Exhaust velocity ($v_e$)

Exhaust velocity is the velocity of expelled propellant measured relative to the engine. In a simple straight jet it points backward, while the engine's thrust points forward. Rocket calculations often use its positive magnitude, as in rocket exhaust speed. The relative frame matters: the propellant can be moving forward relative to a distant observer while still leaving the engine backward.

For a steady rocket nozzle with outward mass flow rate $\dot{m}>0$, exit speed $u_e$, exit pressure $p_e$, ambient pressure $p_a$ and exit area $A_e$, the axial thrust isThe pressure term motivates an effective exhaust velocity $v_e=F/\dot{m}$, which includes both momentum flux and nozzle pressure effects. When exit and ambient pressures match, $v_e=u_e$. Specific impulse in seconds is $I_{\mathrm{sp}}=v_e/g_0$, where $g_0=9.80665\mathrm{m}{\mathrm{s}}^{-2}$ is conventional standard gravity; using this unit does not imply that the engine must operate near Earth.^[ref]

For an illustrative pressure-matched engine, $\dot{m}=2\mathrm{kg}{\mathrm{s}}^{-1}$ and $u_e=3000\mathrm{m}{\mathrm{s}}^{-1}$ give $F=6000\mathrm{N}$. At the same effective exhaust velocity, halving the mass flow rate halves thrust. For an ideal burn without external forces and with constant $v_e$, the rocket equation gives $\Delta v=v_e\ln (m_0/m_f)$: a mass ratio of $2$ yields approximately $2079\mathrm{m}{\mathrm{s}}^{-1}$. Thus higher effective exhaust velocity increases the velocity change available from a specified propellant fraction; it does not by itself specify a large thrust.