Finite-time collision in an inverse-cube central force ($h^2=2k⟹\ddot{r}=0$)

For potential energy $\Phi (r)=-mk/r^2$, the inward central force has magnitude $2mk/r^3$. With conserved specific angular momentum $h=r^2\dot{\theta }$, the radial equation is $\ddot{r}=(h^2-2k)/r^3$. At the critical value $h^2=2k$, this reduces to $\ddot{r}=0$. An inward initial radial velocity $-v$, with $v>0$ and $r(0)=a$, therefore gives $r=a-vt$ until collision at $t_{\ast }=a/v$.

For $h>0$, integration of $\dot{\theta }=h/(a-vt{)}^2$ gives $\theta -\theta (0)=(h/v)(1/r-1/a)$. Thus the orbit is a hyperbolic spiral and the particle completes infinitely many turns before the finite collision time. The tangential speed $h/r$ diverges. The potential energy and kinetic energy individually diverge with opposite signs, while their sum remains the finite conserved mechanical energy $m{v}^2/2$.