= Solution
Choose downward <velocity> $v$ positive. The remaining rocket <mass> is $m(t)=m_0-\alpha t$, and the upward <rocket thrust> has magnitude $\alpha u$, where $u>0$ is the relative <rocket exhaust speed>. The <rocket equation> including gravity is
$$
m\dot v=mg-\alpha u,\qquad
\boxed{\dot v=g-\frac{\alpha u}{m_0-\alpha t}.}
$$
This follows by applying <momentum conservation> to the rocket and its just-expelled <rocket propellant>, with gravity supplying the external <impulse>. Applying $d(mv)/dt=mg$ to the rocket alone would omit the exhaust <momentum flux>.
Back to article page
