Solution
= Solution
Height decreases at rate $\dot h=-v$. Set $m=m_0-\alpha t$; an antiderivative needed for the <rocket equation> is
$$
\int_0^t\log\frac{m_0}{m_0-\alpha s}\,ds
=t-\frac m\alpha\log\frac{m_0}{m}.
$$
Consequently
$$
\boxed{h(t)=h_0-(v_0-u)t-\frac12gt^2
-\frac{u(m_0-\alpha t)}{\alpha}\log\frac{m_0}{m_0-\alpha t}.}
$$
Differentiating this expression recovers $\dot h=-v$ and substituting $t=0$ gives $h(0)=h_0$, which check both the sign and the integration constant.