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Sandbox Physics

Gravity · central forces

Orbit Lab

Fire a world into a live, curved gravity well. Its luminous trail flares through periapsis while teal velocity, amber gravity, and the full predicted conic turn every slingshot into readable physics.

Interactive modelOrbit Lab
OrbitCircle
Radius rr180 px
Speed vv0.00
Energy ε\varepsilon+0.0
Angular momentum LL0
Eccentricity ee0.000

Physics tutorial

Orbits: read conic sections through energy

BackgroundInverse-square gravity always points toward the central body. Initial position and velocity turn that force into circles, ellipses, parabolas, or hyperbolas.

Why it mattersThe same reasoning predicts satellite maneuvers, planetary orbits, and escape without first solving the full trajectory.

Start with the essentials

Focus question
From position and velocity alone, how can we tell whether a body returns or escapes?
One-sentence intuition
Specific energy decides whether the orbit is bound; angular momentum sets its shape and periapsis.

Core mathematical model

Central gravity

r¨=−GMr3r\ddot{\mathbf r}=-\frac{GM}{r^3}\mathbf r

Acceleration always points inward and its magnitude falls as 1/r21/r^2.

Specific mechanical energy

ε=v22−GMr\varepsilon=\frac{v^2}{2}-\frac{GM}{r}

ε<0\varepsilon<0 is a bound ellipse, ε=0\varepsilon=0 is the parabolic boundary, and ε>0\varepsilon>0 escapes hyperbolically.

Common difficulties

Gravity does not guarantee return

Typical misconceptionA body must return because gravity never disappears.

Better mental modelGravity weakens with distance. Positive total energy allows escape even while speed decreases.

Circular motion still accelerates

Typical misconceptionConstant speed means zero acceleration.

Better mental modelVelocity direction changes continuously, requiring inward centripetal acceleration.

Run the experiment

  1. 01

    Identify a circular orbit

    Choose Circular and compare velocity with gravity.

    What to observe: The arrows are nearly perpendicular and radius stays stable.
  2. 02

    Change the energy

    Hold launch radius fixed and increase speed.

    What to observe: The ellipse lengthens toward the escape boundary.
  3. 03

    Cross the escape threshold

    Choose Escape and inspect the classification.

    What to observe: Specific energy becomes positive and the predicted curve opens.