The Verification Venue · pointed at a thing everyone gets wrong
The Cannonball That Never Lands
"Why doesn't the Moon fall to Earth?" It does. It's falling right now, and has been for four billion years — it just keeps missing. Here is the whole trick, the one Newton drew in 1687.
Drop a cannonball and it falls straight down. Fire it sideways and it falls in an arc, landing farther off. Fire it faster and the arc stretches — until, at one particular speed, the ground curves away beneath it exactly as fast as it falls. Now it's still falling, with nothing left to hit. That's an orbit. Drag the speed below and find it.
Launch speed
6.00 km/s
Gravity on the ball now
8.96 m/s²
Lands downrange
—
The cannon sits 300 km up — the altitude where real satellites orbit, above essentially all the air. (Newton drew his mountain reaching past the sky for exactly this reason: to give the ball room to fall.) Below about 7.7 km/s the ball's path is part of an ellipse whose far side dips below the ground, so it hits the planet — a cannon shot, or a sub-orbital rocket, landing farther downrange the faster you fire. At 7.73 km/s the ellipse closes into a circle and it falls forever, never landing: that's a low orbit, and 7.73 km/s is genuinely how fast the satellites up there move. (From the ground it would be the textbook 7.91 km/s — a few hundred kilometres barely changes how far you are from Earth's centre.) Push past that and the orbit bulges into a tall ellipse. At 10.93 km/s — escape speed, exactly √2 times the circular speed — the curve never closes at all, and the ball leaves Earth for good (11.19 km/s from the ground).
Notice the middle readout: gravity is still pulling on the ball the entire time, even far out. It never switches off — it only weakens with distance (as 1/r²). The ball doesn't "escape gravity." It just gets going fast enough that gravity can bend its path but never reel it back in. Which brings us to the two things almost every explanation online gets wrong.
The check — every number recomputed from G, M⊕, R⊕
Myth 1: "There's no gravity in space." Astronauts on the ISS aren't beyond gravity — they orbit just 420 km up, a twentieth of an Earth-radius. The inverse-square law puts the gravity there at:
They float for the same reason a cannonball does: they're in continuous free fall, the whole station dropping around the Earth together. "Weightless" means "falling freely," not "no gravity."
Myth 2: "The Moon is too far for gravity to matter / it just floats." Newton's own test (1666–1687): the Moon's pull-toward-Earth should be the surface value scaled down by the inverse square of its distance. The Moon sits 60.3 Earth-radii out, so gravity there should be 60.3² ≈ 3 640 times weaker:
| quantity | from period & distance | from G·M⊕/r² |
|---|---|---|
| Moon's acceleration toward Earth | 0.00272 m/s² | 0.00270 m/s² |
| vs. surface gravity (9.82) | ÷ 3 610 | ÷ 3 640 |
The two columns agree to under 2% — the left measured purely from the Moon's 27.32-day month and its distance, the right predicted from Earth's surface gravity and the inverse-square law. That agreement is how Newton knew the force holding the Moon is the same force that drops an apple.
And "falling forever" is literal. Each second, the Moon:
The Moon drops 1.36 mm and the Earth curves out from under it by the same 1.36 mm. It falls and falls and the ground is never there. That equality is the orbit. Run it all yourself: node research/moon-cannonball/verify.mjs (30/30 checks pass).
What's exact here, what's idealised, and the "centrifugal force" answer
Exactly true. The simulator integrates the real two-body law F = G·M·m/r² with a symplectic velocity-Verlet step, so the path you see is the genuine Newtonian trajectory — a conic section (ellipse, parabola or hyperbola). The circular speed √(GM/R), the escape speed √(2GM/R) = √2 × circular, the apogee and perigee, and the period are the exact closed-form values; the verifier confirms the integrator returns a launch-at-circular-speed ball to its start after one period with its radius constant to 0.5% and energy conserved to 0.01%.
Idealised — exactly as Newton idealised it. No air drag (a real surface cannonball would burn up and decelerate); a perfectly spherical, non-rotating, uniform Earth of mean radius 6 371 km (real surface gravity varies 9.78–9.83 with latitude and altitude, and Earth's spin gives a real launch a ~0.46 km/s head start at the equator); and only Earth's gravity (no Sun or Moon). These change the precise numbers, never the picture.
The 9.82 vs 9.807 gap is real, not a bug. GM/R² at the mean radius gives 9.82 m/s²; the defined standard gravity is 9.807. The 0.13% difference is Earth's rotation (which subtracts apparent weight) and its equatorial bulge — both averaged away in this uniform-sphere model.
"Isn't it centrifugal force balancing gravity?" That's the half-right answer you'll see most often. In the rotating frame that turns with the orbit, yes — a fictitious centrifugal force balances gravity and the satellite sits still. But that's a bookkeeping choice, not a cause: in the ordinary (inertial) frame there is no outward force at all, only gravity, and it does not balance — it's a net inward (centripetal) pull, the unbalanced force that constantly bends the straight-line path into a curve. If the forces truly balanced, the Moon would coast off in a straight line. The honest one-sentence answer is the one this page shows: it's falling, and moving sideways fast enough to keep missing.