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 $t(={\tau }_1+{\tau }_2={\tau }_3=3/\sqrt{2g})=0.67$ 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.

5 pages and 3 figures