On the least uncomfortable journey from A to B

TL;DR

This paper revisits the least uncomfortable journey problem, providing simple solutions using time as the independent variable for measures based on acceleration and jerk, and critiques previous boundary condition choices.

Contribution

It offers exact solutions for the least discomfort journey problem with acceleration and jerk, simplifying previous methods and challenging earlier boundary condition assumptions.

Findings

Exact solutions for acceleration-based discomfort using time as variable.

Exact solutions for jerk-based discomfort with derived boundary conditions.

Critique of previous boundary conditions as physically unrealistic.

Abstract

The problem of the "least uncomfortable journey" between two locations on a straight line, originally discussed by Anderson {\it et al.} (2016 {\it Am. J. Phys.} {\bf 84} 6905) is revisited. When the integral of the square of the acceleration is used as a measure of the discomfort, the problem is shown to be easily solvable by taking the time, instead of the position, as the independent variable. The solution is quite simple and avoids not only complicated differential equations and the computation of cumbersome integrals, but also the inversion of functions by solving cubic equations. Next, the same problem, but now with the integral of the square of the jerk as a measure of the discomfort, is also exactly solved with time as the independent variable and the appropriate boundary conditions, which are derived. It is argued that the boundary conditions imposed on the velocity in Anderson {\it et al.} (2016 {\it Am. J. Phys.} {\bf 84} 6905) are inappropriate not only because they are not always physically realizable but also because they do not lead to the minimum discomfort possible.