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For Students · Orbital Mechanics Lab

Orbital Mechanics Lab for a roller-coaster loop

Built for students learning it for a class or exam. See the concept move instead of memorizing formulas — and check your homework intuition. Simulate a roller-coaster loop live below — adjust the inputs and watch it respond, right in your browser.

Kepler

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Kepler Orbit StudioLive

two-body gravity · conic-section orbits

Controls

Speed1×

Set the eccentricity to trace Kepler's orbits — a circle, ellipse, or (past e=1) an escape hyperbola. The star sits at the focus. Drag the planet on the canvas to reposition its starting point and reshape the orbit.

Presets

Eccentricity e

Semi-major axis

▶ Run in Python

Data Inspector

Eccentricity0.60

Orbitelliptical

LawKepler / 1-r²

Governing equation

r = \frac{a ( 1 - e ^{2} )}{1 + e cos θ} = \frac{140 ( 1 - 0.6 0 ^{2} )}{1 + 0.60 cos θ}

Reading this result: At e=0.60 the orbit is an ellipse with the star at one focus. The planet sweeps equal areas in equal times, so it runs fastest at perihelion and slowest at aphelion. A higher e means a more elongated ellipse. The period grows as a^(3/2) (Kepler's third law), so a=140 sets the year length.

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Orbital Transfer

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Orbital Transfer (Hohmann)Live

minimum-energy orbit change

Controls

Speed1×

The Hohmann transfer is the fuel-cheapest way between two circular orbits: one burn to enter the transfer ellipse, one to circularize. See the two Δv costs.

Presets

Inner orbit radius

Outer orbit radius

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Data Inspector

Δv burn 11.64

Δv burn 21.34

Total Δv2.99

Transfer time61.3

Governing equation

Δ v = \frac{μ}{90} (\frac{2 \cdot 200}{90 + 200} - 1) + \frac{μ}{200} (1 - \frac{2 \cdot 90}{90 + 200}) = 2.99

Reading this result: A moderate transfer (90 → 200, ~2.2×): burn 1 (1.64) enters the ellipse, burn 2 (1.34) circularizes. Bigger ratios cost more total Δv (2.99) and coast longer (61.3).

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Escape Velocity

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Escape & Orbital VelocityLive

how fast to leave a world

Escape velocity — Earth

11.19km/s

7.91

orbital km/s

1.00

surface g (× Earth)

Controls

Custom body

Presets

Escape velocity v = √(2GM/R) is the speed needed to break free of a body's gravity from its surface, ignoring air resistance. Circular orbital velocity is √(GM/R) — a factor of √2 smaller. Both depend only on mass and radius, not on the mass of the escaping object.

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Data Inspector

Escape velocity11.19 km/s

Orbital velocity7.91 km/s

Surface gravity9.82 m/s²

Governing equation

v_{e} = \frac{2 GM}{R} = \frac{2 ( 6.674 \times 1 0 ^{- 11} ) ( 5.97 e + 24 )}{6.37 e + 6} = 11.19 km/s

Reading this result: Leaving Earth takes 11.2 km/s no matter the spacecraft mass, while a circular orbit needs only 7.9 km/s — smaller by exactly √2 (about 1.41×).

Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.

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Lagrange Points

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Lagrange PointsLive

restricted three-body problem

Controls

Presets

Mass ratio μ

Show effective potential

In a two-body system, five points let a small object stay fixed in the rotating frame. L1-L3 sit on the line through the two masses (unstable); L4 and L5 lead and trail the secondary by 60° and are stable for μ below 0.0385 — where Jupiter's Trojan asteroids live. JWST orbits the Sun-Earth L2.

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Data Inspector

μ = m₂/(m₁+m₂)0.150

L4/L5 stable?no

Points5

Governing equation

Ω (x, y) = \frac{1}{2} (x^{2} + y^{2}) + \frac{1 - μ}{r _{1}} + \frac{μ}{r _{2}}, μ = 0.150

Reading this result: At μ=0.150 the masses are too comparable: even L4 and L5 turn unstable, so every Lagrange point here needs active station-keeping to hold a spacecraft in place.

Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.

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Frequently asked questions

Is this good for students?: Yes — this version of "Orbital Mechanics Lab for a roller-coaster loop" is framed for students learning it for a class or exam. See the concept move instead of memorizing formulas — and check your homework intuition.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.