upsilon_(max)=38.1 ms^(-1)`