Aplly the formula v= root Rg(u+tandeta)/1-u(tandeta).Givven in ncert.If you find diffulcity ask me i solve it

Optimum Speed to Avoid Wear and Tear on Tyres:
To determine the optimum speed of the race-car to avoid wear and tear on its tires, we need to consider the forces acting on the car while it is moving along the banked circular track. The two main forces involved are the gravitational force and the frictional force.

1. Gravitational Force:
The gravitational force acts vertically downwards and can be resolved into two components: the normal force (N) acting perpendicular to the track's surface and the force due to gravity (mg) acting vertically downwards. The normal force can be calculated using the formula: N = mgcosθ, where θ is the angle of banking.

2. Frictional Force:
The frictional force acts horizontally along the track's surface and opposes the car's motion. It can be calculated using the formula: f = μN, where μ is the coefficient of friction between the wheels and the road.

3. Centripetal Force:
The centripetal force is responsible for keeping the car moving in a circular path and is provided by the horizontal component of the normal force. It can be calculated using the formula: Fc = Nsinθ.

Equilibrium of Forces:
For the car to avoid wear and tear on its tires, the net force acting on the car should be zero. This can be achieved when the centripetal force and the frictional force are balanced. Therefore, we can equate these two forces: Fc = f.

Calculating Optimum Speed:
From the equilibrium of forces, we can substitute the values of Fc and f:
Nsinθ = μN

Simplifying the equation, we get:
sinθ = μ

Given that θ = 15 degrees and μ = 0.2, we can calculate sinθ:
sin15 = 0.2

Using a scientific calculator or trigonometric table, we find that sin15 = 0.2588.

Therefore, the optimum speed of the race-car to avoid wear and tear on its tires is when the car is moving at a speed that allows the centripetal force to be equal to or less than the frictional force. This occurs when sinθ ≤ μ.

Maximum Permissible Speed to Avoid Slipping:
To determine the maximum permissible speed of the race-car to avoid slipping, we need to consider the forces acting on the car while it is moving along the banked circular track. The two main forces involved are the gravitational force and the frictional force.

1. Gravitational Force:
The gravitational force acts vertically downwards and can be resolved into two components: the normal force (N) acting perpendicular to the track's surface and the force due to gravity (mg) acting vertically downwards. The normal force can be calculated using the formula: N = mgcosθ, where θ is the angle of banking.

2. Frictional Force:
The frictional force acts horizontally along the track's surface and opposes the car's motion. It can be calculated using the formula: f = μN, where μ is the coefficient of friction between the wheels and the road.

3. Centripetal Force:
The centripetal force is responsible for keeping the car moving in a circular path and is provided by