Rocket thrust, specific impulse and rocket equation

Thrust equation

A rocket expels propellant at a mass flow rate $\dot m_p$ with exhaust velocity $C_e$ (relative to the rocket) through a nozzle exit of area $A_e$ at pressure $p_e$, into an ambient pressure $p_a$. Applying the momentum equation to the control volume around the engine:

$$F = \dot m_p C_e + (p_e - p_a)A_e$$

The first term is the momentum thrust; the second is the pressure thrust, zero when the nozzle is perfectly expanded ($p_e = p_a$). The thrust is greatest in vacuum and increases as the vehicle climbs.

Effective exhaust velocity

$$C_{eff} = \frac{F}{\dot m_p} = C_e + \frac{(p_e - p_a)A_e}{\dot m_p}$$

Specific impulse

Thrust per unit weight flow of propellant:

$$I_{sp} = \frac{F}{\dot m_p g_0} = \frac{C_{eff}}{g_0}\ \text{(seconds)}$$

It is the main figure of merit of a propellant: about 200 to 250 s for solid propellants, 300 to 450 s for liquid propellants (hydrogen/oxygen near 450 s in vacuum).

Propulsive efficiency

For vehicle speed $v$, the useful power is $Fv$; the jet leaves with absolute velocity $(C_{eff} - v)$, so the kinetic energy lost in the jet is $\tfrac12 \dot m_p (C_{eff} - v)^2$:

$$\eta_p = \frac{F v}{F v + \tfrac12 \dot m_p (C_{eff} - v)^2} = \frac{2(v/C_{eff})}{1 + (v/C_{eff})^2}$$

It is maximum (1.0) when $v = C_{eff}$.

Rocket equation

Take a rocket of instantaneous mass $M$ with velocity $v$, in vacuum and no gravity. Momentum balance over a short time $dt$:

$$M\frac{dv}{dt} = C_{eff}\left(-\frac{dM}{dt}\right) \;\Rightarrow\; dv = -C_{eff}\frac{dM}{M}$$

Integrating from initial mass $M_0$ to final (burnout) mass $M_f$:

$$\boxed{\Delta v = C_{eff}\ln\frac{M_0}{M_f}}$$

With gravity and drag the velocity gain is reduced by $g\,t_b$ (gravity loss) and drag loss. A larger mass ratio $M_0/M_f$ is obtained by staging, which is why launch vehicles have several stages.