A single-stage rocket rises vertically from its launchpad by burning liquid fuel in its combustion chamber; the gases escape with a net momentum downward, while the rocket, in reaction, accelerates upward. The gravitational field is \(g\).

(a) Pretending that air resistance is negligible, show that the rocket's equation of motion is

where \(m\) is the instantaneous mass of the rocket at time \(t, v\) is its upward velocity, and \(u\) is the speed of the exhaust relative to the rocket.

(b) Assume that \(g\) and \(u\) remain constant while the fuel is burning, and that fuel is burned at a constant rate \(|d m / d t|=\alpha\). Integrate the rocket equation to find \(v(m)\).

(c) Suppose that \(u=4.4 \mathrm{~km} / \mathrm{s}\) and that all the fuel is burned up in one minute. If the rocket achieves the escape velocity from earth of \(11.2 \mathrm{~km} / \mathrm{s}\), what percentage of the original launchpad mass was fuel?

Transcribed Image Text:

dv dm m- =-u- mg dt dt