Velocity of Bead on Brachistochrone

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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:

$x = a \left({\theta - \sin \theta}\right)$
$y = -a \left({1 - \cos \theta}\right)$

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:

$4 a v^2 = g \left({ {s_0}^2 - s^2}\right)$


Proof

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



Sources