Thrust equation F = ṁv_e + (P_e − P_a)A_e — derivation: Visual walkthrough

Before any symbol appears, let us name our cast of characters in plain words.

Step 1 — Throwing mass backward pushes you forward

WHAT. Picture a person standing still on a frictionless skateboard, holding a heavy ball. They throw the ball backward. The person rolls forward.

WHY. This is the entire heart of a rocket. Nothing about rockets is more fundamental than this. It is Newton's Third Law: every push comes in a pair — if you push the ball back, the ball pushes you forward with an equal, opposite shove. We start here because if you understand this one throw, you understand thrust.

PICTURE.

Look at the two arrows. The magenta arrow is the ball flying back. The violet arrow is the person gliding forward. They are equal in strength, opposite in direction — that is the Third-Law pair.

Step 2 — A rocket is a machine that throws mass continuously

WHAT. A rocket does not throw one ball; it throws a stream of gas, endlessly, every second. So instead of "mass thrown," we care about mass thrown per second.

WHY. A single throw gives a single kick. A continuous stream gives a steady force. To turn "momentum thrown" into a steady force, we ask: how much momentum leaves each second?

PICTURE.

Each little orange puff is a chunk of gas leaving in one tiny slice of time. In one second a whole ribbon of these puffs streams out the back.

Step 3 — Bookkeeping the momentum honestly (variable mass)

WHAT. We now prove $\overset{m}{˙} v_{e}$ carefully, because the rocket's mass keeps shrinking as it burns fuel, and we must not cheat.

WHY. The simple $F = ma$ assumes constant mass. A rocket's mass changes every instant, so we track the total momentum of a fixed lump of matter (rocket + the fuel it is about to burn) across one tiny time slice $d t$ . This is Conservation of Momentum done properly.

PICTURE.

Two snapshots. Before (top): one lump, mass $m$ , moving at velocity $v$ . After (bottom): the rocket, now lighter, moves a touch faster; a small puff of exhaust flies backward.

Momentum before — one lump, so just mass $\times$ velocity: $p_{1} = whole mass m its speed v$

Momentum after — two pieces, added up: $p_{2} = lighter rocket, faster (m - d m_{e x}) (v + d v) + the puff, seen from the ground d m_{e x} (v - v_{e})$

Step 4 — Let the tiny terms fall away

WHAT. Subtract the two momenta and simplify.

WHY. We only want the leading effect. Products of two tiny things (like $d m_{e x} d v$ ) are tiny-times-tiny — utterly negligible, so we drop them. This is the standard "keep first order" move.

PICTURE.

The bar chart shows the sizes: the surviving terms are full-height; the $d m_{e x} d v$ term is a sliver you can barely see — that is the one we throw away.

Expand $p_{2}$ : $p_{2} = m v + m d v - v d m_{e x} - tiny \times tiny, drop it d m_{e x} d v + v d m_{e x} - v_{e} d m_{e x}$

The $- v d m_{e x}$ and $+ v d m_{e x}$ cancel. Subtract $p_{1} = m v$ : $d p = p_{2} - p_{1} = rocket speeds up m d v - momentum carried off by gas v_{e} d m_{e x}$

Divide by $d t$ and use $d m_{e x} = \overset{m}{˙} d t$ : $\frac{d p}{d t} = m \frac{d v}{d t} - v_{e} \overset{m}{˙}$

Step 5 — Now zoom out: air is pressing on the whole rocket

WHAT. We stopped pretending the rocket floats in nothing. It sits inside air, and air presses on every square centimetre of its skin.

WHY. That pressing is a real force. We must check whether it changes the thrust — and it does, but only in one surprising place.

PICTURE.

Little navy arrows push inward on the rocket from all sides — this is atmospheric pressure $P_{a}$ squeezing the body from every direction.

Step 6 — Why all that pressure cancels — except at the hole

WHAT. Add up the ambient pressure over the entire closed surface of the rocket. The total is zero.

WHY. For any fully sealed body sitting in still air, the push from the left cancels the push from the right, top cancels bottom, and so on — otherwise a sealed can of air would spontaneously fly across the room. So ambient pressure alone can never produce a net force on a closed shape.

