Escape Velocity Calculator

How Escape Velocity Is Calculated

Escape velocity is the minimum speed an object needs to break free from a planet’s gravitational field without further propulsion.

Escape Velocity Formula: v_esc = √(2GM / r)

Where:

• v_esc = escape velocity in m/s
• G = gravitational constant = 6.674 × 10⁻¹¹ N·m²/kg²
• M = mass of the planet/body in kg
• r = radius (distance from center) in meters

Worked Example, Earth:

• M = 5.9722 × 10²⁴ kg
• r = 6.371 × 10⁶ m (volumetric mean radius)
• v_esc = √(2 × 6.6743×10⁻¹¹ × 5.9722×10²⁴ / 6.371×10⁶)
• v_esc = √(1.2512×10⁸) = 11,186 m/s ≈ 11.19 km/s

Escape Velocity Comparison:

That third column is not padding. Which radius you pick changes the answer, and the two gas giants are quoted differently from everything else. Jupiter spins once every ten hours, and the resulting bulge puts its equator 4,500 km further from the centre than its poles. Published escape velocities for Jupiter and Saturn use that equatorial figure, which is the largest radius and therefore the lowest escape velocity. Use Jupiter’s volumetric mean radius of 69,911 km instead and you get 60.2 km/s, which is not wrong so much as answering a different question. The presets here follow the published convention.

Key Facts:

• Direction doesn’t matter. Escape velocity is the same straight up or at any angle, because gravitational potential energy depends only on distance.
• It is the speed, not the acceleration, that determines escape.
• Low Earth orbit needs about 7.8 km/s, not the 11.19 km/s of escape. A rocket burns roughly 9.4 km/s of delta-v to get there, and the extra 1.6 km/s is spent fighting gravity and air drag on the way up. Those are three different numbers that get confused constantly.
• Escape velocity assumes a single unpowered impulse. A rocket that keeps thrusting can leave at any speed it likes; it just costs more fuel.
• The Moon holds almost no atmosphere because hydrogen and helium molecules at lunar surface temperatures reach a good fraction of 2.38 km/s in the tail of their speed distribution, so they leak away over geological time. Heavier molecules leak more slowly, which is why the Moon has a wisp of an exosphere rather than nothing at all.

The black-hole limit, or the Schwarzschild radius. Push the formula to its extreme: what happens if you compress a mass until its escape velocity reaches the speed of light? Solving √(2GM/R) = c for R gives the Schwarzschild radius R_s = 2GM/c², the radius at which even light cannot escape, which defines the event horizon of a non-rotating black hole. For Earth’s mass that radius is about 8.9 mm; for the Sun it is about 2.95 km. Anything compressed below its Schwarzschild radius becomes a black hole, and that is what the escape-velocity formula is telling you when it predicts a value greater than c.

The Newtonian derivation gets the right answer here by a genuine coincidence. Escape velocity is a Newtonian idea and light does not obey it, so the agreement with general relativity’s event horizon is a happy accident that Michell and Laplace stumbled onto in the 1780s, more than a century before Schwarzschild solved Einstein’s equations properly.