Escape Velocity Calculator

What escape velocity measures

Throw a baseball straight up and it comes down. Throw it harder and it climbs higher before gravity pulls it back. Escape velocity is the cutoff speed: launch a projectile upward at that speed with no engine running after liftoff, and the Earth's gravity slows it forever but never stops it. It would coast away and never return.

Two features of the idea surprise people. First, the number ignores air: it is the speed for a world with no atmosphere, a clean physics cutoff. Second, it ignores the mass of the projectile entirely. A paper clip and a cargo ship need the same 11.19 km/s to leave Earth, because the craft's own mass cancels out of the equation before the answer appears.

The speed is not a speed limit, either. A spacecraft can leave Earth moving slower than 11.19 km/s at any given moment if it burns its engines along the way, trading thrust for height. Escape velocity is the number that matters for the coast: reach it once, engine off, and gravity alone cannot bring you home. NASA's planetary fact sheets list the mass and radius figures this page builds on.

Where the formula comes from

The derivation is one line of energy accounting. A projectile of mass m sitting on a world of mass M and radius r carries gravitational potential energy that, in Newton's gravity, equals GMm/r, negative by convention. Kinetic energy is 1/2 x m x v^2. To escape, the projectile must bring its total energy up to exactly zero:

1/2 m v^2 = GMm / r, so v = sqrt(2GM / r)

The projectile's mass m sits on both sides and cancels, which is why a probe and a pebble share the same escape velocity. What survives in the formula is GM, the gravitational parameter, a single number per world that astronomers measure to extraordinary precision. G itself, the gravitational constant of 6.6743 x 10^-11 m^3/(kg s^2), is tabulated by NIST as part of the SI, and multiplying it by each body's mass gives the GM that the calculator uses.

The radius r is measured from the center of the world, not the surface. That single point causes most of the wrong answers people get by hand: Earth's surface sits 6,371 km from the center, so that is the number that goes in, not the altitude of the launch pad.

Worked example: leaving Earth

Earth's mass is 5.972 x 10^24 kg and its mean radius is 6,371 km. Feed those into v = sqrt(2GM/r) and the steps run like this:

So Earth's escape velocity is 11.19 km/s, which is 40,270 km/h or about 25,022 mph. Nothing launched from the surface coasts to the stars slower than that. The Apollo flights, the Voyager probes and every deep-space mission since had to find that energy somewhere, whether in one enormous rocket stage or in a gravity assist that borrowed speed from a passing planet.

For contrast, run the Moon: its GM is 4.90 x 10^12 m^3/s^2 over a 1,737.4 km radius, which gives 2.38 km/s. That gap, a factor of 4.7, is why the lunar lander needed a rocket only a few meters tall to get home while the Saturn V that lifted the crew stood 111 meters.

Escape velocity across the solar system

The table below runs the same formula over twelve worlds, using NASA's mass and mean-radius figures. Values are rounded to the digits shown; different references quote slightly different numbers because a gas giant has no solid surface and tables disagree on which radius to use (mean, equatorial or 1-bar level).

The spread is enormous. Ceres, the largest asteroid-belt dwarf planet, lets you leave at highway speeds: 0.52 km/s, about the pace of a jet fighter. Jupiter, at the other end, holds onto its cloud tops at 60.20 km/s, more than five times Earth's figure, which is one reason its atmosphere keeps hydrogen that lighter worlds lose. The Sun tops everything at 617.77 km/s from its visible surface.

Climb first, and the bar drops

Escape velocity shrinks as you start higher, because r in the denominator is the distance from the center of the world, not the surface. The formula still applies at altitude, and the savings are immediate. At the International Space Station's orbital height of 408 km, a coasting escape needs 10.84 km/s instead of 11.19, about 3 percent less. At geostationary altitude, 35,786 km up, the requirement falls to 4.35 km/s.

The orbital column in the table is not a coincidence: circular orbital speed at any radius is escape speed there divided by sqrt(2), a fixed ratio that falls out of the same energy equation. That is why the two columns fall together as you climb.

Escape velocity vs orbital speed

Orbiting and escaping answer different questions, and the sqrt(2) between them is the whole relationship. To orbit, you need enough sideways speed that the ground curves away as fast as you fall, which works out to v = sqrt(GM/r) with no factor of 2. To escape, you need enough total energy to reach infinity with nothing left over, and that demands the factor of 2 under the square root, so escape velocity is always 1.414 times orbital velocity at the same spot.

In low Earth orbit that pairs 7.91 km/s with 11.19 km/s. A rocket that reaches the first number and tips its nose sideways becomes a satellite; one that keeps burning to the second never comes back. Real missions live in between: a spacecraft bound for the Moon aims for a path that exceeds orbital speed but approaches escape speed, and a Mars transfer does roughly the same with a small reserve on top.

The ratio also explains a classic physics puzzle. Slow a satellite in circular orbit and it falls inward and speeds up as it goes; speed it up and the orbit stretches away. At exactly sqrt(2) times the circular speed the ellipse stops closing and becomes a parabola, the mathematical edge between captured and free.

Traps: radius, drag and the rocket myth

Three mistakes cover most of the trouble.

• Using the altitude instead of the radius. The formula wants the distance from the center of the body. For Earth that is 6,371 km plus your altitude, never the altitude alone.
• Expecting the atmosphere in the answer. The formula is a gravity-only cutoff. Drag costs a real rocket more, and this page's number is the ideal coasting case, which is exactly what the term escape velocity means.
• Believing rockets must hit 11.19 km/s at liftoff. They do not, and no launch vehicle could survive it in the lower atmosphere. Rockets build speed gradually while burning fuel; escape velocity describes a ballistic object, and a rocket is never ballistic while its engines run.

The formula has one more trick worth knowing. Shrink any mass and its escape velocity rises, because the same GM sits over a smaller r. Compress the Sun to a 2.95 km radius and its surface escape velocity reaches 299,792 km/s, the speed of light, which is the definition of that radius: the Schwarzschild radius of a black hole. The same equation that moves spacecraft marks the boundary where not even light escapes.

Where to go next with your speed numbers

The neighboring tools on this site carry the rest of the mechanics. The velocity calculator converts any speed between m/s, km/h, mph, ft/s and knots, and the acceleration calculator covers the rate at which a launch builds toward these numbers. The kinetic energy calculator works the other half of the energy equation, and the terminal velocity calculator shows what happens when drag, ignored on this page, gets a vote. For the force behind a real launch, the force calculator applies F = ma in both directions of the unit table.