= Finite-time collision in an inverse-cube central force
{title2=$h^2=2k\ \Longrightarrow\ \ddot r=0$}
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 $t_*=a/v$.
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> $mv^2/2$.
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