Chapter 3.4, Problem 24E

Rocket Flight. 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 of $\beta m/sec$ relative to the rocket. Assume that the magnitude of the gravitational force is proportional to the mass with proportionality constant $g$. Because the mass is not constant, Newton’s second law leads to the equation

$\left(m_0-\alpha t\right)\frac{dv}{dt}-\alpha \beta \equiv -g\left(m_0-\alpha t\right)$,

where $v\equiv dx/dt$ is the velocity of the rocket, $x$ is its height above the ground, and $m_0-\alpha t$ is the mass of the rocket at $t\sec$ after launch. If the initial velocity is zero, solve the above equation to determine the velocity of the rocket and its height above ground for $0\le t<m_0/\alpha$.