GRAVITY & ORBITS
Orbital Motion Laboratory Launch a satellite. Find the balance between falling and flying away.
Set up the experiment
Planet Earth Mars Moon Jupiter Custom planet
Set circular speed Set escape speed
Direction: 0° is radially outward, 90° is tangential counterclockwise, −90° clockwise, ±180° inward. Drag the launch arrow to aim.
Physical edits reset motion, predictions, trails and measurements.
Follow the invariants Simulated time (min) →
The orbit at this instant
TRY A CHALLENGE
One safe revolution. Launch into a bound orbit that avoids the planet and completes one full revolution.
Earth and launch altitude (1,000 km) are fixed. Find a speed and direction with negative orbital energy and periapsis at least 300 km above the surface.
Start challenge → Leave challenge Circular and elliptical solutions are possible, in either direction.
Why?
Why is an orbit continuous free fall? Gravity continually bends the satellite’s path toward the center. With enough sideways velocity, the surface curves away as the satellite falls. Nothing holds it up.
Why does gravity weaken with distance? F = GMm/r² uses distance from the center, r = R + h. Doubling r gives one quarter of the force and acceleration. Doubling altitude alone does not usually double r.
Why must circular velocity be tangential? A circle requires v = √(μ/r) and velocity perpendicular to the radius. A radial component changes distance from the center. The right speed with the wrong direction can produce an ellipse or an impact.
Why faster near periapsis? As the satellite falls inward, gravitational potential energy becomes kinetic energy. It is fastest at periapsis and slowest at apoapsis. On a safe ellipse these exchanges repeat.
Why are energy and angular momentum constant? Gravity is conservative, so E = Eₖ + Eₚ stays constant, with Eₚ = 0 at infinity. Gravity acts radially: τz = (r × F)z = 0, so Lz = m(xvᵧ − yvₓ) stays constant. Positive Lz means counterclockwise; negative means clockwise.
Why does satellite mass not change its orbit? F grows with m, but a = F/m = −μr/r³ does not depend on m. Identical launch states therefore follow identical paths. Force, total energy and angular momentum all scale with satellite mass.
What separates bound motion from escape? Specific energy ε = v²/2 − μ/r is negative for bound motion. Zero is the parabolic threshold, vescape = √(2μ/r); positive energy gives hyperbolic escape when the future path avoids the surface. Leaving the screen does not establish escape.
Can an escaping trajectory still hit the planet? Yes. Escape speed is an energy threshold, not protection from obstacles. An inward launch above escape speed can have a periapsis inside the planet and collide before it ever turns outward.
What do equal swept areas show? The radius line sweeps equal areas in equal simulated times: dA/dt = |Lz|/(2m). The overlay uses triangles along the actual numerical path. The last six completed time windows are shaded; a narrow fast-moving arc and a broad slow-moving arc can sweep the same area.