An interesting track for the Brachistochrone

TL;DR

This paper investigates the time of descent for a frictionless particle on a family of tracks defined by a parameter, revealing conditions where the track's shape significantly affects the descent time, including cases faster than the cycloid.

Contribution

It introduces a new class of track shapes defined by a parameter and analyzes their impact on descent time, revealing non-intuitive results and a discontinuity at a critical parameter value.

Findings

For certain parameter ranges, the descent time exceeds that of a linear inclined plane.

When the parameter is below a critical value, the track becomes very steep and the descent time is less than the cycloid.

A jump discontinuity in the time function occurs at the critical parameter value.

Abstract

If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, it takes a time $t(=\tau_1+\tau_2=\tau_3=3/\sqrt{2g})=0.67$ s. Under gravity and without friction, if it sides down on a linear track inclined at $45^0$ between two points A and B of 1 m height, it takes time $t(=\tau_4=2/\sqrt{g})=0.63$ s. Between these two extremes, historically, Bernoulli (1718) proved that the fastest track between these points A and B is cycloid with the least time of descent $t=\tau_B=0.58$ s. Apart from other interesting cases, here we study the frictionless motion of a particle/bead on an interesting track/wire between A and B given by $y(x)=(1-x^{\nu})^{1/\nu}.$ For $\nu > 1$ the track becomes convex and $t>>\tau_4$, and when $\nu >1.22$, the motion with zero initial speed is not possible. We find that when $\nu \in (0.09653, 0.31749), \tau_4<t <\tau_3$ and when $\nu \in ( 0.31749, 1),\tau_B < t < \tau_4$. But most remarkably, the concave curve becomes very steep/deep if $\nu \in (0, \nu_c=0.09653)$, then $t=0.2258$ s $< \tau_B$, this is as though a particle would travel 1 meter horizontally with a speed equal $\sqrt{2g}$ m/sec to take the time ($=1/\sqrt{2g}=\tau_2) < \tau_B$. The function $t(\nu$) suffers a jump discontinuity at $\nu=\nu_c$, we offer some resolution.