Escape Velocity Calculator

What the escape velocity calculator does

Escape velocity is the speed something needs to break free of a body's gravity for good. This calculator finds it from the body's mass and radius, and it also reports the first cosmic velocity, the speed needed to orbit at the surface. Presets let you work in Earth and Sun units as well as ordinary ones.

Below is what escape velocity is, the equation behind it, why it does not depend on the escaping object, and a worked example.

How to use it

1. Enter the body's mass, in kilograms, Earth masses, Sun masses, or other units.
2. Enter its radius, the distance from the centre to the surface.
3. Press Calculate for the escape velocity and the first cosmic velocity, or Reset to clear them.

What escape velocity is

Escape velocity is the minimum speed an object must have to escape a body's gravitational pull completely, coasting away forever without any further push. Throw a ball upward and it falls back, because it was too slow to escape. Throw it faster and it goes higher before returning. Throw it at escape velocity, and it never comes back: gravity slows it forever but never quite stops it. For Earth, that speed is about 11.2 kilometres per second, roughly 40,000 kilometres per hour.

The idea is central to spaceflight and astronomy. A spacecraft leaving for another planet must reach escape velocity to break free of Earth, and the escape velocities of other worlds set how hard they are to leave. It also governs which gases a planet can keep: if a molecule's typical speed approaches the escape velocity, the planet slowly loses that gas to space, which is part of why small, low-gravity worlds have thin atmospheres or none at all. This calculator computes escape velocity for any body from its mass and size.

The equation it uses

Escape velocity comes from balancing kinetic energy against the energy needed to climb out of the gravitational well:

v_e = √(2 G M ÷ R)

Here v_e is the escape velocity, G is the gravitational constant, M is the mass of the body, and R is the distance from its centre, which for a surface launch is the body's radius. The escape velocity grows with the body's mass and shrinks with its radius, so a more massive or more compact body holds on more tightly. The calculator multiplies, divides, and takes the square root to give the result.

Why it does not depend on what is escaping

One of the surprising features of escape velocity is that it does not depend at all on the mass of the object trying to escape. A pebble, a person, and a spaceship all need exactly the same speed to break free from the same point in a gravitational field. The escaping object's mass simply cancels out of the physics, because gravity pulls more strongly on a heavier object but that same heaviness makes it harder to accelerate, and the two effects balance exactly.

What does change with mass is the energy required. Reaching escape velocity takes far more energy for a spaceship than for a pebble, which is why launching heavy payloads is so demanding even though the target speed is the same. So escape velocity is a property of the body being escaped and the distance from its centre, not of the traveller. The calculator therefore needs only the mass and radius of the body, never anything about what is leaving it.

Escape velocity and orbital velocity

The calculator also gives the first cosmic velocity, which is the speed needed to circle the body in a low orbit just above its surface, as opposed to escaping it entirely. These two speeds are simply related: the escape velocity is always the square root of two, about 1.41 times, larger than the orbital velocity at the same distance. So a body's escape velocity is roughly 41 percent higher than the speed needed to orbit it.

This neat relationship explains a common point of confusion. A satellite in low Earth orbit travels at about 7.9 kilometres per second, the first cosmic velocity, not the 11.2 kilometres per second of escape velocity. To merely orbit, you need the lower speed; to break free and leave entirely, you need the higher one. The gap between them, that factor of the square root of two, is the difference between circling a world and departing it, and the calculator reports both so you can see it directly.

From planets to black holes

Escape velocity varies enormously across the cosmos. The Moon, small and light, has an escape velocity of only about 2.4 kilometres per second, which is why its ascent stages need far less power than Earth's launches. Jupiter, massive and large, demands about 59.5 kilometres per second, and the Sun a staggering 617.5 kilometres per second. The deeper the gravitational well, the faster you must go to climb out.

Pushed to its extreme, this idea leads to one of the most remarkable objects in physics. Imagine a body so massive and compact that its escape velocity reaches the speed of light. Then not even light can escape it, and the object is a black hole. The distance at which the escape velocity equals light speed is the black hole's event horizon, and it follows directly from the same formula this calculator uses. So the humble escape velocity equation, applied to ordinary planets here, also marks the boundary of the darkest objects in the universe.

Units and precision

The calculator takes the mass in kilograms, Earth masses, Sun masses, or other units, and the radius in metres, kilometres, Earth radii, or Sun radii, which makes it easy to work at planetary and stellar scales. It returns the escape velocity and first cosmic velocity in your choice of speed units, including a fraction of the speed of light. The calculation is exact for a spherical body, and results carry several significant figures.

A worked example

Take the Earth, with a mass of about 5.97 × 10²⁴ kilograms and a radius of about 6,371 kilometres.

The escape velocity is v_e = √(2 G M ÷ R) ≈ 11.2 kilometres per second, the speed a rocket must reach to leave Earth for good. The first cosmic velocity, the speed to orbit just above the surface, comes out at about 7.9 kilometres per second, lower by the factor of the square root of two. A satellite in low orbit travels at the lower speed; a probe bound for another planet must reach the higher one.

Questions people ask

How do you calculate escape velocity?

Use v_e = √(2GM/R), where G is the gravitational constant, M is the body's mass, and R is the distance from its centre, usually its radius for a surface launch.

What is Earth's escape velocity?

About 11.2 kilometres per second, or roughly 40,000 kilometres per hour. This is the minimum speed needed to leave Earth's gravity entirely without further propulsion.

Does escape velocity depend on the object's mass?

No. A pebble and a spaceship need the same escape velocity from the same point. The object's mass cancels out; only the energy required, not the speed, depends on it.

Why is orbital velocity lower than escape velocity?

Because orbiting only requires circling the body, not leaving it. Orbital velocity is escape velocity divided by the square root of two, about 41 percent lower at the same distance.

References

A quick note on where the physics comes from. Escape velocity and its relationship to orbital velocity are standard gravitation, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The gravitational constant follows the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 13.4, Satellite Orbits and Energy. https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy
2. HyperPhysics, Escape Velocity. http://hyperphysics.phy-astr.gsu.edu/hbase/vesc.html
3. National Institute of Standards and Technology (NIST), Fundamental Physical Constants, Newtonian constant of gravitation. https://physics.nist.gov/cgi-bin/cuu/Value?bg