Escape velocity calculator for planets, stars and orbits

Escape velocity is the speed an object needs, starting from a distance r from the centre of a planet or star, to coast away and never fall back, ignoring air resistance: v = √(2GM/r). From the Earth's surface it is about 11.2 km/s. This escape velocity calculator takes the mass of the body and the distance from its centre and gives the escape velocity with each step of the working, in m/s and in km/s, km/h, mph and multiples of c, the speed of light.

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Escape Velocity Calculator

Calculate escape velocity from a body

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How to use the escape velocity calculator

1. Open the Escape Velocity Calculator. It opens on the Earth: 5.972 × 10²⁴ kg and 6,371,000 m.
2. Type the Mass of the planet, moon or star as a number and a power of ten, such as 6.42 and 23 for Mars. Mass can be in kg, g, tonnes, lb or solar masses.
3. Type the Radius: the distance from the centre you are escaping from. For the surface, that is the body's radius; from an orbit, add the height. Pick m, km, miles or AU.
4. Read the escape velocity in m/s, with Also equal to listing km/s, km/h, mph and the fraction of the speed of light, c. The working shows 2GM, then 2GM ÷ r, then the square root.
5. The graph shows escape velocity falling as the distance grows, with your distance marked. For practice, press Random problem.

The escape velocity formula

It comes from energy: the object's kinetic energy, ½mv², must be enough to climb out of the gravitational potential well, GMm/r. The object's own mass m cancels, so a pebble and a rocket need the same speed. The derivation, with more examples, is in the escape velocity guide.

Worked examples you can check in the calculator

Example 1: Mars

Mars has a mass of 6.42 × 10²³ kg and a radius of 3,396 km. Type 6.42 and 23, then 3396 with the unit set to km.

About 5.0 km/s, less than half the Earth's, which matches NASA's value for Mars. A rocket leaving Mars needs much less fuel than one leaving the Earth.

Example 2: Jupiter

Jupiter: 1.898 × 10²⁷ kg and a radius of 71,492 km. The calculator gives 59,530 m/s, or 59.5 km/s, more than five times the Earth's.

Example 3: escaping from orbit

A spacecraft in orbit 420 km above the Earth is 6,371 + 420 = 6,791 km from the centre. Type 6791 km with the Earth's mass: the escape velocity there is 10.83 km/s. Moving in a circular orbit at that height, the craft already has v = √(GM/r) = 7.66 km/s, so it needs about 3.2 km/s more, not the full amount. Escape velocity is always √2, about 1.41, times the circular orbit speed at the same distance.

Example 4: how small would the Earth have to be to trap light?

Set v = c, the speed of light, and rearrange: r = 2GM ÷ c². The calculator only finds v, so do this step by hand: r = 2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ ÷ (2.998 × 10⁸)² = 0.00887 m. Now check it: type 0.00887 m as the radius, and the calculator gives 299,789,177 m/s, which Also equal to shows as 1.0000 c. Squeeze the whole Earth into a ball with a radius under 9 mm and not even light could leave it: that radius is the Earth's Schwarzschild radius.

How escape velocity depends on mass and radius

v is proportional to √(M/r). Double the mass and the escape velocity goes up by √2. Halve the radius, squeezing the same mass smaller, and it also goes up by √2. That is how a collapsing star can end up with an escape velocity above the speed of light. Try a question on it, then press Mark to check your answer in the Black Holes course.

Here x is the distance from the Earth's centre in thousands of kilometres and y is the escape velocity in km/s. Four times as far out, the escape velocity is half as big.

What the calculator doesn't do

• Solve for the mass or the radius. Rearrange by hand: M = v²r ÷ 2G, or r = 2GM ÷ v², as in Example 4.
• Air resistance. It gives the ideal speed with no atmosphere. Real rockets don't reach escape velocity at launch; they keep firing their engines as they climb.
• The Earth's spin. A launch eastwards near the equator gets a head start from the Earth's rotation, which it doesn't include.
• Escaping the Sun too. Leaving the Earth isn't the same as leaving the solar system; that needs the Sun's escape velocity at the Earth's distance.

Common mistakes

• Using the height above the surface as r. r is from the centre: add the planet's radius.
• Kilometres in the formula. In SI, r is in metres. Set the unit in the calculator, or convert.
• Forgetting the 2. √(GM/r) is the orbit speed; escape needs √(2GM/r).
• Thinking a heavier rocket needs a higher speed. The rocket's mass cancels. It needs more energy, not more speed.
• Forgetting the square root when you scale: 4 times the mass gives 2 times the speed.

Read next

Physics · Fields and gravityEscape velocity: formula, derivation and examplesThe escape velocity formula v = √(2GM/r), how to derive it, Earth's 11.2 km/s worked out, values for the planets, black holes and practice questions.