A particle of mass \(m\) is subject to the central attractive force \(\mathbf{F}=-k \mathbf{r}\), like that of a Hooke's-law spring of zero unstretched length, whose other end is fixed to the origin. The particle is placed at a position \(\mathbf{r}_{0}\) and given an initial velocity \(\mathbf{v}_{0}\) that is not colinear with \(\mathbf{r}_{0}\).

(a) Explain why the subsequent motion of the particle is confined to a plane containing the two vectors \(\mathbf{r}_{\mathbf{0}}\) and \(\mathbf{v}_{\mathbf{0}}\).

(b) Find the potential energy of the particle as a function of \(r\).

(c) Explain why the particle's angular momentum is conserved about the origin, and use this fact to obtain a first-order differential equation of motion involving \(r\) and \(d r / d t\).

(d) Show that the particle has both an inner and an outer turning point, and solve the equation for \(t(r)\), where the particle is located at an outer turning point at time \(t=0\).

(e) Invert the result to find \(r(t)\) in this case.