Mechanics
How To Calculate An Escape Velocity
Escape velocity is the speed at which an object's kinetic energy exactly cancels the gravity holding it down, and it depends only on the body you are leaving.
Quick Answer
v = sqrt(2 x g x r), where g is surface gravity and r is the body's radius
- v
- Escape velocity
- g
- Surface gravity
- r
- Body radius
Take the surface gravity in metres per second squared and the radius in metres, double the gravity, multiply by the radius, and take the square root. For Earth with g of 9.81 and a radius of 6,371 km, the answer is 11,180 m/s.
What Is An Escape Velocity?
Escape velocity is the minimum speed an object needs at the surface of a body, with no further propulsion, to coast out to infinity and arrive there with nothing left over. It is a statement about energy rather than about thrust: at exactly this speed the object's kinetic energy equals the work gravity does pulling it back.
The form that makes the physics transparent is one half m v squared equals G M m over r. The mass of the object cancels from both sides, which is why a feather and a boulder need the same escape velocity. What is left is one half v squared equals G M over r, and since surface gravity is G M over r squared, substituting gives v squared equals 2 g r.
That substitution is the practical shortcut. Rather than reach for the gravitational constant and a mass in the trillions of kilograms, take the surface gravity you can look up and the radius you can measure. Earth's 9.81 m/s squared times a radius of 6,371 km gives 11,180 m/s, which is about 11.18 km/s or 40,249 km/h.
The square root of two appears because of the difference between escaping and merely orbiting. An object in a low circular orbit travels at the square root of g r, so escape velocity is that figure times the square root of two, or about 1.414 times as fast. This is why launching from orbit is so much cheaper than launching from the ground: the velocity you still have to add is under 42 percent of what a ground launch requires.
Escape velocity falls as you get further from the body, which is a point worth being careful about. The value usually quoted is the surface value, and a rocket starting from the ground has to climb through a deepening gravity well while shedding the atmosphere on the way. Real vehicles never simply coast at the surface number; they accelerate continuously and are still climbing when they pass that speed.
The number also does not scale the way people expect. Doubling a planet's radius doubles the escape velocity, but doubling its mass does as well only if the radius stays fixed. A body that is twice as dense with the same diameter is much harder to leave than one that is simply twice the mass at twice the radius, and that is why the direction of the scaling matters as much as the factor.
It is worth being explicit about what the figure does not mean. Escape velocity is not the speed a rocket engine must produce, and it is not a threshold below which an object falls straight back down. A cannonball fired at 90 percent of escape velocity still rises a very long way before turning around, and its total energy simply is not enough to coast out forever.
The same arithmetic explains a good deal about the solar system. Mercury, small and dense, sits near 4.3 km/s and has almost no atmosphere. Titan, with modest gravity but a very cold environment, retains one that is thicker than Earth's at ground level. Venus sits close to Earth in escape velocity and keeps an atmosphere ninety times denser, which shows how much temperature and chemistry matter alongside the raw speed.
Formula
v = sqrt(2 x g x r)
Use the surface gravity and radius, which avoids the gravitational constant entirely.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| v | Escape velocity | metres per second | Required speed at the surface. |
| g | Surface gravity | metres per second squared | Gravitational acceleration at the surface. |
| r | Radius | metres | Distance from the centre of the body to its surface. |
v = sqrt(2 x G x M / r)
The original form, useful when the mass is known but the surface gravity is not.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| v | Escape velocity | metres per second | Required speed at the surface. |
| G | Gravitational constant | newton metre squared per kilogram squared | Universal constant, about 6.6743e-11. |
| M | Body mass | kilograms | Total mass of the body. |
| r | Radius | metres | Distance from the centre to the surface. |
v escape = sqrt(2) x v orbital
Escape velocity is the square root of two times the circular orbital speed at the same radius.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| v escape | Escape velocity | metres per second | The speed to escape entirely. |
| v orbital | Orbital speed | metres per second | Speed for a circular orbit at that radius. |
How To Calculate An Escape Velocity
- 1
Convert the radius to metres
The formula needs metres. A radius quoted in kilometres must be multiplied by 1,000, so Earth's 6,371 km becomes 6,371,000 m.
- 2
Double the surface gravity
Multiply g by two. For Earth that is 9.81 times 2, giving 19.62.
- 3
Multiply by the radius
Multiply the doubled gravity by the radius in metres. This gives 19.62 times 6,371,000, which is about 124,998,000.
