Rocket Launch Physics (2D)

Side view of a two-stage rocket flying a gravity turn around a round Earth, with the rocket equation, drag, max Q and the resulting orbit calculated live.

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Presets

Playback

Stage 1

First stage: thrust, burn rate (sets Isp) and masses.

Stage 2 and payload

Second stage, payload and coast before ignition.

Guidance and air

Pitch-over kick and stage-2 steering, drag.

Layers

Model checks

The science

The rocket equation

A rocket accelerates by throwing mass backwards. With exhaust speed ve = F / ṁ = Isp g0 the momentum balance gives Tsiolkovsky's equation for the velocity gained by a stage:

Δv = ve ln(m0 / m1),    m0 = mdry + mprop + mabove,   m1 = m0 − mprop

Staging drops the dead mass of stage 1 so that stage 2 starts with a much better mass ratio; the page adds the two terms. The actual flight is integrated with RK4 (time step 0.05 s while burning) from

m dv/dt = Fthrust u − ½ ρ(h) Cd A |v| v − m G M r / r3,    ρ(h) from the US Standard Atmosphere 1976 table (1.225 kg/m3 at sea level, 5.6×10-7 at 100 km)

Thrust grows slightly as the air pressure falls: F = Fsl (1 + k (1 − p/p0)), with k the vacuum gain (about 0.1 for a real engine). Dynamic pressure q = ½ ρ v2 peaks at "max Q", which is where a vehicle is structurally loaded most.

Gravity turn and orbit

Flying straight up wastes propellant against gravity (the gravity loss ∫ g cosγ dt). After a short vertical rise the rocket is tipped by a small kick angle and then thrusts along its own velocity vector (zero angle of attack, so no side loads); gravity then bends the path over by itself. To stay in orbit the rocket needs not only altitude but about 7.8 km/s of horizontal speed at 150 km: vcirc = √(GM/r). The stage-2 steering option points the thrust slightly above or below the local horizontal in order to null the climb rate (a crude stand-in for real guidance). The page then reports the osculating orbit from the energy and angular momentum:

ε = v2/2 − GM/r,    a = −GM/(2ε),    e = √(1 + 2εh2/(GM)2),    perigee = a(1 − e) − RE

Honest simplifications

  • Planar flight over a non-rotating Earth: the 400 m/s boost from Earth's rotation is ignored, and so are winds, oblateness (J2) and the Moon and Sun.
  • Constant drag coefficient (real Cd varies strongly with Mach number) and a static standard atmosphere (no weather, no solar-activity changes of the thermosphere).
  • Thrust and mass flow are constant while a stage burns (no throttling, no fixed g-limit); thrust-vector control is perfect and instant.
  • The sounding-rocket preset uses the numbers of the 3D page (36 kN, 800 kg of propellant) with a realistic burn rate; the 3D page itself uses a game-style burn rate. The small orbital launcher preset has the mass and thrust class of a small commercial launcher but is not a model of any real vehicle.

Checks run in the page

(1) The integrator is run in a gravity-free vacuum and its final speed must equal the sum of ve ln(m0/m1) of the two stages. (2) After burnout the page propagates the orbit for one full revolution without drag or thrust: the time of that revolution must equal Kepler's third law, 2π√(a3/GM). (3) The specific energy and the angular momentum must stay constant during it. Suborbital flights have no closed orbit and these two checks are skipped.

What to try
  • Run Small orbital launcher at 15 s/s: the speed (graph 2) rises to meet the dashed circular-speed line and the perigee in the readout turns positive.
  • Compare Straight up: it climbs to a high apogee but has almost no horizontal speed, so the orbit's perigee is inside the Earth.
  • Watch the drag and gravity losses in the readout: the ideal vacuum delta-v is much more than the speed you end up with.
  • Switch to No air: the max Q becomes zero and the losses shrink to gravity only.
  • Try Heavy payload and then add propellant to stage 1 to recover.
Altitude against time
Inertial speed against the local circular orbital speed
Thrust acceleration in g (jumps at staging)
Dynamic pressure q (max Q marked)