Foundations — Thrust equation F = ṁv_e + (P_e − P_a)A_e — derivation

This page has one job: build every letter of the thrust equation from zero, one at a time, each earning the next. We will only write the full formula at the very end, once every symbol in it has a meaning and a picture. Nothing here assumes you have seen the parent derivation.

First, a promise about direction. Everywhere on this page we agree that forward (the way the rocket travels) counts as positive, and backward as negative. Keep that fixed — it is the single rule that decides whether a term adds to thrust or eats into it.

0 — What "force" and "thrust" even mean

Thrust is just the special name for the forward force an engine makes. So our whole target is one question: how long is the forward arrow the rocket engine draws? Because forward is our positive direction, a thrust that helps the rocket is a positive $F$.

Figure s01 (below) shows exactly this: the same rocket with a short mint arrow and a long coral arrow, so you can see that "more force" simply means "longer forward arrow" — that length is the number $F$ we are chasing.

Why we need it: the entire topic computes one number — the size of that forward arrow. Everything else is bookkeeping to get it right.

1 — Velocity $v$ and the two frames

Here is the subtle part the parent leans on constantly: velocity depends on who is watching.

• A ground observer sees the exhaust gas moving at some speed.
• The rocket itself sees the gas leave much faster (the rocket is running away from it).

To talk about "who is watching" we need a name for the rocket's own motion.

Figure s02 (below) puts both viewpoints in one scene: a mint arrow shows the rocket moving forward at $v$; a coral arrow shows the gas puff shooting backward at $v_e$ as the rocket sees it. This is the picture behind the frame confusion — read it before the callout.

Why we need it: thrust depends on how fast gas leaves the rocket, so $v_e$ (not the ground speed $v$) is the honest measure of "how fast we throw".

2 — Mass $m$ and mass flow rate $\dot{m}$

A rocket does not throw all its gas at once; it throws a steady stream. To describe a stream we need "how much per second".

Figure s03 (below) contrasts a lavender trickle (small $\dot{m}$) with a coral fire-hose (big $\dot{m}$) so that the abstract symbol becomes a picture of flow rate — how many kilograms cross the nozzle mouth each second.

Why we need it: "throw more gas" means a bigger $\dot{m}$. The product $\dot{m}{v}_e$ = (how much per second) × (how fast) = momentum thrown per second = the main part of thrust.

3 — Momentum $p$ and why we multiply $\dot{m}$ by $v_e$

Newton's insight, which powers this whole chapter, is that force equals momentum thrown per second. Throw momentum out the back at a steady rate and you feel an equal, opposite force forward — this is exactly Newton's Third Law combined with Conservation of Momentum.

$\text{momentum per second}=\dot{m}{v}_e=\left(\frac{\text{kg}}{\text{s}}\right)\left(\frac{\text{m}}{\text{s}}\right)=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^2}=\text{N}$

The units come out as newtons — a force. That is our sanity check that $\dot{m}{v}_e$ really is a thrust, and (by the sign argument above) a positive, forward one.

Why we need it: momentum is the bridge from "throwing gas" to "feeling a force". Without it, $\dot{m}{v}_e$ is just letters multiplied together.

4 — Pressure $P$ and area $A$

Now the key link: if a pressure $P$ pushes on an area $A$, the total force is

$F=P\times A\text{(force = pressure}\times \text{area).}$

That is why pressure ever becomes a force in the equation: $P$ alone is not a push, but $P$ acting over the exit hole $A_e$ is.

Figure s04 (below) shows the tug-of-war at the exit plane: coral arrows are the exhaust pushing out ($P_e$), mint arrows are the air pushing back in ($P_a$), both acting over the butter-coloured area $A_e$. The picture makes the subtraction visible — the net force is whatever is left after the two opposing pushes fight.

Why we need it: the "bonus" term in the equation is nothing but $\text{pressure difference}\times \text{area}$, and its sign follows straight from our forward-positive rule.

Assembling the equation

Now — and only now — every symbol has a meaning, a picture, and a sign. We can finally write the whole thing:

How the pieces feed the equation

Read this map top to bottom, following the arrows: each box is one idea from this page, and an arrow means "this idea is needed to build the box it points to". Everything funnels into the single box "Thrust equation" at the right. The two grey ideas at the bottom-left, Newton's Third Law and Conservation of Momentum, are the physical laws that justify the "momentum per second" step.

Follow any arrow into "Thrust equation" and you are reading the meaning of one term in $F=\dot{m}{v}_e+(P_e-P_a)A_e$.

Where these lead next

Once these symbols are solid, the same pieces reappear across the chapter:

• $v_e$ and $\dot{m}$ drive the Tsiolkovsky Rocket Equation (how fast a rocket can ever go).
• The combination $F/\dot{m}$ becomes Specific Impulse (a fuel-efficiency score).
• The nozzle that sets $P_e$ and $A_e$ is the De Laval Nozzle, governed by Bernoulli & Compressible Flow.

Equipment checklist

Test yourself — cover the right side of each line.