A circular racetrack of radius 100 m is banked at an angle of `45^(@)`. What is the
(i) Optimum speed of race car to avoid wear and tear of its tyres?
(ii) Maximum permissible speed to avoid slipping if the coefficient of friction is 0.2?

Explanation

To solve the problem, we will break it down into two parts:

### Part (i): Optimum Speed of Race Car

1. **Understanding the Forces**: When a car moves on a banked track, the forces acting on it are the gravitational force (mg), the normal force (N), and the frictional force (f). The banking angle (θ) helps in providing the necessary centripetal force for circular motion.

2. **Centripetal Force Requirement**: The centripetal force required to keep the car moving in a circle of radius \( r \) is given by:
\[
...