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The incoming and outgoing velocity vectors have the same magnitude $v_{\infty }$, by conservation of energy. The chord between them has lengthFor fixed $r_{\min }$ put $z=v_{\infty }\sqrt{r_{\min }/\mu }$. Then $|\Delta \mathbf{v}|=\sqrt{\mu /r_{\min }}2z/(1+z^2)$, and $(z-1{)}^2\ge 0$ proves $2z/(1+z^2)\le 1$. Equality occurs at $v_{\infty }=\sqrt{\mu /r_{\min }}$.

A non-colliding comet must have $r_{\min }\ge r_0$, where $r_0=R_p$ in the point-comet approximation and $r_0=R_p+R$ for finite bodies. Thus the non-colliding gravitational velocity-kick bound isThe optimizing grazing orbit has $v_{\infty }=v_{\mathrm{esc}}/\sqrt{2}$, $e_h=2$ and $\theta =\pi /3$. If contact itself is excluded, this is a supremum approached from larger $r_{\min }$. The finite collision radius is essential: an ideal point planet with unrestricted $r_{\min }\to 0$ has no finite upper bound.

A constant Galilean boost by the planet's stellar orbital velocity leaves the vector difference unchanged. It changes the comet's incoming and outgoing heliocentric energies, which is the basis of a gravitational assist.