Escape Velocity Calculator

Compute escape velocity from any celestial body with the formula v = √(2GM/r). Presets for Earth, Moon, Mars, Sun, Jupiter. Four modes with full steps.

Calculation parameters
Scegli quale grandezza vuoi calcolare.
Usato solo in modalità preset.
Massa totale del corpo attraente. Per la Terra ≈ 5,972·10²⁴ kg.
Distanza dal centro di massa al punto di lancio. Per la superficie terrestre ≈ 6,371·10⁶ m.
Velocità di fuga desiderata (usata nelle modalità inverse).
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What escape velocity is

The escape velocity is the minimum speed needed by a projectile to leave the gravitational influence of a massive body, without further propulsion and ignoring drag. It is derived by requiring the total mechanical energy at launch — kinetic plus gravitational potential — to be zero, so the body just reaches infinity with vanishing residual speed. Setting ½·m·v² − G·M·m/r = 0 gives the closed-form result v = √(2·G·M/r), which is independent of the mass m of the escaping body.

The formula and constants

The fundamental relation is:

vesc = √(2·G·M/r)

where:

  • G = 6.67430·10⁻¹¹ m³/(kg·s²) is the gravitational constant (CODATA 2018);
  • M is the total mass of the attracting body in kg;
  • r is the distance from the centre of mass to the launch point, in metres.

Escape speed is isotropic: the direction of launch does not matter as long as it is not aimed into the body. The expression equals √(2·|U|/m), meaning the kinetic energy exactly cancels the gravitational potential.

The four modes

  1. Given M and r → v: enter mass and radius, get escape velocity.
  2. Preset body → v: pick Earth, Moon, Mars, Venus, Mercury, Jupiter, Saturn, Sun or Ceres; the tool uses NASA fact-sheet values.
  3. Given v and r → M: inverse form to find the mass required for a given escape speed at distance r.
  4. Given v and M → r: finds the radius where the escape speed equals the target value.

Escape velocities in the Solar System

  • Earth: ≈ 11.186 km/s (≈ 40 270 km/h).
  • Moon: ≈ 2.376 km/s — low enough that small ascent stages could leave the Moon.
  • Mars: ≈ 5.027 km/s — less than half Earth's, a bonus for return missions.
  • Jupiter: ≈ 59.5 km/s — highest among planets; gravity assists are essential.
  • Sun: ≈ 617.7 km/s from the photospheric surface.

Escape velocity vs. first cosmic velocity

The first cosmic velocity, the speed of a circular orbit at the surface, is v₁ = √(G·M/r), exactly the escape velocity divided by √2. For Earth this is about 7.905 km/s. Escape velocity is therefore √2 ≈ 1.414 times the first cosmic velocity — a satellite in low orbit needs an additional 41.4 % speed boost to leave Earth's gravity well.

Connection to the Schwarzschild radius

Setting v = c in the classical formula yields rs = 2·G·M/c², which happens to coincide with the Schwarzschild radius from general relativity. For Earth it is 8.87 mm, for the Sun 2.95 km, for Sgr A* (M ≈ 4.15·10⁶ M☉) about 1.22·10¹⁰ m. The calculator reports rs for reference, though the Newtonian formula is not valid in strong-field regimes.

Model limitations

The formula ignores planetary rotation (rotational speed at the equator slightly reduces the required launch speed), atmospheric drag (which practically raises the needed speed), the presence of third bodies (Moon, Sun) distorting the potential, and relativistic effects (relevant only near compact objects). Real missions use Hohmann transfers and flybys rather than direct escape-speed launches.

Escape velocity in real space travel

In practice a rocket need not reach escape velocity all at once: as long as the engines keep firing, it can climb slowly and build up energy little by little. Escape velocity is the speed a projectile would need if launched only once, with no further propulsion. Real launchers exploit staging (dropping empty tanks to lighten the vehicle) and the gravity assist, which steals a small share of a planet's orbital energy to accelerate a probe without burning fuel. Interplanetary missions combine these effects to reach distant planets with launchers of limited power, turning a modest boost into a journey across the Solar System.

Extreme cases: black holes and planetary atmospheres

If a body is compressed enough, its escape velocity can exceed the speed of light: this is what happens in a black hole, where the Schwarzschild radius marks the horizon beyond which not even light can escape. At the opposite extreme, escape velocity explains why the Moon and Mercury retain no atmosphere: their low escape velocities cannot stop gas molecules, agitated by heat, from drifting off into space. Gas giants like Jupiter, with escape velocities of nearly 60 km/s, can hold on even to hydrogen, the lightest and most elusive gas.

Updated July 2026

Frequently Asked Questions

Why does escape velocity not depend on the escaping body's mass?

Because the formula v = √(2GM/r) contains only the mass M of the celestial body, not that of the departing object. The physical reason is that both kinetic and gravitational potential energy are proportional to the projectile's mass, which therefore cancels out of the equation. A stone and a spacecraft have the same escape velocity from Earth: about 11.2 km/s.

Does launch direction matter?

No, escape velocity is a scalar: only the speed matters, not the direction, provided the body does not fall back toward the planet. Without an atmosphere, launching horizontally or vertically requires the same speed to escape. In practice, however, direction matters for atmospheric drag and for exploiting the planet's rotation to advantage.

What is Earth's escape velocity?

The escape velocity from Earth's surface is about 11.2 km/s, that is over 40,000 km/h. It follows from v = √(2GM/r) with Earth's mass (5.97×10²⁴ kg) and radius (6371 km). It is the minimum speed at which an object, with no further thrust, would permanently escape the gravitational pull of our planet.

How does escape velocity relate to the Schwarzschild radius?

The Schwarzschild radius is the radius to which a body would have to be compressed for its escape velocity to equal the speed of light: r_s = 2GM/c². At that point not even light can escape and a black hole forms. For Earth it is about 9 millimetres, for the Sun about 3 kilometres: it is the same escape-velocity formula with v = c.

Which value of G is used?

The universal gravitational constant G = 6.674×10⁻¹¹ m³/(kg·s²) is used, determined experimentally since Cavendish's 1798 experiment. It is one of the fundamental constants of physics, measured today with great precision. Its tiny value reflects the intrinsic weakness of the gravitational force compared with the other fundamental interactions.

Does planetary rotation help launches?

Yes, launching eastward exploits Earth's rotation speed, which at the surface at the equator is about 0.46 km/s. This is why spaceports near the equator (such as Kourou) are favoured: the rotation provides a free boost that reduces the fuel needed. The help is greatest at the equator and zero at the poles, where the rotation speed is nil.

What is the escape velocity from the Solar System?

To escape the Sun's pull from Earth's orbit requires about 42 km/s relative to the Sun. Since Earth already moves at about 30 km/s around the Sun, exploiting this motion means only about 16.6 km/s relative to Earth is needed: this is the third cosmic velocity, with which probes like the Voyagers left the Solar System.

Why do some planets keep an atmosphere and others not?

It depends on the comparison between escape velocity and the speed of gas molecules, which rises with temperature and falls with molecular mass. If the typical molecular speed is close to the escape velocity, the gas drifts off into space over geological time. This is why the Moon and Mercury have no atmosphere, while gas giants retain even hydrogen.

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