Rocket Thrust with the Pressure Term

The momentum term is only most of the thrust. The gas leaving the nozzle has a pressure of its own at the exit plane, and the atmosphere outside pushes back with the ambient pressure, and the difference acting over the exit area either adds to the thrust or subtracts from it: $F = \dot m v_e + (p_e - p_a) A_e$. This second term is small compared with the momentum term — usually 10 to 20% — and it is responsible for almost everything that seems odd about how rocket engines behave.

It is why the same engine gets stronger as it climbs. At sea level $p_a$ is 101.325 kPa; at 20 km it is about 5.5 kPa; in vacuum it is exactly zero. Nothing about the engine changes, but the term that was subtracting starts adding, and a first stage typically gains 10 to 20% thrust between the pad and staging. Both pressures must be ABSOLUTE for this to work — a gauge reading, which is already referenced to the atmosphere, silently double-counts $p_a$ and gives an answer that is wrong at every altitude except sea level.

The nozzle is called perfectly expanded when $p_e = p_a$, the pressure term vanishes entirely, and thrust is maximised for that ambient condition. A nozzle can only be perfect at one altitude, which forces a design compromise that has never been solved: sea-level nozzles are made deliberately short and stubby so they will not over-expand on the pad, and they are correspondingly inefficient higher up, while vacuum nozzles are made enormous and could not be fired at sea level at all. Aerospike engines were invented precisely to escape this compromise by letting the ambient pressure shape the outer boundary of the exhaust, and they have never flown operationally.

Over-expansion is the dangerous direction. When $p_e$ falls well below $p_a$, the exhaust is being squeezed by the outside air, and past roughly 40% of ambient the flow separates from the nozzle wall. Separation is not gentle: it is unsteady, it moves around the bell, and the side loads it generates have destroyed nozzles on test stands. This is why sea-level engines are designed with only modest over-expansion and why altitude-compensating hardware keeps being proposed. The mirror-image mistake is bookkeeping rather than mechanical — quoting a vacuum thrust figure for a sea-level lift-off, which overstates a first stage by 10 to 20% and an upper-stage engine by far more.