= Complete-flyby gravitational impulse
{title2=$|\Delta v_\perp|=2GM/(Vb),\quad\Delta v_{\parallel,\rm reduction}=2G^2M^2/(V^3b^2)$}
Integrate the transverse force along a complete straight path: $\int_{-\infty}^{\infty}(1+s^2)^{-3/2}ds=2$. This gives the transverse kick. Conservation of relative speed gives longitudinal speed reduction $\Delta v_\perp^2/(2V)$ at second order. As an independent check, exact weak hyperbolic scattering has angle $2\arctan\eta$, where $\eta=GM/(bV^2)$, and reduction $V(1-\cos(2\arctan\eta))=2V\eta^2/(1+\eta^2)$. The small-$\eta$ coefficient agrees. Keeping only one half of the encounter in the transverse kick makes this quadratic reduction four times too small.
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