A projectile is fired in a uniform gravitational field \(g\) with initial speed \(v_{0}\) and angle \(\theta_{0}\) relative to the horizontal. Note that the Hamiltonian is

\[H=\frac{p_{x}^{2}}{2 m}+\frac{p_{y}^{2}}{2 m}+m g y\]

Find the projectile's motion \(x(t)\) and \(y(t)\) using the Hamilton-Jacobi method.