Constant-rate retro-rocket landing

With downward speed $v$, downward relative exhaust speed $v_E>0$ and mass $M=M_0-\alpha t$, the ideal vertical equation is $\dot{v}=g-\alpha v_E/M$. Integration gives $v=v_0+gt+v_E\log (M/M_0)$. Height follows from $\dot{h}=-v$; its touchdown condition must be imposed together with the final fuel fraction.