But a rocket is not closed — it has a hole at the back where the nozzle opens. At that hole there is no skin for the air to push on. That is the one spot where the cancellation breaks.

PICTURE.

On the left, the sealed body: every inward arrow has an equal opposite partner → net zero. On the right, the same body with the nozzle mouth open (dashed magenta ring): the arrow that would have pushed on the sealed back is missing. Its partner on the front is now unbalanced.

Step 7 — The two pushes that survive at the exit

WHAT. At the open exit plane, area $A_{e}$ , two things differ from the "sealed" pretend-body:

The exhaust gas pushes forward on the engine with force $P_{e} A_{e}$ .
The ambient air is not there to push back (there is no skin), so we must remove the backward push $P_{a} A_{e}$ that our "sealed everywhere" pretend-body wrongly included.

WHY. Force is pressure $\times$ area. The gas at pressure $P_{e}$ presses on the exit area $A_{e}$ → $P_{e} A_{e}$ forward. The air at $P_{a}$ would have pressed on that same $A_{e}$ backward → subtract $P_{a} A_{e}$ .

PICTURE.

The big orange arrow $P_{e} A_{e}$ pushes the engine forward. The navy arrow $P_{a} A_{e}$ pushes backward. Their difference is the net pressure push.

Step 8 — Add the two pushes: the full thrust

WHAT. Total thrust = momentum thrust (Step 4) + pressure thrust (Step 7).

WHY. They are two independent forward pushes acting on the same engine, so they simply add.

PICTURE.

Two stacked arrows: the long violet momentum arrow $\overset{m}{˙} v_{e}$ , then the shorter orange pressure arrow $(P_{e} - P_{a}) A_{e}$ on top, summing to the total magenta thrust $F$ .

Step 9 — All the cases: sign of the pressure term

WHAT. The pressure term can be positive, negative, or zero. We walk every case.

WHY. A reader must never meet a nozzle condition we didn't show. There are exactly three, plus the vacuum limit.

PICTURE.

Three nozzle exits side by side, and the vacuum case.

The one-picture summary

One diagram, the whole story: gas thrown back (momentum thrust $\overset{m}{˙} v_{e}$ ), pressure difference at the mouth (pressure thrust $(P_{e} - P_{a}) A_{e}$ ), and their sum $F$ pushing the rocket forward.

Recall Feynman retelling — the whole walkthrough in plain words

Stand on a skateboard and throw balls backward: each throw shoves you forward (Step 1). A rocket does this nonstop, so we count balls-per-second times ball-speed — that is $\overset{m}{˙} v_{e}$ (Steps 2–4). Then remember the rocket sits in air. Air squeezes it from all sides, but on a closed shape that squeeze cancels out completely (Step 6). The only place it can't cancel is the open nozzle mouth: there the hot gas pushes the engine forward at $P_{e} A_{e}$ , and the missing air-push means we subtract $P_{a} A_{e}$ (Step 7). Add the throwing-push and the mouth-push and you have the whole thrust $F = \overset{m}{˙} v_{e} + (P_{e} - P_{a}) A_{e}$ (Step 8). If the gas leaves at exactly the outside pressure, the mouth-push is zero and only throwing matters; in space there's no air to push back, so the mouth-push is biggest (Step 9).

Recall Quick self-check

Which velocity goes in the momentum term — ground speed or relative speed? ::: The exhaust speed relative to the rocket, $v_{e}$ . Why does $P_{a}$ get subtracted, not added? ::: Because at the exit the outside air pushes the engine backward; over the rest of the closed body it cancels. When is the pressure term exactly zero? ::: Perfect expansion, $P_{e} = P_{a}$ .

Connections

Newton's Third Law — the throw-and-be-pushed pair behind momentum thrust (Step 1).
Conservation of Momentum — the honest bookkeeping of Steps 3–4.
De Laval Nozzle — the hardware that sets $P_{e}$ and $A_{e}$ in Steps 7 & 9.
Bernoulli & Compressible Flow — why exit pressure and speed come out the way they do.
Specific Impulse — how thrust turns into a fuel-efficiency number.
Tsiolkovsky Rocket Equation — what this thrust does to the rocket's speed over time.