- 4
Take the square root
The square root of that product is the escape velocity, 11,180 m/s for Earth, or about 11.18 km/s.
Examples
Example 1: Earth from the surface
- Surface gravity
- 9.81 m/s2
- Body radius
- 6,371 km
| Step | Calculation | Result |
|---|---|---|
| Radius in metres | r in metres | 6371 x 1000 |
| Twice g times r | 2 x g x r | 2 x 9.81 x 6371000 |
| Square root | v = sqrt(2 g r) | sqrt(124998000) |
| Escape velocity | v | 11180 m/s |
Result: Earth's escape velocity is 11,180 metres per second, which is 11.18 kilometres per second or about 40,249 kilometres per hour.
Example 2: The Moon
- Surface gravity
- 1.62 m/s2
- Body radius
- 1,737 km
| Step | Calculation | Result |
|---|---|---|
| Radius in metres | r in metres | 1737 x 1000 |
| Twice g times r | 2 x g x r | 2 x 1.62 x 1737000 |
| Escape velocity | v = sqrt(5627880) | 2372.3 |
Result: The Moon's escape velocity is about 2372.3 metres per second, roughly a fifth of Earth's, which is why it kept no atmosphere of its own.
Example 3: Jupiter at its cloud tops
- Surface gravity
- 24.79 m/s2
- Body radius
- 69,911 km
| Step | Calculation | Result |
|---|---|---|
| Radius in metres | r in metres | 69911 x 1000 |
| Twice g times r | 2 x g x r | 2 x 24.79 x 69911000 |
| Escape velocity | v = sqrt(3469200000) | 58899.9 |
Result: Jupiter's escape velocity is about 58899.9 metres per second, over five times Earth's and the reason capture missions to the Jovian system need so much propellant.
Calculator
Escape velocity
11,180.2961
- Escape velocity in km/h
- 40,249.0658
- Escape velocity in km/s
- 11.1803
- Circular orbital speed
- 7,905.6632
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the An Escape Velocity calculator page.
Common Mistakes
Leaving the radius in kilometres
Using 6,371 instead of 6,371,000 gives a velocity about 31.6 times too small, because the radius enters under a square root.
Using the gravitational constant and forgetting a mass
The G M over r squared form needs the body's total mass. If you do not have it, use the surface gravity form instead, which is why this calculator is built on g.
Adding the mass of the launched object
The projectile's mass cancels out of the energy balance, so escape velocity does not depend on it at all. A 1 kg probe and a 1 tonne probe need the same speed.
Treating escape velocity as a force
It is a speed, not a push. An object at escape velocity coasts the whole way with engines off, which is what makes the energy argument work.
Expecting a spacecraft to reach the surface value instantly
Real launches accelerate continuously through the atmosphere. The escape velocity figure is a target speed, not a requirement at launch.
Confusing escape velocity with circular orbital speed
The orbital speed at the same radius is escape velocity divided by the square root of two, about 0.707 times as fast. Using the smaller figure means the object reaches orbit and comes no further.
FAQ
How do you calculate escape velocity?
Use the surface gravity and radius: take the square root of two times gravity times radius in metres. For Earth that gives about 11,180 metres per second.
What is the escape velocity of Earth?
About 11,180 metres per second, or 11.18 kilometres per second, which is roughly 40,249 kilometres per hour.
Does escape velocity depend on the mass of the object?
No. The object's mass cancels out of the energy balance, so a grain of sand and a spacecraft need exactly the same speed to escape.
Why is escape velocity the square root of two times orbital velocity?
Because escaping takes twice the energy of reaching a circular orbit, and kinetic energy goes as speed squared. Twice the energy means the square root of two times the speed.
Does escape velocity decrease with altitude?
Yes, it falls with distance from the centre of the body. The commonly quoted figure is the surface value, and it gets smaller the higher you start.
Can anything travel slower than escape velocity and still leave?
Yes, if it has continuous propulsion. Escape velocity applies to an unpowered coast, which is why rockets can climb out more slowly while burning fuel the whole way.
References
- [1]Wikipedia, Escape velocity — https://en.wikipedia.org/wiki/Escape_velocity
- [2]NASA, Gravitation and escape speed — https://science.nasa.gov/
- [3]Khan Academy, Orbital mechanics for engineering students — https://www.khanacademy.org/science/physics