Solve
$$
m \; dv = -dm \; v_{ex}
$$
for $v(m)$. (Initial speed and mass are $v_0$ and $m_0$ respectively.)
If thrust is given by: $-\overset{\bullet}{m}v_{ex}$, then we have from the 2nd law:
$$m \overset{\bullet}{v} = - \overset{\bullet}{m}v_{ex} - mg $$
(Near the surface of the Earth)
Assuming that $\overset{\bullet}{m} = -k $ ($k$ is a positive constant), solve this equation of motion for $v(t)$