Rocket Propulsion & Launch Vehicles

Leaving Earth. Learn the rocket thrust equation, specific impulse and exhaust velocity, the Tsiolkovsky rocket equation, mass ratio and staging, delta-v needed for orbit (with gravity and drag losses), nozzle expansion and altitude effects, liquid (bipropellant cycles), solid, hybrid and electric propulsion, and reusability — with Python for single- and two-stage Δv.

A rocket carries both fuel and oxidiser, so it works anywhere — including the vacuum of space. It is also brutally mass-limited: a typical launcher is about 90% propellant by mass at lift-off, and only 2–4% reaches orbit as payload.

The rocket equation explains why: the velocity you gain grows only with the logarithm of the mass ratio. Staging — dropping empty tanks and engines on the way up — and high-performance engines are the answers.

An analogy

A rocket is like someone on a frictionless skateboard throwing bricks backwards: the faster each brick is thrown (exhaust velocity) and the more bricks carried (mass ratio), the faster they go — but carrying more bricks makes it harder to accelerate at first.

How rockets work

Propellants burn in a combustion chamber at high pressure and temperature; the hot gas expands through a converging-diverging nozzle to supersonic speed. Thrust comes from the momentum of the exhaust (and a pressure term). Specific impulse — seconds of thrust per unit weight of propellant — is the rocket's 'fuel economy'.

Propellant types

Typical specific impulses (vacuum):

Nozzles and altitude

Thrust is maximal when the exit pressure equals ambient pressure. A sea-level nozzle is under-expanded in vacuum; a vacuum nozzle (large ε) would be over-expanded and could separate at sea level. That's why upper-stage engines have huge nozzle bells and why aerospikes and extendible nozzles are studied.

Engine cycles, losses and reusability

Liquid engines pump propellants with turbopumps driven by: gas generator (simple, some waste), expander (heated hydrogen), staged combustion (all flow through the chamber — high pressure, high Isp) or full-flow staged combustion (Raptor), or electric pumps (Rutherford). Ascent loses 1.5–2 km/s to gravity (burning vertically), drag and steering. Reusable boosters spend propellant on boost-back and landing burns in exchange for recovering the most expensive hardware.

Formulas

Worked example: Exhaust velocity

Given: I_sp = 450 s

Find: v_e.

Answer: 4413 m/s

Worked example: Mass ratio for 9.4 km/s

Given: Single stage, I_sp 350 s

Find: m₀/m_f.

Answer: ≈ 15.4 (93.5% propellant)

Worked example: Thrust-to-weight

Given: 750 kN thrust, 60 t vehicle

Find: T/W at lift-off.

Answer: 1.27 (needs > 1 to lift off)

Rocket equation and staging

import math
g0 = 9.80665
def dv(isp, m0, mf):
    return isp * g0 * math.log(m0 / mf)
# Single stage: Isp 300 s, 90% propellant
print(f"single stage, mass ratio 10: dv = {dv(300, 100, 10)/1000:.2f} km/s")
# Two stages (payload 1 t): stage masses (propellant, structure)
payload = 1000.0
s2p, s2s = 12000.0, 1500.0
s1p, s1s = 90000.0, 8000.0
m0 = payload + s2p + s2s + s1p + s1s
dv1 = dv(290, m0, m0 - s1p)
m0_2 = payload + s2p + s2s
dv2 = dv(340, m0_2, m0_2 - s2p)
print(f"two stages: dv1 = {dv1/1000:.2f} km/s, dv2 = {dv2/1000:.2f} km/s, total {(dv1+dv2)/1000:.2f} km/s")
# Thrust and exhaust velocity
mdot, ve = 250.0, 3000.0
print(f"thrust (optimally expanded) = {mdot*ve/1000:.0f} kN, Isp = {ve/g0:.0f} s")
# single stage, mass ratio 10: dv = 6.77 km/s
# two stages: dv1 = 4.58 km/s, dv2 = 5.86 km/s, total 10.44 km/s
# thrust (optimally expanded) = 750 kN, Isp = 306 s

Even with 90% propellant a single stage falls short of the ~9.4 km/s needed for low orbit; two stages reach 10.4 km/s with a 1 t payload on a 112.5 t rocket (0.9% payload fraction). This is why nearly every orbital launcher is multi-stage.

Common mistakes

Applications

Key takeaways

Going further