= Solution
Moving radially outward from the quasar, the standard <active-galactic-nucleus wind bubble> contains: the freely expanding fast wind; a reverse shock that thermalizes it; a hot shocked-wind bubble; a contact discontinuity; a dense swept-up interstellar shell behind a forward shock; and finally the undisturbed <interstellar medium>.
Let $M_s(R)$ be the swept-up shell mass. If inverse-Compton and atomic radiative cooling remove the shocked wind's thermal energy faster than the bubble expands, the reverse shock is <momentum-driven outflow>[momentum driven] and
$$
\boxed{\frac d{dt}(M_s\dot R)=\dot M_wv_w
-\frac{GM_sM_{\rm grav}(<R)}{R^2}
\simeq\frac Lc-\frac{GM_sM_{\rm grav}(<R)}{R^2}.}
$$
If cooling is slow, the shocked wind remains hot and the bubble is <energy-driven outflow>[energy driven]:
$$
\boxed{\frac d{dt}(M_s\dot R)=4\pi R^2P_b
-\frac{GM_sM_{\rm grav}(<R)}{R^2},}
$$
supplemented by the bubble energy equation
$$
\frac d{dt}\left(\frac32P_bV_b\right)
=\frac12\dot M_wv_w^2-P_b\frac{dV_b}{dt}.
$$
The hot bubble stores wind energy and performs work for much longer than the direct photon momentum-crossing time. Equating wind power with shell kinetic power gives the characteristic <momentum boost>
$$
\dot p_s\sim\frac{v_w}{2\dot R}\frac Lc\gg\frac Lc
$$
when the shell is much slower than the nuclear wind.
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