Final answer:
To calculate the maximum velocity on a turn that isn't banked, the formula v_max = sqrt(mu_s * r * g) is used, where mu_s is the coefficient of static friction, r the radius of curvature, and g the gravitational acceleration. The minimum velocity depends on vehicle capabilities and driver behavior. For banked curves, the ideal speed is calculated using v_ideal = sqrt(r * g * tan(theta)), where theta is the banking angle.
Step-by-step explanation:
When calculating the minimum and maximum velocity of a vehicle on a turn that is not banked, one must consider the forces acting on the vehicle. For a vehicle to navigate the curve without slipping, the centripetal force required to keep the vehicle on its path must be provided fully by the frictional force between the tires and the road. The maximum velocity is limited by the maximum static frictional force that can be developed. The formula to find the maximum velocity v_max is derived from setting the maximum static frictional force equal to the required centripetal force:
v_max = sqrt(mu_s * r * g), where mu_s is the coefficient of static friction, r is the radius of curvature of the turn, and g is the acceleration due to gravity.
The minimum velocity is not necessarily governed by physics in the same way and can vary widely based on conditions like vehicle capabilities and driver behavior. However, it could be defined as the velocity at which the kinetic frictional force equals the required centripetal force if one were considering the threshold of sliding.
For banked turns, the concept of an ideal speed is often used. The ideal speed for a banked curve is the speed that allows a vehicle to negotiate the curve solely based on the banking angle and gravity without any reliance on friction. The formula for the ideal speed v_ideal can be derived from the balance of the gravitational component along the banking surface and the required centripetal force, such as:
v_ideal = sqrt(r * g * tan(theta)), where theta is the banking angle of the curve.