How can falling keep a spacecraft above the ground?

Start above the ground

Our generic spacecraft is placed above a spherical Earth with an assigned velocity. This is the start of a thought experiment, not a simulation of a rocket getting there. Its size is enlarged so you can follow it.

Choose how it moves

The velocity arrow has both length and direction. Start with the local tangent and compare slightly different speeds. Then keep the speed and tilt the arrow. The next position depends on both choices.

Gravity keeps changing the velocity

In this model, acceleration points toward Earth’s center. With no force from the selected instant, the same velocity would carry the craft along a straight line. With gravity, its direction changes. In a circle its speed stays constant even though its velocity changes.

Missing the ground takes a special balance

A circle needs circular speed and a perpendicular velocity at the chosen radius. A lower tangential speed gives a different ellipse. Some ellipses clear the sphere; others meet it. A spacecraft cannot finish the attractive drawn loop if the ground is in the way.

Look at the terminal event

A trial stops at its first calculated sphere contact. Exact tangency counts as contact. In a comparison, the other trial continues on the shared clock. We do not animate the contacting craft through Earth or invent a bounce.

Ask what unbound means

At the escape boundary, specific orbital energy is zero; above it, energy is positive. Gravity still acts. An inward-directed unbound path can reach the sphere before it has any chance to leave.

Watch the ground turn

A separate view rotates the displayed surface and maps the point below the spacecraft. The projection moves across longitude and latitude. Rotating the display frame does not supply a new force to the spacecraft.

Test an explanation two ways

First compare mathematics with exact cases and numerical refinement. Then ask which real forces and observations the model omits. A stable numerical answer can still be an incomplete account of nature.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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°.

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.

Satellites and mission reasoning

Position and velocity together define a starting state. Real missions add atmospheric, nonspherical and many-body effects to the underlying gravitational ideas.

Understanding apparent weightlessness

Shared free fall separates gravitational acceleration from supporting contact. A changing direction can be evidence of acceleration even when speed stays constant.

Checking any numerical model

Use exact special cases, conserved quantities, event checks and refinement. Then ask whether the physical assumptions match the question and available evidence.

Why can occupants float in an orbiting spacecraft?

• Gravity has disappeared
• They and the craft approximately fall together
• Their mass becomes zero

If no force acted from this instant, what would the same velocity do in fixed axes?

• Stop immediately
• Continue along a straight tangent at constant velocity
• Point outward along the radius

At400 km, circular speed tilted outward by 10° produces which result?

• A circle
• An ellipse that contacts the sphere
• An escape trajectory

At400 km with tangent launches, which clears the sphere: k=.98 or k=.99?

• Both
• Only k=.99
• Neither

What does the exact tangential √2-speed preset mean?

• Gravity becomes zero
• Speed becomes zero immediately
• A zero-specific-energy unbound conic

At k=1.5, the ±45° trials have the same energy and periapsis. Which has future surface contact?

• Both must have the same future
• Only the inward-directed trial
• Neither because both are unbound

Equal time intervals on a Kepler ellipse sweep equal what about the Earth focus?

• Angles
• Areas
• Path distances

Almost constant numerical energy proves which statement?

• All computed positions are exact
• The real satellite prediction is validated
• One consistency check passed

Orbiting as continuing free fall

NASA Basics of Space Flight, How Orbits Work and Freefall. The lesson uses a placed spacecraft and explicitly omits atmospheric effects and rocket ascent.

Motion, force and gravity foundations

NASA gravity/mechanics introduction. The force-free tangent is an explicitly labeled counterfactual from the selected state.

Shared falling explains floating

NASA Glenn account of microgravity; the 8.694 m/s² value is calculated from this lesson’s declared model.

Two-body equations and orbital invariants

Richard Battin, MIT 16.346 Lecture 1, pp. 2–5, relative motion, angular momentum, eccentricity, conics and period. No MIT diagrams are reproduced.

Energy and velocity relations

MIT Lecture 2, pp. 2 and 5; per-unit-mass energy and velocity identities. Numeric lesson results are independent calculations.

Time and orbital anomaly

MIT Lecture 3, pp. 2–4, anomaly/time relations used to independently check terminal events and noncircular states.

Universal-variable propagation foundations

Wisdom and Hernandez 2015, sections 2–4. The local f/g and Stumpff implementation is separately authored and bounded by its own checks.

Parameter provenance and units

NAIF DE440 GM kernel, BODY399 in km³/s², converted to SI. Original header and bytes preserved; this is not an ephemeris.

Mean radius basis

JPL lists Earth mean radius 6371.0084 km; the lesson deliberately uses a rounded 6371 km spherical collision boundary.

Sidereal rotation period

JPL astrodynamic parameters provide the mean sidereal day. Precise Earth-orientation terms are not modeled.

Original Earth composite and its credits

NASA Blue Marble collection, 2001 MODIS land/coastal composites and USGS topographic shading. The selected 2048 derivative has about 19.5 km equatorial pixels; the old variant item-page redirects while the NASA asset still resolves.

Educational image reuse permission

NASA factual educational simulation/web-page guidance with source attribution and third-party exceptions. No invented CC0 license or NASA endorsement is claimed.

Unchanged kernel redistribution

NAIF rules permit kernel use and unmodified redistribution. The source header and content remain intact.

Ephemerides have an observational basis

Park et al. 2021 describe DE440/DE441 dynamics and observations including lunar laser ranging. No Horizons response is republished or treated as a raw spacecraft measurement here.