  In galaxy formation simulations, it is increasingly common to represent
supernovae (SNe) at finite resolution (when the Sedov-Taylor phase is
unresolved) via hybrid energy-momentum coupling with some 'terminal momentum'
$p_{\rm term}$ (depending weakly on ambient density and metallicity) that
represents unresolved work from an energy-conserving phase. Numerical
implementations can ensure momentum and energy conservation of such methods,
but these require that couplings depend on the surrounding gas velocity field
(radial velocity $\langle v_{r} \rangle$). This raises the question of whether
$p_{\rm term}$ should also be velocity-dependent, which we explore analytically
and in simulations. We show that for simple spherical models, the dependence of
$p_{\rm term}$ on $\langle v_{r} \rangle$ introduces negligible corrections
beyond those already imposed by energy conservation if $\langle v_{r} \rangle
\ge 0$. However, for SNe in some net converging flow ($\langle v_{r}
\rangle<0$), naively coupling the total momentum when a blastwave reaches the
standard cooling/snowplow phase (or some effective cooling
time/velocity/temperature criterion) leads to enormous $p_{\rm term}$ and
potentially pathological behaviors. We propose an alternative $\Delta$-Momentum
formulation which represents the differential SNe effect and show this leads to
opposite behavior of $p_{\rm term}$ in this limit. We also consider a more
conservative velocity-independent formulation. Testing in numerical
simulations, these directly translate to large effects on predicted star
formation histories and stellar masses of massive galaxies, explaining
differences between some models and motivating further study in idealized
simulations.
