= Attractive and repulsive inverse-square closest approaches
{title2=$p_-=\sqrt{b^2+q^2}-q,\quad p_+=\sqrt{b^2+q^2}+q$}
For unit <mass>, incoming <speed> $v>0$, <impact parameter> $b>0$, and <potential energy> $\mp\kappa/r$, put $q=\kappa/v^2>0$. At closest approach, <conservation of energy> and <conservation of angular momentum> give the displayed radii and tangential <speeds> $u_\mp=bv/p_\mp$. Consequently $p_-p_+=b^2$ and $u_-u_+=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.
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