Keep falling. Keep missing.: sources & model

Draw a velocity arrow, predict a path and compare two spacecraft on one clock. Follow a real NASA Earth map, inspect the next moment of free fall and discover why speed alone cannot make a circle.

What supports the explanation?

What this model assumes

1. Authored spacecraft initial conditions around an airless spherical Earth. No rocket ascent, atmospheric drag or heating, thrust, terrain, J2, other bodies or relativity.
2. The NASA composite is image context. Shaded terrain does not change the 6371 km collision sphere or supply current weather.
3. Specific energies are per unit mass with potential zero at infinity; marker sizes and vector lengths have explicit independent display scales.
4. The main state uses bounded universal-variable propagation. Numerical verification is separate from observational validation of real missions.
5. The 12,000-second window is a display limit. Future contact can lie beyond it; a large bound period is not shortened to fit the animation.
6. A contacting record freezes at its terminal position and time. No post-contact behavior, bounce or through-Earth trajectory is inferred.
7. Equal-area and numerical-method views use explicitly fixed reference trials. They do not silently grade arbitrary live settings against default answers.
8. The rotating map is geocentric and uses an authored epoch alignment. It omits precise Earth-orientation modeling and is not navigation guidance.
9. Paper vectors use dimensionless mathematical units. The activity has not been learner-trialed and does not require throwing or spinning any object.
10. Declared physical model: The core is Newtonian two-body motion outside a spherical Earth, with negligible spacecraft mass and no thrust after initialization. It omits atmosphere, terrain, nonspherical gravity, other bodies, radiation pressure and relativity. It is not an operational orbit or lifetime prediction.
11. Parameters and units: Earth μ=3.9860043550702266×10¹⁴ m³/s² comes from BODY399_GM in the DE440 parameter kernel. The sphere radius is 6371 km, an authored rounding of the published mean radius. Starting altitude is 400–2000 km. SI calculations are separate from metric or imperial displays. Published digits support reproducibility, not equivalent real-world accuracy.
12. Position, velocity and direction: With fixed physical axes and +z north, r₀=(R+altitude,0,0). v₀=k√(μ/r₀)(sinγ, cosγ cosi, cosγ sini). γ is measured from the local tangent, positive outward; inclination i rotates the plane. Changing inclination leaves radial dynamics unchanged in a spherical potential.
13. Continuous free fall: The acceleration is r″=−μr/|r|³. At the default400 km altitude its magnitude is 8.69425035 m/s², about 88.53% of this model’s surface value. Spacecraft and occupants approximately fall together; a lack of supporting contact is not a lack of gravity or mass.
14. Conserved quantities before contact: Specific angular momentum h=r×v is constant. Specific orbital energy ε=|v|²/2−μ/|r| is constant with potential zero at infinity. These are per-unit-mass quantities. The kinetic and potential terms can change while their sum stays fixed.
15. A circle is a particular initial condition: Circular speed is √(μ/r), but the velocity must also be tangent. At400 km, speed is 7.67259859 km/s and period is 5544.85514 s. Tilting that same speed outward by 10° keeps the initial energy unchanged but produces an ellipse that contacts the sphere at 3376.113634 s.
16. Shape versus future contact: For nonradial motion, evec=(v×h)/μ−r/|r|, p=|h|²/μ and periapsis radius is p/(1+e). Bound paths are circular or elliptic; zero-energy nonradial motion is parabolic; positive-energy motion is hyperbolic. The k=0 case is radial fall and needs separate handling. A buried mathematical periapsis does not mean an outbound unbound trajectory will return to it.
17. Compare neighboring tangential speeds: At400 km, k=.99 has a calculated periapsis about 135.772 km above the sphere. k=.98 instead contacts after 1821.009546 s. An atmosphere-free clearing result at such a low periapsis does not predict a real satellite’s lifetime.
18. The exact grazing and escape presets: Tangential grazing uses k=√[2R/(R+r₀)]≈.984664020052 at the default altitude and contacts after 2650.502322 s. The exact escape preset uses k=√2. These named values retain their identity separately from rounded slider values; a neighborhood of speeds is not declared exactly parabolic.
19. A finite view is not an infinite conclusion: The investigation displays at most 12,000 elapsed seconds. First future contact is classified from the initial conic and direction; a numerical contact time is reported only inside that horizon. A bound trial may be labeled as contacting later. Reaching the time cap or leaving the near-Earth window does not establish escape or a completed orbit.
20. Propagation and terminal events: The main calculation uses universal-variable f/g propagation with stable small-argument Stumpff series and bracketed time solving. Surface contact uses the first future periapsis and an inward-radius root in universal anomaly, with a separate radial-fall formula. A scale-limited numerical tangent tolerance handles floating roundoff, not a physical safety margin.
21. Equal time sweeps equal area: For a central force, swept area rate is |h|/2. The fixed k=1.1 reference ellipse starts at 400 km and has period 7896.771431 s. Each one-eighth-period window sweeps 2.82044452151×10¹³ m². Angles and distances differ. The shaded polygon follows sampled orbit points; the displayed area uses the analytic invariant.
22. The rotating surface frame: The surface rotates with constant sidereal period 86164.09054 s and authored initial alignment. rground=Rz(−θ)r; vground=Rz(−θ)[v−Ω×r]. Longitude is atan2(y,x), latitude is geocentric asin(z/r), and longitude is undefined at a pole. The map breaks at its seam. After one default circular revolution, ground longitude is approximately −23.166819°.
23. Numerical error is not missing physics: Velocity Verlet combines a half velocity kick, position drift and another half kick. After one default circle, a 1 s step has about 18.21 m position error despite very small energy error. Halving the step reduces that position error to about 4.55 m. At exact grazing a 1 s Verlet path misses the tangent radius by about 4.366 m. The main terminal-event calculation does not infer clearance from that drifting path.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Original offline rendering of the lesson’s spherical Earth and calculated path. NASA Blue Marble composite: NASA Goddard; Reto Stöckli; Robert Simmon; MODIS and USGS data. The unchanged original texture and exact provenance remain in the lesson. Geometry, spacecraft scale and lighting are teaching choices; terrain shading does not alter the collision sphere. No mission prediction or NASA endorsement implied.