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For equal $|k|=\kappa$ in the two cases, their closest distances are conjugate roots expressed with the same $q=\kappa /v^2$. ConsequentlyAt either central-force radial turning point, the velocity is tangential and conservation of angular momentum gives $pu=\tilde{p}\tilde{u}=bv$. ThusThese attractive and repulsive inverse-square closest approaches pair a shorter, faster attractive passage with a longer, slower repulsive one. The speed-product conclusion assumes $b>0$; at $b=0$, the attractive orbit hits the singular centre with unbounded speed, so the formal product with the repulsive zero speed is not defined.

For the scattering calculation, take the heavy target as fixed and define $\kappa =2Z{e}^2/(4\pi {ϵ}_0)>0$. The incoming kinetic energy and angular momentum are $E=m{v}^2/2$ and $J=mvb$. In the original sketch, the horizontal dashed incoming asymptote has perpendicular distance $b$ from the nucleus, while $d$ is the radius from the nucleus to the point where $\dot{r}=0$.

At closest approach, the velocity is tangential and $v_t=J/(md)=vb/d$. conservation of energy givesChoose the positive root and put $s=\kappa /(m{v}^2)$:Head-on incidence gives $d_{\mathrm{rep}}=2s$, and repulsion always gives $d_{\mathrm{rep}}>b$. With an attractive Coulomb force, change the sign of the potential term:For $b>0$, the attractive and repulsive inverse-square closest approaches obey $d_{\mathrm{att}}{d}_{\mathrm{rep}}=b^2$. For $b=0$ attraction causes a collision with the point centre, so $d=0$ is a singular endpoint, not a regular scattering turning point. Finite nuclear size is excluded by the stated point-force model.

For the repulsive inverse-cube central force $\mathbf{F}={\kappa }_3\hat{\mathbf{r}}/r^3$, the torque vanishes, $\mathbf{r}\times \mathbf{F}=0$, and angular momentum is still conserved. Its potential energy is $+{\kappa }_3/(2r^2)$, since $-dV/dr=+{\kappa }_3/r^3$. ThusThe new coefficient is denoted ${\kappa }_3$ because its units are $\mathrm{N}{\mathrm{m}}^3$, unlike the $\mathrm{N}{\mathrm{m}}^2$ units of the inverse-square coefficient. Conserving angular momentum depends on centrality, not on a particular radial power law.