Solution
= Solution
Integrating the <rocket equation> with $v(0)=v_0$ gives
$$
\boxed{v(t)=v_0+gt-u\log\frac{m_0}{m_0-\alpha t}.}
$$
This formula holds during the constant-flow burn, while $m_0-\alpha t>0$ and fuel remains. The negative logarithmic term represents the slowing due to upward <rocket thrust>.