Escape velocity calculator

Escape velocity is the lowest speed a coasting projectile must have at a given distance from a massive body if Newtonian gravity is not to pull it back from infinity. For a spherical source of mass M evaluated at radius r the energy argument yields v equals the square root of two G M over r, with G the Newtonian gravitational constant. This calculator accepts mass in kilograms and radius in metres, both required to be strictly positive finite numbers, and returns that speed in metres per second together with a kilometres-per-second convenience field.

What escape velocity is and how v equals sqrt of 2GM over r

In classical mechanics a test particle of mass m moving in the gravitational field of a fixed spherical body of mass M has mechanical energy equal to one half m v squared minus G M m over r. Escape from the well in the Newtonian sense means the total energy is at least zero, so the particle can reach infinite separation with leftover kinetic energy that is not negative. Setting energy to zero and solving for speed produces v equals the square root of two G M over r.

The test mass m cancels, which is why a grain of dust and a spacecraft share the same numerical threshold at the same radius. The gravitational constant G is the CODATA 2018 value six point six seven four three times ten to the minus eleven cubic metres per kilogram per second squared. Radius is the centre-to-point distance: at the surface of a sphere it is the body's own radius; at altitude h above that surface it is R plus h.

The result is a scalar speed, not a required heading; any trajectory that does not strike the body and whose speed meets the threshold can leave in the ideal model. Atmosphere, planetary rotation, continuous thrust after departure, and general relativity are outside this capability. Those effects matter for real launches, but homework, first-order mission sketches, and physics engines still start from this closed-form number before they add drag tables or rotating-frame corrections.

Circular orbital speed at the same radius is the square root of G M over r, so escape speed is exactly the square root of two times that circular value—a useful classroom check when both formulas appear on the same worksheet.

Where an escape-velocity step earns its keep

Exam generators and LMS plugins need an oracle that rejects non-positive mass or radius instead of returning NaN that a buggy grader might treat as success. Game and simulation engines that place spherical planets often need a single scalar they can compare against a projectile speed each tick; calling a tiny parse step during content authoring, then baking the constant into the build, avoids shipping a physics library just for one square root.

Laboratory write-ups that start from tabulated planetary mass and mean radius can pin the published velocity in a report or in continuous-integration goldens that must not depend on network services. Prefer this capability when you already have mass and radius in SI and only need the classical speed. It is not an orbital-period tool, not a Hohmann-transfer planner, not a rocket-equation delta-v budget, and not a general-relativity geodesic solver.

If your source quotes density and diameter, convert to mass and radius first. Keep mass-over-radius ratios inside double range: an absurdly large M with a tiny r can overflow two G M over r into a non-finite value that the solver rejects rather than inventing an infinite speed. Round-trip checks are straightforward: recompute two times G times M over r by hand, take the square root, divide by one thousand for kilometres per second, and confirm both published fields match after the same twelve-decimal rounding.