Escape Velocity Calculator
Compute escape velocity using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
v_escape = √(2GM / r)
Escape velocity equals the square root of (2 × gravitational constant × body mass ÷ body radius). G = 6.674 × 10⁻¹¹ N·m²/kg². This is derived from setting kinetic energy equal to gravitational potential energy. Orbital velocity is v_orbit = √(GM/r), which is exactly v_escape / √2.
Worked Examples
Example 1: Earth's Escape Velocity
Problem:Calculate the escape velocity from Earth's surface.
Solution:v_e = sqrt(2GM/r)\n= sqrt(2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / 6.371×10⁶)\n= sqrt(1.249×10⁸)\n= 11,186 m/s\n= 11.19 km/s
Result:11.19 km/s (40,270 km/h or 25,020 mph) — about Mach 32.6
Example 2: Mars Escape Velocity
Problem:Calculate the escape velocity from Mars for mission planning.
Solution:v_e = sqrt(2 × 6.674×10⁻¹¹ × 6.417×10²³ / 3.3895×10⁶)\n= sqrt(2.527×10⁷)\n= 5,027 m/s\n= 5.03 km/s
Result:5.03 km/s — about 45% of Earth's, making Mars launches easier
Frequently Asked Questions
What is escape velocity?
Escape velocity is the minimum speed an object must reach to break free from a celestial body's gravitational pull without further propulsion. It is calculated using v_e = sqrt(2GM/r), where G is the gravitational constant, M is the body's mass, and r is its radius. For Earth, escape velocity is approximately 11.19 km/s (about 40,270 km/h or 25,020 mph). Note that rockets don't actually need to reach escape velocity all at once — they can escape with continuous thrust at lower speeds.
Why doesn't escape velocity depend on the object's mass?
Escape velocity depends only on the celestial body's mass and radius, not on the escaping object's mass. This comes from energy conservation: kinetic energy (½mv²) must equal gravitational potential energy (GMm/r). The object's mass m cancels on both sides, giving v = sqrt(2GM/r). A feather and a rocket have the same escape velocity — though practically, heavier objects need more energy (force × distance) to reach that speed.
What is the relationship between escape velocity and orbital velocity?
Escape velocity is exactly sqrt(2) ≈ 1.414 times the circular orbital velocity at the same altitude. Orbital velocity keeps an object in a circular orbit (v_o = sqrt(GM/r)), while escape velocity provides enough energy to reach infinity. To go from orbit to escape, you only need to increase speed by about 41.4%. This relationship holds at any altitude.
What happens if you exceed escape velocity?
If an object exceeds escape velocity, it follows a hyperbolic trajectory and will never return to the body. The excess speed above escape velocity determines the 'hyperbolic excess velocity' — the speed the object retains at infinite distance. This is crucial for interplanetary missions: spacecraft must leave Earth with enough hyperbolic excess to reach their target planet on a transfer orbit.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy