Rockets · trajectories · the road to the Moon

The math that leaves the ground.

From the rocket equation to a full Earth–Moon trajectory you can scrub through, with every burn, velocity and timeline annotated. Built around the two programs that defined lunar flight: Apollo and Artemis.

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How rockets work

Launch physics — throwing mass to go fast

A rocket moves by throwing propellant backwards. Every result below follows from that single idea, formalised by Konstantin Tsiolkovsky in 1903.

Δvvelocity change the rocket can produce — the currency of spaceflight
v_eeffective exhaust velocity
I_spspecific impulse (s) — engine efficiency
g₀9.80665 m/s², standard gravity
m₀wet mass (fuelled)
m_fdry mass (empty)

The logarithm is the tyranny of the rocket equation: doubling your mass ratio adds only a fixed Δv, so reaching orbit (~9.4 km/s to LEO) demands either ferocious exhaust velocity or throwing away empty tanks — i.e. staging.

Fthrust = mass flow rate × exhaust velocity
propellant mass flow rate
TWRthrust-to-weight ratio — must exceed 1 to lift off

Launch lab ?

Tune an engine and a rocket, see the Δv.

350 s
12.0
Exhaust vel.
km/s
Delta-v
km/s

Will it lift off?

35 MN
2900 t
TWR
at liftoff

Getting to orbit and beyond

Orbital mechanics — falling, forever

Orbit is not about height; it is about sideways speed. Go fast enough horizontally and you fall past the planet. The vis-viva equation ties speed, distance and orbit size together.

vspeed at distance r
μGM, the body's gravitational parameter
rcurrent distance from the centre
asemi-major axis of the orbit

Two special cases drop out immediately:

To change orbits efficiently you burn at the right point — the Hohmann transfer, the cheapest two-burn route between circular orbits.

Hohmann transfer ?

From a 300 km parking orbit to a target altitude.

35,786 km (GEO)
Burn 1 (Δv₁)
km/s
Burn 2 (Δv₂)
km/s
Transfer time
hours

The main event

Earth–Moon trajectory simulator

Scrub through a full lunar mission. Watch the spacecraft climb out of Earth's gravity well, coast for days, and arrive at the Moon — with every phase, burn and number exposed. Switch programs to compare how Apollo and Artemis fly.

Trajectory geometry is schematic (not a numerical integration); milestones, burns and durations are representative real-mission values. Velocities/distances shown update continuously as you scrub.

Mission time
Phase
Speed (Earth frame)
km/s
Dist. from Earth
×1000 km

Program I

Apollo — humans on the Moon, 1969–1972

Six crewed landings, built on the largest rocket ever flown to operational service: the Saturn V. Apollo 11 carried Armstrong, Aldrin and Collins to the Sea of Tranquility, landing on 20 July 1969 at mission-elapsed time 102:45:40.

Saturn V — three stages to the Moon ?

110.6 m tall · ~2,970 t at liftoff · ~34.5 MN sea-level thrust.

Apollo 11 by the numbers

Launch
16 Jul 1969
Parking orbit
~190
km altitude
TLI Δv
~3.1
km/s
Coast
~3
days
Landing MET
102:45:40
Mission
~8d 3h
to splashdown

After surface operations the ascent stage of the Lunar Module (Eagle) lifted off, rendezvoused with the Command Module (Columbia), and a trans-Earth injection burn started the three-day coast home, ending in splashdown on 24 July 1969.

Program II

Artemis — returning to stay

NASA's Artemis program aims for a sustainable lunar presence, targeting the south pole and building the Gateway station. It flies the Space Launch System (SLS) and the Orion spacecraft with its European Service Module.

SLS Block 1 — boosters, core, upper stage ?

~98 m tall · ~2,600 t at liftoff · ~39.1 MN liftoff thrust (~8.8 million lbf).

The three opening flights

Artemis I · 2022

Uncrewed. Orion flew to a distant retrograde orbit, then home — a 25.5-day, ~2.0-million-km shakedown that ended with splashdown on 11 Dec 2022.

Artemis II · planned

First crewed flight: a free-return lunar flyby with four astronauts (Wiseman, Glover, Koch, Hansen). ~10 days; no lunar orbit.

Artemis III · planned

The crewed landing — near the lunar south pole, using a SpaceX Starship Human Landing System. Crew transfers from Orion to the lander in a near-rectilinear halo orbit (NRHO).

Artemis II and III dates and architecture have shifted over time. For the current schedule and crew, check up-to-date NASA sources (enable web search).

Apollo vs Artemis

DimensionApollo (11)Artemis (I–III)
Launch vehicleSaturn VSLS Block 1
Liftoff thrust~34.5 MN~39.1 MN
Crew capsuleApollo CM · 3 crewOrion + ESM · 4 crew
Lunar orbitLow lunar orbit (~110 km)DRO (I) / NRHO (Gateway)
LanderLunar ModuleStarship HLS (III)
Return profileDirect entry, ~11 km/sSkip entry, ~11 km/s
GoalReach the Moon firstSustainable presence, south pole

Beyond the Moon

Other missions worth simulating

The same equations send probes across the Solar System and beyond. Click any mission for its key numbers and the physics trick that made it possible.

Quick reference

Formulas & glossary

Key formulas

Glossary

Δv (delta-v)
Total velocity change a vehicle can achieve; the budget every mission is planned around.
Specific impulse
Thrust per unit propellant weight flow (s). Higher = more efficient.
Staging
Shedding empty mass mid-flight to keep accelerating efficiently.
TLI
Trans-lunar injection — the burn that leaves Earth orbit for the Moon.
LOI
Lunar orbit insertion — braking into orbit around the Moon.
Free return
A trajectory that uses lunar gravity to swing back to Earth with no capture burn.
Gravity assist
Stealing a tiny bit of a planet's orbital momentum to change speed for free.
DRO / NRHO
Large, stable lunar orbits used by Artemis and the Gateway.