A ball of mass \(m\) is dropped from rest above the surface of an airless moon in essentially uniform gravity \(g\).

(a) If \(y\) is the vertical axis, show that the Hamilton-Jacobi equation for the ball is

\[\frac{1}{2 m}\left(\frac{\partial S}{\partial y}\right)^{2}+m g y+\frac{\partial S}{\partial t}=0\]

where \(S\) is Hamilton's principal function.

(b) Then show that

\[S= \pm \frac{2 \sqrt{2}}{3 g \sqrt{m}}(C-m g y)^{3 / 2}-C t\]

where \(C\) is a constant.

(c) Then show, using a judicious choice of constants, that the equation of motion of the ball can be written

\[y=y_{0}-\frac{1}{2} g\left(t-t_{0}\right)^{2}\]