How this instrument works
Escape velocity is where a projectile's kinetic energy exactly cancels its gravitational binding energy: set ½mv² equal to GMm ⁄ r, and m drops out of both sides. That cancellation is the whole surprise — a marble, a boulder and a crewed vehicle share one threshold at a given distance from a given body. Direction never enters either. Sideways, straight up or at 45°, anything unpowered leaving above this figure coasts away forever, so escape speed would be a truer name than the one that stuck.
Newton sketched it in 1687 with a cannon on an impossibly tall mountain: fire gently and the ball lands nearby, fire harder and it circles, harder still and it departs. A century on, in an 1783 letter read to the Royal Society, John Michell asked what a star would be like if √(2GM ⁄ r) at its surface exceeded light's speed — inventing dark stars, an idea Laplace repeated in 1796. Put v = c into this same expression and out comes r = 2GM ⁄ c², numerically identical to the Schwarzschild radius general relativity delivered in 1916. How much of that agreement is luck remains a good argument to start among physicists.
Each assumption behind that square root deserves naming. Gravity comes from one spherically symmetric body, so an Earth answer strands you in solar orbit rather than interstellar space. There is no atmosphere, no thrust after release, and no relativity — which is precisely why the black-hole coincidence above stays a coincidence. And r runs from the centre of mass outward, not from ground level; feed in an altitude by mistake and your figure will be wildly wrong. Escape carries a strict meaning too: arriving infinitely far away with nothing left over, on a parabolic path that slows forever without quite stopping.
- Set Mass of the body in kilograms or tonnes — Earth's 5.9722×10²⁴ kg is preloaded.
- Set Radius / launch distance in metres or kilometres, measured from the centre outward rather than from ground level.
- Read Escape velocity in m/s, then flip that field's unit menu to km/h or mph for a figure you can picture.
- Open the working block to see √(2GM ⁄ r) with your own numbers substituted.
Worked example — leaving Earth from sea level
Earth first: M = 5.9722×10²⁴ kg, r = 6,371,000 m, its mean radius. Then 2GM = 2 × 6.6743×10⁻¹¹ × 5.9722×10²⁴ = 7.97205×10¹⁴, and dividing by 6,371,000 gives 1.2513029×10⁸ m²/s². Take a square root: v = 11,186.17 m/s. Call it 11.19 km/s, or 40,270 km/h, or 25,023 mph — the figure every astronautics text quotes.
No launch vehicle has ever hit that speed on the pad, and none tries to. This formula governs a single ballistic kick with nothing to follow it, whereas a rocket burns for minutes, clawing through thick air before adding real speed where drag has thinned. Apollo's translunar injection stopped deliberately short, near 10.8 km/s, so a free-return path would swing its crew home should an engine quit. New Horizons went hard the other way in 2006, departing at roughly 16.26 km/s — quickest Earth departure on record, and still nine and a half years to Pluto.
Questions
Does a heavier spacecraft need more speed to get away?
No. Mass of whatever is leaving cancels out of the energy balance, so a pebble and a space station share one threshold at equal distance from equal M. What mass does change is your fuel bill: energy scales as ½mv², so a heavier vehicle costs proportionally more to push up to that identical speed.
How does escape velocity compare with orbital velocity?
Larger by exactly √2, about 41%, at any radius you choose. Circling needs √(GM ⁄ r); leaving needs √(2GM ⁄ r). Just above Earth's atmosphere that reads 7.91 km/s to stay versus 11.19 km/s to go, meaning low orbit already buys roughly 70% of a departure in speed terms — though the remaining 30% is where mission budgets go to suffer.
Should I enter altitude or distance from the centre?
Distance from the centre of mass, without exception. For a sea-level start on Earth that means 6,371,000 m, not zero. Type 400,000 m while thinking of a 400 km orbit and you get about 44.6 km/s, which is nonsense; the correct entry there is 6,771,000 m, returning 10.85 km/s. Values fall as you climb, one quiet argument for starting high.
Why does the Moon have no atmosphere?
Because 2.38 km/s is too low a bar. A body keeps a gas across geological time roughly when its escape figure beats six times that molecule's mean thermal speed, and lunar daytime heat pushes light molecules straight past the line. Earth's 11.19 km/s holds nitrogen and oxygen comfortably while still leaking hydrogen and helium into space at a few kilograms per second.
Does launch direction change the answer?
Not for this formula — it states an energy condition, so any heading that misses solid ground serves equally. Reality adds two footnotes: air near the bottom is dense enough to shred a horizontal shot at 11 km/s, and Earth's spin already hands you up to 465 m/s eastward at the equator, free speed that sites from Kourou to Cape Canaveral sit low in latitude to collect.
Does reaching 11.19 km/s get me out of the solar system?
No — you only shake off whichever body you entered. Leave Earth at 11.19 km/s and the Sun still holds you, carrying you around at some 30 km/s alongside the planet you left. Breaking free of solar gravity from that orbit takes about 42.1 km/s heliocentric, near 16.6 km/s measured from the ground. Voyager and New Horizons both borrowed Jupiter's gravity to cover their shortfall.