Burnout Velocity of Upward Rocket under Constant Gravity

Theorem
Let $R$ be a rocket whose structural mass is $m_1$.

Let $R$ contain fuel of initial mass $m_2$.

Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant.

Let $R$ burn fuel at a constant rate $a$, producing a constant exhaust velocity $b$ relative to $R$.

Let all forces on $R$ except that due to the gravitational field be neglected.

Then the burnout velocity of $R$ is given by:
 * $v_b = b \ln \left({1 + \dfrac {m_1} {m_2} }\right) - \dfrac {g m_2} a$