= Solution
The smaller galaxy is treated as a <test particle> released from rest at $r_0$. <Conservation of energy> in the fixed <gravitational potential> gives
$$
\frac12\dot d^2+\psi(d)=\psi(r_0),\qquad
\dot d^2=2A\left(d^{-1/2}-r_0^{-1/2}\right).
$$
The galactic gas is stationary in its hydrostatic model, so this is the relative speed squared. Write $\Delta(d)=d^{-1/2}-r_0^{-1/2}\geq0$ on the inward path. The astrophysical <ram pressure> is the incident normal momentum flux $\rho(d)\dot d^2$, giving
$$
\boxed{p_{\rm ram}(d)=\frac{2A\mu m_up_0}{k_BT}\,\Delta(d)
\exp\!\left[\frac{A\mu m_u}{k_BT}\Delta(d)\right],\qquad0<d\leq r_0.}
$$
It vanishes at release and increases during infall. This <ram pressure> convention is $\rho v^2$, not the $\rho v^2/2$ sometimes called dynamic <pressure>. Gas drag and the object's response are neglected consistently with the prescribed ballistic motion.
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