---
title: "Rocket thrust, specific impulse and rocket equation"
subject: "Turbo Machines"
chapter: 5
key: "semester_7/turbo_machines:rocket-thrust-specific-impulse-derivation"
source: https://www.bctengineer.com/q/semester_7/turbo_machines/rocket-thrust-specific-impulse-derivation
site: BCT Engineering
---

# Rocket thrust, specific impulse and rocket equation

Turbo Machines · Chapter 5: Theoretical Jet Engine

- Asked in: year not given, 8 marks
- Video: [Deriving the Ideal Rocket Equation](https://www.youtube.com/watch?v=dqyGOvokRsw) — Madaydude
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/turbo_machines/rocket-thrust-specific-impulse-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_7/turbo_machines/chapter-5

## Question

Derive the thrust equation of a rocket engine. Define effective exhaust velocity, specific impulse, and propulsive efficiency, and derive the ideal rocket (Tsiolkovsky) equation for the velocity gain of a rocket in a gravity-free vacuum.

## Short answer

Thrust is F = mdot_p C_e + (p_e - p_a)A_e; effective exhaust velocity is C_eff = F/mdot_p and specific impulse is I_sp = C_eff/g0. Propulsive efficiency is eta_p = 2(v/C_eff)/(1 + (v/C_eff)^2). The ideal rocket equation is Delta v = C_eff ln(M0/Mf).

## Answer

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The rocket thrust equation is
   A. F = m_p C_e - (p_e - p_a)A_e
   B. F = m_p C_e + (p_e - p_a)A_e
   C. F = m_p (C_e - v)
   D. F = p_e A_e only
   Answer: B. Thrust = momentum thrust m_p C_e plus pressure thrust (p_e - p_a)A_e, zero for a perfectly expanded nozzle. (Key point: Thrust equation)

2. Specific impulse is defined as
   A. I_sp = F m_p
   B. I_sp = C_e g0
   C. I_sp = F/(m_p g0) = C_eff/g0, in seconds
   D. I_sp = m_p/F
   Answer: C. It is thrust per unit weight flow of propellant, equal to C_eff/g0. (Key point: Effective exhaust velocity and Isp)

3. Rocket propulsive efficiency is maximum (1.0) when
   A. vehicle speed equals effective exhaust velocity
   B. vehicle speed is zero
   C. exhaust velocity is zero
   D. ambient pressure is zero
   Answer: A. At v = C_eff the jet leaves with zero absolute velocity so no kinetic energy is wasted. (Key point: Propulsive efficiency maximum)

4. The ideal rocket equation in vacuum with no gravity is
   A. dv = C_eff M0/Mf
   B. dv = C_eff (M0 - Mf)
   C. dv = C_eff ln(Mf/M0)
   D. dv = C_eff ln(M0/Mf)
   Answer: D. Integrating dv = -C_eff dM/M from M0 to Mf gives Delta v = C_eff ln(M0/Mf). (Key point: Rocket equation)

5. Why do launch vehicles use several stages?
   A. To reduce exhaust velocity
   B. To get a larger mass ratio M0/Mf and so a larger velocity gain
   C. To cancel pressure thrust
   D. To avoid the rocket equation
   Answer: B. Dropping empty structure raises the effective mass ratio, and Delta v grows with ln of it; gravity and drag losses reduce the real gain. (Key point: Staging)

## Related in Turbo Machines

- [Rocket engine thrust and specific impulse calculation](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Arocket-thrust-specific-impulse-numerical) — Turbo Machines, chapter 5
- [Turbojet cycle, thrust and propulsive efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Aturbojet-cycle-thrust-propulsive-efficiency) — Turbo Machines, chapter 6

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