When a car negotiates a curved road, several factors come into play to determine the maximum safe velocity. These factors include the radius of the road, the angle of banking, and the coefficient of friction between the car's tires and the road surface.



To understand the maximum safe velocity, we need to consider the forces acting on the car as it goes around the curved road. The centripetal force, provided by the friction between the tires and the road, keeps the car moving in a circular path.

The maximum frictional force that can be exerted between the tires and the road is given by the equation:

Frictional Force (Ff) = Coefficient of Friction (μ) * Normal Force (N)

where the normal force is the force exerted by the road surface on the car, perpendicular to the surface.



When the road is banked at an angle, it helps to provide a component of the normal force in the inward direction, reducing the required frictional force. This allows the car to safely negotiate the curve at a higher velocity.

The component of the normal force acting in the inward direction is given by:

Normal Force (N) = Weight of the Car (mg) * cos(θ)

where θ is the angle of banking and m is the mass of the car.



To calculate the maximum safe velocity, we need to equate the frictional force to the centripetal force required to keep the car on the curved path. The centripetal force is given by:

Centripetal Force (Fc) = (mass of the car) * (velocity)^2 / Radius of the curve (R)

By equating the frictional force and the centripetal force, we can solve for the maximum safe velocity:

μ * mg * cos(θ) = (m * v^2) / R

Simplifying the equation, we get:

v^2 = μ * g * R * cos(θ)

Taking the square root of both sides, we obtain:

v = √(μ * g * R * cos(θ))

where g is the acceleration due to gravity.



The maximum safe velocity on a curved road depends on the radius of the curve, the angle of banking, and the coefficient of friction between the tires and the road. By considering the forces involved and equating the frictional force to the centripetal force, we can calculate the maximum safe velocity. The angle of banking plays a crucial role in reducing the required frictional force, allowing the car to safely negotiate the curve at a higher velocity.