How it falls

A bead, under the influence of gravity, sides down a frictionless wire whose $y$ -coordinate is changing with the $x$ -coordinate as shown in the figure.

Assume that at position $O$ the wire is vertical and that the bead passes this point with a given speed of ${ v }_{ 0 }$ downward. If the shape of the wire is such that the vertical component of the velocity remains ${ v }_{ 0 }$ at all time, find $(a + b + c -1)$ in the shape function of the wire given by $y=\frac { { (ag{ v }_{ 0 }x) }^{ \frac { b }{ c } } }{ 2g },$ where $g$ is gravitational acceleration. Here $b$ and $c$ are coprime positive integers.