An interesting track for the Brachistochrone

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 $t(=τ_1+τ_2=τ_3=3/\sqrt{2g})=0.67$ 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 $t(=τ_4=2/\sqrt{g})=0.63$ 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 $t=τ_B=0.58$ 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 $y(x)=(1-x^ν)^{1/ν}.$ For $ν> 1$ the track becomes convex and $t>>τ_4$, and when $ν>1.22$, the motion with zero initial speed is not possible. We find that when $ν\in (0.09653, 0.31749), τ_4<t <τ_3$ and when $ν\in ( 0.31749, 1),τ_B < t < τ_4$. But most remarkably, the concave curve becomes very steep/deep if $ν\in (0, ν_c=0.09653)$, then $t=0.2258$ s $< τ_B$, this is as though a particle would travel 1 meter horizontally with a speed equal $\sqrt{2g}$ m/sec to take the time ($=1/\sqrt{2g}=τ_2) < τ_B$. The function $t(ν$) suffers a jump discontinuity at $ν=ν_c$, we offer some resolution.