Orbit isn't about going “high” — it's about going fast: you have to fly sideways so quickly that Earth keeps curving away beneath the falling rocket. Launch from the rotating Earth, survive Max Q, drop a stage, and reach orbit — then hop over to geostationary orbit.
Challenges
Do them in the simulator — they complete themselves.
Flight log
Key moments of the mission (mission time from ignition).
- Waiting for ignition…
The Tsiolkovsky equation — how much speed does fuel buy?
A rocket doesn't push against the air: it throws gas backward, so it moves forward itself (conservation of momentum). The gain in speed Δv depends only on the exhaust velocity ve and on how many times heavier the fueled rocket is than the empty one.
Why do rockets have stages?
An empty tank is dead weight. Once a stage burns out, the rocket drops it and keeps accelerating only what is still needed. Compare rockets with the same launch mass, the same propellant (ve from the calculator next to this) and the same structural fraction.
Stages sized optimally (each has the same mass ratio). Gold line: about 9.4 km/s — what it takes to reach low orbit, including gravity and drag losses.
Cosmic velocities
The higher a satellite orbits, the slower it moves and the longer one lap takes. The numbers below come from the same model as the simulator.
A circular orbit just above the surface (if there were no air): v = √(GM/R).
Escape velocity from the surface: kinetic energy equals binding energy. That is √2 times the first cosmic velocity.
What you need near Earth to escape the Solar System, using Earth's own motion around the Sun.
Kepler's third law: T² ∝ r³. A geostationary orbit takes as long as one rotation of Earth relative to the stars (23 h 56 min), so the satellite “hangs” over one point on the equator.
How does it work?
A few formulas are enough to understand any flight in the simulator.
1. An orbit is a never-ending fall
Gravity weakens with the square of the distance from Earth's center. A rocket in orbit is falling all the time, but it moves sideways so fast that the surface “curves away” beneath it at the same rate.
g(r) = GMr² vcirc = √GMr T = 2π √r³GMGM = 3.986 · 10¹⁴ m³/s², R = 6,371 km. At 400 km: v ≈ 7.67 km/s, T ≈ 92.4 min.
2. Energy determines the shape of the orbit
From the position and velocity, the simulator computes at every moment the ellipse (a conic section) the rocket would follow with the engine off. That's where apoapsis and periapsis come from.
v² = GM (2r − 1a) ε = v²2 − GMrε < 0 — closed orbit (ellipse), ε ≥ 0 — escape. Hence the escape velocity vesc = √2 · vcirc.
3. The engine and the Tsiolkovsky equation
Thrust is mass flow times exhaust velocity. Specific impulse Isp tells you how “economical” the propellant is; in a vacuum an engine has a higher Isp than at ground level.
F = ṁ · ve ve = Isp · g0 Δv = ve ln m0mkThe rocket in the simulator: stage I 7.6 MN, Isp 282 s (sea level) / 311 s (vacuum); stage II 0.98 MN, 345 s.
4. Atmosphere, drag, and Max Q
The air thins out exponentially. Drag grows with the square of speed, so the worst moment comes partway up: the rocket is already fast, but the air is still dense — that's Max Q, the peak of dynamic pressure.
ρ = ρ0 e−h/H q = ½ ρ v² FD = q · CD · Aρ0 = 1.225 kg/m³, H = 8.5 km. That's why a rocket first flies straight up and then performs a gravity turn: it tilts slightly and lets gravity gradually bend its path, always pointing its nose along its velocity.
5. The Hohmann transfer
The cheapest way to move between circular orbits: a prograde burn stretches the orbit into an ellipse whose apoapsis reaches the target orbit. Half an orbit later, a second burn “rounds out” the orbit.
Δv1 = √GMr1 (√2r2r1+r2 − 1) Δv2 = √GMr2 (1 − √2r1r1+r2)From 400 km to GEO: Δv1 ≈ 2.40 km/s, Δv2 ≈ 1.46 km/s, transfer time ≈ 5 h 17 min.
6. G-load
G-load is the acceleration the crew feels: from thrust and drag, without gravity (in free fall — including in orbit — it is 0 g). The flight program throttles the engine so as not to exceed 4 g.
n = |Fthrust + FD|m · g0 g0 = 9.806 65 m/s²On the launch pad the ground holds the rocket up, which is why the g-load is 1 g at that point.
Model simplifications
- The flight is two-dimensional, in the equatorial plane; Earth is a uniform sphere with a radius of 6,371 km, rotating once per sidereal day (launch from the equator toward the east).
- Exponential atmosphere with no wind and a constant drag coefficient; above 140 km we ignore drag, so orbits don't decay.
- The rocket is fictional (parameters similar to today's medium-lift rockets); the stage II engine can be restarted any number of times, and thrust can be throttled from 0 to 100%.
- The flight program is simple automation (vertical, an 8° pitch-over, gravity turn, then holding an altitude of about 190 km), not real rocket guidance.
- GEO altitude (35,786 km) is really measured from the equator (radius 6,378 km); on our spherical Earth the period comes out about 20 s shorter than a sidereal day.
Did you know…?
At an altitude of 400 km, Earth's pull still has almost 90% of its strength at the surface. Astronauts float because they and the station are constantly falling — they just keep “missing” Earth.
A point on the equator races east at about 465 m/s. That's why rockets launch eastward, and why Europe's spaceport is in Kourou, French Guiana, just 5° from the equator.
Konstantin Tsiolkovsky — the son of a Polish father — published his equation in 1903, and Walter Hohmann described the transfer between orbits in 1925. The first rocket didn't reach space until 1944.