The speed that just frees a body from a gravity well — and the Newtonian shadow of the black hole. Set the kinetic energy equal to the depth of the well and the answer falls out: v = √(2GM/r). It does not depend on what is escaping, it is √2 times the speed of a circular orbit, and — pushed to v = c — it lands on r = 2GM/c², exactly the Schwarzschild radius. Down the center, data flows: mass and radius go in, the engine computes, the escape speed comes out. The blue team builds and defends it; the red team tries to break it.
source Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1687), Motte tr. — archive.org/details/newtonspmathema00newtrich. The escape-speed formula is derived from Newton's gravitation and energy conservation, not stated verbatim there — AMBER. Rendered, not quoted.
The derivation is one equation. To just barely escape, a body of mass m needs enough kinetic energy to climb out of the potential well to infinity, where both terms vanish:
The escaping mass m cancels from both sides. Solve for v:
The circular-orbit speed comes from balancing gravity against centripetal need, v²/r = GM/r², giving vorb = √(GM/r). The two differ by exactly the factor √2 — the 2 in the escape formula is the whole difference between orbiting and leaving.
Set vesc = c in Newton's own formula and solve for the radius: c = √(2GM/r) gives r = 2GM/c². That is the-schwarzschild-radius — derived in 1916 from general relativity, yet already sitting inside a 1687 equation.
Newton could not know the light would truly be trapped — that needs the metric — but the coincidence is exact. Escape velocity is the classical foreshadow of the-event-horizon. Each sphere is the next one's premise.
The blue team's live check: recompute the invariants — the energy balance, the √2 ratio, and the r = 2GM/c² tie — and confirm them against the known truth. If red tampers with the engine, this badge is where it shows.
Escape depends on exactly two numbers about the body you are leaving — never on the thing that leaves. Feed the engine a mass M and a radius r:
| body | M (kg) | r (m) |
|---|---|---|
| Earth | 5.972×10²⁴ | 6.371×10⁶ |
| Moon | 7.342×10²² | 1.737×10⁶ |
| Jupiter | 1.898×10²⁷ | 6.991×10⁷ |
| Sun | 1.989×10³⁰ | 6.963×10⁸ |
Constants are exact-as-measured: G = 6.674×10⁻¹¹, c = 299 792 458 m/s. "Distributed" over the surface, the well's depth is GM/r — that is what you feed the panel below.
At exactly v_esc the trajectory is parabolic — it just reaches infinity with zero speed left.
Change the body or the slider — every number is computed live from v = √(2GM/r), never looked up.
What the machine produces, proven: Earth's surface escape velocity is 11.19 km/s, the Sun's 617.5 km/s; escape is always √2 ≈ 1.41421 times the orbital speed at the same radius; and forcing vesc = c reproduces r = 2GM/c² — the Sun's is 2.95 km. The current body's values are above; these invariants are the output.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
And the r = 2GM/c² coincidence is exactly that — a coincidence of factors. A real black hole horizon requires general relativity; Newton's math cannot say the light is trapped, cannot give the bending of light (twice the Newtonian value), and cannot explain Mercury's 43″/century. The formula foreshadows, it does not prove.
"You need to reach escape velocity to leave Earth." Cut. False for powered flight — any speed escapes given sustained thrust. v_esc is the ballistic, thrust-free minimum only.
"Escape velocity depends on the rocket's mass." Cut. The escaping mass cancels in ½mv² = GMm/r. A feather and a moon leave at the same v_esc.
"Newton predicted black holes." Kept, corrected. Michell (1783) and Laplace noted a "dark star" where v_esc = c — a real Newtonian result — but trapping light needs GR. The math coincides; the physics does not.
The red team's move: drop the 2 in the escape formula — use v = √(GM/r), which is the circular-orbit speed — so escape is confused with orbit. The blue team's witness (window 7) is watching.
Drop the 2 and v_esc collapses onto v_orbital: the √2 ratio becomes 1 and the v_esc = c radius halves to GM/c², breaking the r = 2GM/c² tie to the Schwarzschild radius. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.