What this calculator does
Find out how fast something must be launched to break free of a planet's gravity for good. Throw a ball upward and gravity pulls it back. Throw it harder, and it climbs higher before falling.
Escape velocity is the launch speed that lets an object coast away forever. It slows down the whole time, but never quite stops and never comes back. (This ignores air resistance and engines.) From the Earth's surface, it's about 11.2 kilometers per second, roughly 40,000 km/h.
The answer depends only on two things: the mass of the body you're leaving, and how far you start from its center. The formula is v = √(2GM ÷ r). It doesn't depend on the mass of the object being launched, so a pebble and a spacecraft need the same speed.
Pick a built-in body, or enter your own mass and radius. You can also add a starting height. The calculator shows the escape speed, plus the speed for a circular orbit and the strength of gravity at that distance.
Where it's used
Why this math matters outside the classroom:
- Leaving Earth. Rocket planners use Earth's escape velocity of about 11.2 km/s as the speed a craft would need to leave without more thrust.
- Lunar launches. Mission designers note the Moon's escape velocity is only about 2.4 km/s, which makes launching from its surface far easier.
- Black holes. Astronomy classes show that a body squeezed small enough has an escape velocity above light speed, the idea behind a black hole.
- Planet atmospheres. Planetary scientists compare escape velocity with gas molecule speeds to explain why small bodies like the Moon lost their air.
How to use it
- Pick a planet, moon or the Sun, or choose a custom body and enter its mass in kilograms and its radius.
- If you start above the surface, enter the height. Leave it at 0 for a launch from the surface.
- Choose the speed unit you want for the answer.
- Read the escape velocity and compare it with the orbit speed and local gravity shown beneath it.
What the inputs mean
- Mass of the body (M)
- The mass of the planet, moon or star, not of the object being launched. You can use scientific notation, such as 6.4171e23, meaning 6.4171 × 10²³.
- Radius of the body (R)
- The distance from the center of the body to its surface. The built-in bodies use their volumetric mean radius: the radius of a perfect ball with the same volume.
- Starting height above the surface (h)
- Height above the surface at the moment the object starts coasting. The escape speed drops as you start farther out.
Formula and method
v = √(2 × G × M ÷ r) r = R + h G = 6.6743 × 10⁻¹¹ m³/(kg·s²) v_orbit = √(G × M ÷ r) = v ÷ √2 g = G × M ÷ r²
v is the escape speed. G is the gravitational constant, a fixed number that sets the strength of gravity everywhere. M is the mass of the body you're leaving. R is its radius, h is your starting height, and r is your distance from its center. g is the pull of gravity at that distance.
The formula comes from energy. A moving object has kinetic energy (energy of motion) of ½mv², where m is its own mass. Climbing from distance r all the way out of the body's gravity costs GMm ÷ r of energy. Set the two equal and solve for v to get the escape velocity. The object's own mass m cancels out.
A circular orbit at the same distance needs exactly 1 ÷ √2 (about 70.7%) of the escape speed. The calculator converts your inputs to kilograms and meters first. So the result comes out in meters per second, then it's shown in the unit you choose.
Worked example
What is the escape velocity from the surface of the Earth, using M = 5.9722 × 10²⁴ kg and R = 6,371 km?
- Convert the radius: R = 6,371,000 m.
- G × M = 6.6743e-11 × 5.9722e+24 = 398,602,544,600,000 m³/s².
- v = √(2 × 398,602,544,600,000 ÷ 6,371,000) = 11,186.165197 m/s.
- In km/s: 11,186.165197 ÷ 1,000 = 11.186165 km/s.
Result: The escape velocity from the Earth's surface is about 11.19 km/s (about 40,270 km/h). A circular orbit at the surface would need about 7.91 km/s.
Understanding the result
Escape velocity is a coasting speed. A rocket that keeps firing its engines can leave the Earth at any speed. The figure tells you how much speed an object with no engine needs at the start.
Small bodies are easy to leave. From the Moon, the escape speed is under 2.4 km/s. That's one reason it has almost no atmosphere. Fast-moving gas molecules can drift off into space over time.
Assumptions and limitations
- The body is a sphere with its mass spread evenly in layers, so its gravity acts as if all the mass were at the center.
- No air resistance, no rotation of the body and no pull from other bodies.
- Newton's law of gravity, which is accurate for planets and ordinary stars.
- Real launches get a boost from the planet's spin (about 0.46 km/s at the equator for Earth). They also lose speed to air drag.
- Escaping the Sun's gravity from Earth's orbit is a separate, bigger problem. This one-body formula doesn't combine the two.
- Near extremely dense objects such as neutron stars or black holes, you need Einstein's general relativity instead.
Frequently asked questions
Does escape velocity depend on the mass of the rocket?
No. The energy of motion it needs and the gravity it must overcome both grow with the object's mass, so the mass cancels out. A heavier rocket needs more fuel to reach the speed. But the speed itself is the same.
Does the direction of launch matter?
In the ideal calculation, no. Any direction that doesn't hit the body works, because escape depends on energy. In practice, launching eastward near the equator borrows speed from the planet's spin. And a steep path through the atmosphere loses less to drag.
Why is escape velocity √2 times the orbital speed?
In a circular orbit, the energy of motion is exactly half of the energy needed to escape. Doubling that energy means multiplying the speed by √2, about 1.414.
How does the escape speed change with height?
It falls in proportion to 1 ÷ √r, where r is your distance from the center. Start one Earth radius above the surface, which is twice as far from the center. That cuts the escape speed by a factor of √2, to about 7.9 km/s.
How this calculator is checked
Method: Newtonian energy conservation for a spherically symmetric body: v = √(2GM/r), with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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