Escape Velocity Calculator

Definition of escape velocity

Escape velocity is the smallest speed a body must have at a given point of a gravitational field so that — moving freely from then on — it never returns and recedes to infinity with a speed approaching zero. In the language of energy: the body’s total mechanical energy, the sum of kinetic ½mv² and potential −GMm/r, equals zero. A body with negative energy stays bound and moves on an ellipse, a body with positive energy departs on a hyperbola, and exactly at v₂ its path is a parabola.

Three properties follow from the definition. First, v₂ does not depend on the mass of the escaping body, because m cancels in the energy equation. Second, it does not depend on the direction of launch — only the magnitude of the speed matters, as long as the path does not intersect the surface of the central body. Third, it decreases with distance as 1/√r: escape from geostationary orbit requires 4.35 km/s instead of 11.19 km/s from the surface.

Historical note: from Newton’s cannonball to Luna 1

The thought experiment of a cannonball fired from a high mountain comes from Newton’s De mundi systemate (published 1728): with ever greater speed the projectile lands ever farther, then circles the Earth, and finally flies away forever. In 1783 John Michell, and in 1796 Pierre-Simon Laplace, noted that for a sufficiently massive and compact body the escape velocity would exceed the speed of light — the first hint of the black hole concept; the radius at which v₂ = c is today called the Schwarzschild radius r = 2GM/c².

The names “first, second, and third cosmic velocity” took hold in Russian and Polish literature through the work of Konstantin Tsiolkovsky (1903) and astronautics textbooks; English-language texts speak of escape velocity. The first human-made object to reach escape velocity was the Soviet probe Luna 1 in January 1959; it passed the Moon and entered orbit around the Sun.

Deriving the formula from energy conservation

Step 1. The potential energy of a body of mass m at distance r from the center of mass M in a Newtonian field is E_p = −GMm/r, with the conventional zero at infinity. Step 2. Write energy conservation between the launch point (speed v, distance r) and infinity, where both the potential energy and — in the limiting case — the kinetic energy are zero: ½mv² − GMm/r = 0.

Step 3. Cancel m and solve for v: v² = 2GM/r, hence v₂ = √(2GM/r). Step 4. For Earth, with G = 6.67430·10⁻¹¹ N·m²/kg², M = 5.972·10²⁴ kg, and r = 6.371·10⁶ m, we get GM = 3.986·10¹⁴ m³/s², 2GM/r = 1.251·10⁸ m²/s², and v₂ = 11,186 m/s. The same formula can be written through the surface acceleration g = GM/r² as v₂ = √(2gr): for Earth √(2·9.82·6.371·10⁶) gives the same value.

Three cosmic velocities and the relation v₂ = v₁·√2

The first cosmic velocity v₁ = √(GM/r) is the speed of a circular orbit just above the surface; it follows from equating gravity and the centripetal force, GMm/r² = mv²/r. Comparison with v₂ = √(2GM/r) gives the exact relation v₂ = v₁·√2 ≈ 1.414·v₁: escape needs 41 % more speed than a circular orbit, but twice the kinetic energy. For Earth v₁ = 7.91 km/s and v₂ = 11.19 km/s.

The third cosmic velocity, about 16.7 km/s relative to Earth, allows a departure from the Solar System starting in Earth orbit; it is not a simple sum but results from combining escape from Earth’s field with the surplus needed to escape the Sun while using Earth’s orbital speed (29.8 km/s). This calculator computes only v₂ for a single central body; the escape velocity from the Sun at 1 AU (42.1 km/s) appears as an example, without accounting for Earth’s motion.

Model assumptions and what is left out

The formula assumes the central body is a point mass or has a spherically symmetric mass distribution; by Newton’s shell theorem the field outside such a body is identical to that of a point of the same mass. It also assumes the escaping mass is negligible compared with M (for two comparable masses the total mass M + m would be used) and that nothing acts besides the single central body.

The model leaves out the atmosphere, the rotation of the body, and the way the speed is built up. A real rocket loses about 1.5–2 km/s to drag and gravity during ascent, but gains up to 0.465 km/s from Earth’s rotation when launched eastward from the equator. It also does not need v₂ immediately: it is enough to have that speed at engine cutoff, already outside the atmosphere at a distance r somewhat larger than the planet’s radius, where the required value is slightly lower.

The constant G, the parameter GM, and Solar System values

The gravitational constant G = 6.67430(15)·10⁻¹¹ N·m²/kg² (CODATA 2018) is the least precisely known fundamental constant: its relative uncertainty is 2.2·10⁻⁵. Paradoxically, the product GM for planets is known far more precisely from satellite orbits — for Earth GM = 3.986004418·10¹⁴ m³/s² to ten significant figures. That is why astrodynamics works directly with the parameter GM (denoted μ), and the calculator displays it as a separate value.

Escape velocities from the surface according to NASA data: Mercury 4.25 km/s, Venus 10.36 km/s, Earth 11.19 km/s, Moon 2.38 km/s, Mars 5.03 km/s, Jupiter 59.5 km/s (for the equatorial radius of 71,492 km; the example on this page uses the mean radius of 69,911 km and gives 60.2 km/s), Saturn 35.5 km/s, Sun 617.7 km/s. For small bodies the values are surprisingly low: from Ceres about 0.5 km/s, and from Phobos only about 11 m/s — a thrown ball would leave that moon for good.

Sources

The definition, derivation, and numerical values come from mechanics textbooks and metrology sources: I. Newton, Philosophiae Naturalis Principia Mathematica, 1687, and De mundi systemate, 1728; D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapter on gravitation; H. D. Young, R. A. Freedman, University Physics, chapter “Gravitation”; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapter 7 (The Theory of Gravitation); H. D. Curtis, Orbital Mechanics for Engineering Students, Butterworth-Heinemann; CODATA 2018 (NIST), value of G; NASA Planetary Fact Sheets (planetary parameters and escape velocities); J. Michell, Philosophical Transactions 74 (1784). This page is for educational purposes.