\problem
A model rocket having initial mass $m_0$~kg is launched vertically
from the ground. The rocket expels gas at a constant rate of $\alpha$
kg/sec and at a constant velocity $\beta$~m/sec relative to the
rocket, providing an upward force of $\alpha\beta$. The rocket's
mass is given by $m = m_0 - \alpha t$ at $t$ seconds after launch.
Assume that the force of gravity is $-mg$ where $g = 10\ {\rm m}/{\rm
sec}^2$ and that there are no frictional forces. Use Newton's
second law to derive a differential equation for the velocity $v$
and solve this equation for $0 < t < m_0/\alpha$.

\correct
$v = -g t - \beta\ln(1 - {\alpha\over m_0} t)$
\wrong
$v = ({\alpha\beta\over m} - g)t$
\wrong
$v = -{1\over 2}g t^2 - {\alpha\beta\over m_0} t$
\wrong
$v =-(m_0t - {1\over2}\alpha t^2)g + \alpha\beta t$
\wrong
$v = -g t + \beta\ln(m_0 - \alpha t)$