Escape velocity explained: how fast you need to go to leave a planet

Escape velocity is the speed an object needs, at a given distance from a planet or star, to coast away forever without any further push and never fall back. It is v = √(2GM ÷ r), where G is the gravitational constant, M the body's mass and r the distance from its centre. From Earth's surface it is about 11.2 km/s, roughly 40,300 km/h or 25,000 mph. The escape velocity calculator works it out for any planet, moon or star; this guide explains where the formula comes from and what it does and does not mean.

Where the formula comes from

Throw a ball upwards and gravity slows it until it falls back. Throw it faster and it climbs higher before turning round. Because gravity weakens with distance, following the inverse-square law that the gravitational force calculator applies, there is a speed at which the ball never quite stops: its kinetic energy is just enough to pay the whole gravitational energy bill out to infinity. Kinetic energy is ½mv², and the energy needed to climb from distance r to infinity is GMm ÷ r. Setting them equal gives ½mv² = GMm ÷ r, and solving for v gives v = √(2GM ÷ r).

Two things fall out of the algebra. The escaping object's own mass m cancels, so a pebble and a spacecraft need the same speed. And the direction does not matter, as long as the path does not run into the planet: the energy needed is the same whichever way you go, though in practice an atmosphere and the planet's rotation change things.

Earth's escape velocity

With G = 6.674 × 10⁻¹¹ N·m²/kg², Earth's mass of 5.972 × 10²⁴ kg and a mean radius of 6,371 km, the formula gives 11.19 km/s, matching the 11.186 km/s in NASA's Earth fact sheet; Scientific notation explained, with e-notation and engineering notation helps if numbers like 10²⁴ are unfamiliar. The value falls with height because r grows. At 400 km up, roughly where the International Space Station orbits, it is about 10.85 km/s, and at geostationary altitude, 35,786 km up, about 4.35 km/s.

Escaping the Sun is a separate and harder problem. Starting from Earth's distance from the Sun, it takes about 42.1 km/s relative to the Sun, but a spacecraft launched from Earth already shares Earth's orbital speed of about 29.8 km/s, which provides most of that.

Other worlds

• The Moon: about 2.38 km/s.
• Mars: about 5.03 km/s.
• Jupiter: about 60.2 km/s at the cloud tops.
• The Sun: about 618 km/s at its visible surface.
• Ceres, the largest object in the asteroid belt: about 0.51 km/s.

Low escape velocities matter for atmospheres. Gas molecules are always moving, and over long periods a world with weak gravity loses its lighter gases to space; escape velocity is not the only factor in whether a world keeps an atmosphere, but it is a major one. Surface gravity, which decides how much you would weigh there, is a different quantity, GM ÷ r², and the weight on other planets calculator covers it.

Escape speed versus orbital speed

A circular orbit needs less speed than escape: v = √(GM ÷ r), which is exactly the escape velocity divided by √2. Just above Earth's surface, ignoring the air, that would be 7.9 km/s; at 400 km it is about 7.67 km/s, with one orbit taking about 92 minutes, as the orbital period calculator confirms. A spacecraft travelling between orbital speed and escape speed follows an ellipse; above escape speed its path becomes a hyperbola and it leaves for good. That is why missions to other planets are often placed in a low parking orbit first and then given the extra 3.2 km/s or so needed to reach escape speed from there.

Why rockets do not launch at 11.2 km/s

Escape velocity is the speed for an unpowered object, one given a single kick and then left to coast. A rocket keeps firing its engines as it climbs, so it never needs to be doing 11.2 km/s at ground level; it only needs enough energy by the time the engines stop, and by then it is high up, where escape velocity is lower. Launching at that speed through the lower atmosphere would be disastrous anyway, because drag and heating at such speeds would destroy most objects. Earth's rotation gives a small head start: a launch site on the equator is already moving east at about 0.46 km/s, which is one reason rockets usually launch eastwards.

At the other extreme, if a mass were squeezed so small that its escape velocity reached the speed of light, nothing could leave it. That radius, 2GM ÷ c², is the Schwarzschild radius of a black hole: about 9 mm for a mass equal to Earth's and about 3 km for the Sun's. Time dilation explained: why moving clocks run slow picks up what strong gravity does to clocks.