Systems with Varying Mass: A Rocket — Question 5

Question 5

Hovering requires a declining mass flow

An ideal rocket hovers at fixed height in uniform gravity. Initially $m_0=100\,\mathrm{kg}$, of which $60\,\mathrm{kg}$ is dry mass. Relative exhaust speed is $u=1000\,\mathrm{m/s}$ and $g=9.8\,\mathrm{m/s^2}$. Neglect air effects.

1. Derive the mass as a function of time needed to hover.

2. Find maximum hover time and compare initial and final propellant flow rates.

Question 5 - Solution

Hovering requires a declining mass flow

(a) Derivation and calculation

Zero acceleration requires $uq=mg$, with $q=-\dot m$: $$\frac{dm}{dt}=-\frac gu m\Rightarrow\int_{100}^{m(t)}\frac{dm}{m}=-\frac gu t.$$ $$\boxed{m(t)=100e^{-gt/u}=100e^{-0.0098t}\ \mathrm{kg}}.$$

(b) Result and interpretation

The model applies until the dry-mass limit is reached: $$60=100e^{-gt_{\max}/u}\Rightarrow\boxed{t_{\max}=\frac{1000}{9.8}\ln\frac{100}{60}=52.13\ \mathrm s}.$$ $$\boxed{q_i=100(9.8)/1000=0.980},\qquad\boxed{q_f=60(9.8)/1000=0.588\ \mathrm{kg/s}}.$$ A constant flow would exceed the required thrust as fuel is consumed.

Physical check

The hovering trajectory ends at fuel exhaustion; the exponential cannot continue below dry mass.