The comfortable roller coaster -- on the shape of tracks with constant normal force

TL;DR

This paper investigates the shape of roller coaster tracks that maintain a constant normal force, revealing a connection to Kepler problem trajectories and conic sections in velocity space.

Contribution

It provides an analytical solution to the problem of designing tracks with constant normal force, linking it to classical Kepler problem solutions.

Findings

Tracks with constant normal force are related to conic sections.

The problem is mathematically connected to the Kepler problem.

The solution characterizes track shapes that keep rider normal force constant.

Abstract

A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.