Physics
Physics
Problem
A roller-coaster car of mass is released from rest at a point P that is a vertical distance above the station level. The track is frictionless until the car has cleared a vertical loop of radius whose top is above the station level. Immediately after the loop, the car enters a straight -long brake run in which kinetic friction with coefficient acts until the car comes to rest. (a) Derive an expression for the minimum value of that allows the car to just maintain contact with the track at the top of the loop. (b) Using your result from part (a), determine the distance the car travels along the brake run before coming to rest, expressed in terms of the given quantities and fundamental constants. (c) Suppose the initial release height is times the minimum found in part (a). Determine the speed of the car at the end of the brake run, showing all reasoning. (d) Discuss qualitatively how increasing would affect the required initial height for the car to come to rest exactly at the end of the brake run, keeping all other parameters fixed.Intuitive explanation
This problem chains together three energy transformations: gravitational potential energy converting to kinetic energy as the car falls, the constraint that the car must maintain contact through the loop's top (where normal force ≥ 0), and finally friction dissipating all remaining kinetic energy. The key insight is to use energy conservation separately for each section—from P to the loop top (finding the minimum for the car to complete the loop), then from the loop exit to the brake run's end (relating to the car's speed after the loop). This way you avoid solving for speed at every point and instead focus on what each stage demands.
Intuitive explanation
This problem chains together three energy transformations: gravitational potential energy converting to kinetic energy as the car falls, the constraint that the car must maintain contact through the loop's top (where normal force ≥ 0), and finally friction dissipating all remaining kinetic energy. The key insight is to use energy conservation separately for each section—from P to the loop top (finding the minimum for the car to complete the loop), then from the loop exit to the brake run's end (relating to the car's speed after the loop). This way you avoid solving for speed at every point and instead focus on what each stage demands.