Velocity of Bead on Brachistochrone

Theorem

Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line.

Let $C$ be defined by Equation of Cycloid embedded in a cartesian plane:

$x = a \paren {\theta - \sin \theta}$

$y = a \paren {1 - \cos \theta}$

Let a bead $B$ be released from some point on the wire.

Let $B$ slide without friction under the influence of a constant gravitational field exerting an acceleration $g$.

Let $s_0$ be the arc length along the cycloid.

Let $s$ be the arc length along the cycloid at any subsequent point in time.

Then the speed $v$ of $B$ relative to $C$ is defined by the equation:

$4 a v^2 = g \paren {2 s s_0 - s^2}$

Proof

By Brachistochrone is Cycloid, $C$ is a brachistochrone.

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Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.11$: Problem $4$