Predict → set conditions → observe → record → change one variable. Loading an example clears records and creates a reproducible paused state.
01 · An orbit is continuous free fall
A fast horizontal projectile can keep falling without reaching the surface as the curved ground falls away beneath it.
a=\frac{GM}{r^2}
Try itRecord 6500 m/s at 500 km above Earth, then compare with 8000 m/s. Inspect whether each path intersects the surface.
Prediction. 6500 m/s intersects the ground; 8000 m/s gives an ellipse outside the surface.
Compare and explain. How does continuous free fall differ from gravity disappearing?
Model limit. The body is spherical and airless; drag can matter in real low orbits.
02 · Find circular speed
A circular orbit occurs when gravity supplies the required centripetal acceleration.
v_\mathrm{c}=\sqrt{\frac{GM}{r_0}}
Try itUse Near circular, then play. Inspect the altitude graph and record speeds 100 m/s below and above it.
Prediction. At 500 km above Earth, circular speed is about 7616 m/s; altitude variation is smallest near it.
Compare and explain. Why is velocity not constant even at constant circular speed?
Model limit. The preset rounds to 1 m/s; this alone can produce small altitude changes.
03 · Escape speed and zero energy
At the boundary for reaching infinity, specific energy is zero; launch speed is √2 times circular speed.
v_\mathrm{esc}=\sqrt{\frac{2GM}{r_0}},\qquad\varepsilon=\frac{v^2}{2}-\frac{GM}{r}
Try itRecord the Below escape and Above escape presets, comparing energy signs and classification.
Prediction. Escape speed at 500 km is about 10771 m/s: below it energy is negative, above it positive.
Compare and explain. Does leaving the screen prove that an object has escaped?
Model limit. Escape is classified by energy; reaching infinity cannot be drawn in finite simulation time.
04 · How altitude changes thresholds
Farther circular orbits need lower circular and escape speeds but have longer periods.
v_\mathrm{c}\propto r^{-1/2},\qquad T\propto r^{3/2}
Try itRecord Near circular at 500 km; change altitude to 1500 km, apply Near circular again, and record.
Prediction. The second orbit has lower circular speed and gravity. Changing only altitude while keeping speed does not produce the new circular orbit.
Compare and explain. What goes wrong if altitude h is used in place of center distance r?
Model limit. Center distance is body radius plus altitude; km must also be converted to m.
05 · Launch direction matters
Equal position and speed give equal energy, but angular momentum depends on direction, changing the orbit shape.
\ell=|\vec r\times\vec v|=r_0v_0\cos\alpha
Try itRecord an 8000 m/s tangent launch, then a 30° launch. Compare energy, paths and periapsis.
Prediction. Initial energies match; the second angular momentum is multiplied by cos30°. A negative-energy orbit may still intersect the surface.
Compare and explain. Can two orbits with the same semi-major axis have different eccentricities?
Model limit. A mathematical orbit inside the body is not continued; integration stops at the surface.
06 · Compare Earth and Moon
The Moon has both smaller mass and radius; mass alone does not determine surface gravity or orbital thresholds.
g_\mathrm{surface}=\frac{GM}{R^2}
Try itLoad Near circular for the Moon at 100 km and record. Switch to Earth and apply it again at the same altitude.
Prediction. Circular speed at 100 km above the Moon is about 1633 m/s, lower than Earth’s.
Compare and explain. Why are equal altitude and equal center distance different comparisons?
Model limit. Earth and Moon differ in scale; after changing the body, compare old conditions in the table rather than overlaying their paths.
07 · Two meanings of constant gravity
Here constant-magnitude central gravity always points inward. It differs from the fixed downward field of projectile motion.
\vec a=-g_0\hat r,\qquad U_\mathrm{specific}=g_0r
Try itRecord 12000 m/s with inverse-square gravity, then compare with constant-magnitude central gravity.
Prediction. The inverse-square orbit escapes; the constant-magnitude central model reports no finite escape speed.
Compare and explain. Why do constant-acceleration projectile formulas fail when the acceleration direction changes?
Model limit. Potential conventions differ; do not directly compare energy signs or values across these two models.
08 · Check the integration with invariants
Central forces conserve angular momentum; time-independent conservative forces conserve mechanical energy.
\varepsilon=\mathrm{const.},\qquad\ell=\mathrm{const.}
Try itFor an 8000 m/s orbit, record the start, middle and end; export CSV and compare relative changes in energy and angular momentum.
Prediction. Readouts should appear almost constant; the CSV retains more digits to reveal numerical error.
Compare and explain. If playback speed changed the orbit, what would you suspect about time stepping?
Model limit. Good invariant conservation does not validate omitted atmosphere or many-body effects.