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 = 5.972 × 10²⁴ kg
• r = 6.371 × 10⁶ m (mean radius)
• v_esc = √(2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / 6.371×10⁶)
• v_esc = √(7.972×10⁷) = 11,186 m/s ≈ 11.2 km/s

Escape Velocity Comparison:

Key Facts:

• Direction doesn’t matter — escape velocity is the same straight up or at any angle
• It is the speed, not the acceleration, that determines escape
• SpaceX Falcon 9 reaches ~9.4 km/s in orbit (below escape velocity by design)
• The Moon has no atmosphere because light gases (H₂, He) exceed lunar escape velocity

The black-hole limit — 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’s about 3 km. Anything compressed below its Schwarzschild radius becomes a black hole; that is what the escape-velocity formula is saying when it predicts a value greater than c.