A variable-mass rocket climbs straight up under three forces: thrust, weight and atmospheric drag. While the engine burns, mass drops linearly (ṁ) and thrust comes from momentum plus a pressure term:
Ve = Isp·g0
F = ṁ·Ve + (Pe − Pamb(h))·Ae
Pamb(h) = P0·exp(−h/H), ρ(h) = ρ0·exp(−h/H)
D = ½·ρ(h)·Cd·A·v·|v|
a = (F − D − m·g0) / m, m(t) = m0 − ṁ·t
Ambient pressure and air density fall off exponentially with altitude (scale height H≈8.5 km), so the exhaust always meets thinner, lower-pressure air the higher the rocket climbs. Comparing the fixed nozzle exit pressure Pe to that falling ambient pressure gives the nozzle pressure ratio (NPR): near 1 the plume is perfectly expanded (a clean, near-cylindrical exhaust column); above ~1.2 it is underexpanded and flares wider than the nozzle lip, forming visible shock (Mach) diamonds as the flow over- and under-corrects to match ambient pressure; below ~0.8 it is overexpanded, pinching inward before re-expanding downstream, with pronounced flow separation risk below ~0.4. Because Pamb falls with altitude, the same fixed nozzle drifts from overexpanded at sea level toward increasingly underexpanded near burnout — exactly the effect real launch footage shows as the visible exhaust plume balloons out with height.
- Dynamic pressure (Q) — ½ρv²; the "Max-Q" readout marks the structural stress peak of the ascent, where density and velocity trade off.
- Burnout — thrust cuts to zero once the propellant fraction is spent; the vehicle then coasts under gravity and drag alone to apogee.
- The nozzle-diamond drawing is a simplified schematic (spacing narrows as |NPR−1| grows) illustrating the same regime the NPR number reports, not a CFD-accurate shock solution.