Final answer:
The ideal speed for a car to take a 100 m radius curve banked at 15.0° can be calculated using principles of circular motion, while the minimum coefficient of friction needed to take the same curve at a lower speed involves understanding the forces in play and the role of friction.
Step-by-step explanation:
When a car navigates a banked curve at less than the ideal speed, friction becomes necessary to prevent the car from sliding toward the inside of the curve, especially on icy mountain roads where traction is reduced. To calculate the ideal speed for a car to take a 100 m radius curve with a 15.0° incline, we use the principle that the net external force on the car must provide the necessary centripetal force for circular motion, with friction playing no role (ideally banked curve with no friction). According to physics, the ideal speed (v) can be calculated from the expression v = √(r×g×tanθ), where r is the curve's radius, g is the acceleration due to gravity, and θ is the banking angle of the curve.
To find the minimum coefficient of friction needed for a frightened driver to safely take the same curve at 20.0 km/h, we must consider the forces acting on the car, such as its weight and the normal force from the road, and how these forces interact with the coefficient of friction to provide additional centripetal force. The calculation involves comparing the force of friction that's available with the additional centripetal force required at the lower speed.