A jet aircraft of mass m hovers at a fixed height by blowing…

A jet aircraft of mass $m$ hovers at a fixed height by blowing its exhaust gases straight down. The exhaust opening has area $S$, and the gas density is $\rho$. At what speed must the gases leave the engine?

Explanation of Method

Exhaust speed for a hovering jet.

Step 1: Thrust

Each second a gas mass $\dot m = \rho S v$ leaves at speed $v$, carrying momentum $\dot m v$ downward per second. The reaction on the aircraft is

\[ F = \dot m\,v = \rho S v^2. \]

Step 2: Hovering

\[ \rho S v^2 = mg. \]

\[v = \sqrt{\frac{mg}{\rho S}}\]

$\text{thrust } \rho S v^2$ $mg$ $v$ $S$ $\rho S v^2 = mg$

Each second, a gas mass $\rho S v$ is thrown down at speed $v$; the reaction on the aircraft is $\rho S v^2$ upward, which must equal its weight.

Result

\[\boxed{\,v = \sqrt{\frac{mg}{\rho S}}\,}\]

Remark

Check. The original result is correct (the air taken in by the engine is neglected, and $m$ is treated as constant over a short time).