Buoyant thermal mass balance

If a buoyant thermal receives volume inflow $J$ with reduced gravity $g_p$ and receives no other fluid entrainment, its mass and volume balances imply $\dot{V}=J$ and $d(g_{t}V)/dt=g_{p}J$. If $V\propto t^{9/4}$ and $g_t\propto t^{-5/4}$, then $g_t=9g_p/4$.