Escape Velocity

The formula

To escape gravity from a body of mass M at distance r:

v_escape = √(2 · G · M / r)

where G = 6.674 × 10⁻¹¹ N·m²/kg² is the gravitational constant.

This is the minimum speed needed; any higher and the object also escapes (with leftover speed at infinity). Lower, and it doesn't escape — falls back or settles into an elliptical orbit.

Derivation

Energy conservation. At launch (radius r), assume velocity v:

E_initial = KE + PE = ½·m·v² + (-GMm/r)

At infinity (the boundary of escape), the object should have at least zero kinetic energy and zero PE (PE → 0 as r → ∞):

E_final ≥ 0

For the borderline case (just barely escaping):

½·m·v² + (-GMm/r) = 0
v² = 2GM/r
v = √(2GM/r)

Escape velocities of common bodies

Body	v_escape	From surface or relative to body
Earth (surface)	11.2 km/s	From R_E ≈ 6,371 km
Moon (surface)	2.4 km/s	From R_m ≈ 1,737 km
Mars (surface)	5.0 km/s	From R_Mars ≈ 3,390 km
Mercury (surface)	4.3 km/s	—
Jupiter (cloud tops)	59.5 km/s	—
Saturn (cloud tops)	35.5 km/s	—
Sun (surface)	617 km/s	From solar radius
Solar system (from Earth's orbit)	42.1 km/s	To leave the Sun's gravity
Milky Way (from Sun's location)	~550 km/s	To escape the galaxy
Black hole event horizon	c (≈ 3 × 10⁸ m/s)	By definition of horizon

Escape velocity vs orbital velocity

For a circular orbit at radius r — v_orbit = √(GM/r). For escape — v_escape = √(2GM/r). So:

v_escape = √2 · v_orbit ≈ 1.414 · v_orbit

For Earth at surface: v_orbit = 7.9 km/s (would skim Earth's surface), v_escape = 11.2 km/s.

From low Earth orbit (~400 km up): v_orbit = 7.7 km/s, v_escape = 10.9 km/s.

Speed	Trajectory
v < v_orbit	Falls back (parabolic to surface, or sub-orbital)
v = v_orbit	Circular orbit
v_orbit < v < v_escape	Elliptical orbit
v = v_escape	Parabolic trajectory (just barely escapes)
v > v_escape	Hyperbolic trajectory (escape with leftover energy)

Black holes — when v_escape = c

For a body of mass M, the radius at which escape velocity equals the speed of light:

c² = 2GM/r_s
r_s = 2GM/c²

This is the Schwarzschild radius. Any object compressed within this radius forms a black hole — no escape, even for light.

Object	Mass	Schwarzschild radius
Earth	5.97 × 10²⁴ kg	8.9 mm (less than a marble!)
Sun	1.99 × 10³⁰ kg	2.95 km
Stellar black hole (10 M_sun)	1.99 × 10³¹ kg	30 km
Supermassive (Sagittarius A*)	4.3 × 10⁶ M_sun	~13 million km (~0.09 AU)

JavaScript — escape velocity calculations

const G = 6.674e-11;

function escapeVelocity(M, r) {
  return Math.sqrt(2 * G * M / r);
}

function orbitalVelocity(M, r) {
  return Math.sqrt(G * M / r);
}

// Earth surface
const M_EARTH = 5.972e24;
const R_EARTH = 6.371e6;
console.log(`Earth escape: ${(escapeVelocity(M_EARTH, R_EARTH)/1000).toFixed(2)} km/s`); // 11.18
console.log(`Earth orbit: ${(orbitalVelocity(M_EARTH, R_EARTH)/1000).toFixed(2)} km/s`); // 7.91
console.log(`Ratio: ${escapeVelocity(M_EARTH, R_EARTH) / orbitalVelocity(M_EARTH, R_EARTH)}`); // √2

// Moon
const M_MOON = 7.342e22;
const R_MOON = 1.737e6;
console.log(`Moon escape: ${(escapeVelocity(M_MOON, R_MOON)/1000).toFixed(2)} km/s`); // 2.38

// From low Earth orbit altitude
console.log(`Escape from LEO 400km: ${(escapeVelocity(M_EARTH, R_EARTH + 400e3)/1000).toFixed(2)} km/s`);

// Schwarzschild radius
const c = 3e8;
function schwarzschildRadius(M) {
  return 2 * G * M / (c * c);
}

const M_SUN = 1.989e30;
console.log(`Sun's r_s: ${(schwarzschildRadius(M_SUN)).toFixed(0)} m`);  // ~2950 m
console.log(`Earth's r_s: ${(schwarzschildRadius(M_EARTH)*1000).toFixed(2)} mm`);  // ~8.9 mm

// Tsiolkovsky rocket equation: how much fuel to escape Earth?
function rocketDeltaV(massInitial, massFinal, vExhaust) {
  return vExhaust * Math.log(massInitial / massFinal);
}

// Required Δv = ~11.2 km/s; chemical rocket v_e ~ 4500 m/s
const vEsc = escapeVelocity(M_EARTH, R_EARTH);
const massRatio = Math.exp(vEsc / 4500);
console.log(`Mass ratio for chemical rocket to escape Earth: ${massRatio.toFixed(2)}`); // ~12.3
console.log(`That means ${((1 - 1/massRatio) * 100).toFixed(1)}% must be fuel`);  // ~92%

Where escape velocity matters

• Spaceflight. Every interplanetary or interstellar mission must overcome local escape velocity. Apollo, Voyager, New Horizons all achieved Earth escape (~11 km/s) plus more.
• Astrophysics. Atmospheric retention — Earth retains heavy gases (N₂, O₂) at thermal speeds well below escape; loses light gases (H, He) over time. Mars lost most of its atmosphere because gravity is weaker.
• Black hole physics. Schwarzschild radius defines event horizons. Beyond it, no escape.
• Cosmology. Universe expansion vs gravitational binding — galaxies near each other are gravitationally bound; distant ones receding faster than escape velocity due to dark energy.
• Planetary defense. Asteroid deflection — if you give an asteroid enough Δv to exceed Earth's gravitational pull, it doesn't return.
• Engineering rockets. Multi-stage designs minimize fuel mass; orbital velocity (~7.7 km/s) is "halfway" to escape (~11.2 km/s) in terms of energy.
• Educational physics. Beautiful application of energy conservation — derives a complex result from a simple energy budget.

Common mistakes

• Confusing escape velocity with orbital velocity. Orbital v is to circle; escape v is to leave forever. Escape is √2 × orbital.
• Thinking escape velocity is "the speed needed to fly forever." It's the minimum starting speed to escape WITHOUT further propulsion. With continuous thrust, you can escape at any speed (just takes longer).
• Treating it as a "max speed." Escape velocity is a launch speed. Once moving, the object slows as it climbs (gravity pulls it back). At infinity, ideally zero kinetic energy left.
• Ignoring height of launch. v_escape decreases with altitude (smaller GM/r). Launching from a high mountain or LEO requires less escape velocity than from sea level.
• Forgetting the radius is from CENTER, not surface. v_escape = √(2GM/r) where r is distance from central body's center, not "altitude above surface." For Earth surface, r = R_Earth ≈ 6,371 km.
• Using Newton's formula for high gravity. Near black holes, general relativity changes things. Newton's escape velocity formula is good for low-gravity regimes but breaks down at extreme gravity.