 Integrated tidal tensor of a straight-line flyby

For a perturber following $(p,0,vt)$, the impulse approximation integrates its tidal tensor to $2G{m}_p\mathrm{diag}(1,-1,0)/(v{p}^2)$. A fixed target position therefore receives $\Delta \mathbf{v}=2G{m}_p(x,-y,0)/(v{p}^2)$. The longitudinal kick vanishes after integrating the symmetric trajectory.