Escape Velocity Calculator Formula
Understand the math behind the escape velocity calculator. Each variable explained with a worked example.
Formulas Used
Escape Velocity
escape_vel = sqrt(2 * 6.674e-11 * mass / radius)Escape Velocity (km/s)
escape_vel_kmps = sqrt(2 * 6.674e-11 * mass / radius) / 1000Variables
| Variable | Description | Default |
|---|---|---|
mass | Body Mass(kg) | 5.972e+24 |
radius | Body Radius(m) | 6371000 |
How It Works
Deriving Escape Velocity
Escape velocity is the minimum launch speed at the surface for an object to reach infinity without further propulsion.
Formula
v_esc = √(2 G M / r)
This is exactly √2 times the circular orbital velocity at the same radius. It comes from setting kinetic energy equal to gravitational potential energy: 1/2 m v² = G M m / r.
Worked Example
Find Earth's escape velocity (M = 5.972e24 kg, r = 6 371 km).
- 01v_esc = √(2 G M / r)
- 022 G M = 2 × 6.674e-11 × 5.972e24 = 7.972e14
- 037.972e14 / 6.371e6 = 1.251e8
- 04v_esc = √(1.251e8) ≈ 11 186 m/s ≈ 11.19 km/s
Frequently Asked Questions
Does escape velocity depend on the escaping object's mass?
No. The mass cancels out in the energy balance. A pebble and a spacecraft need the same speed.
Why is escape velocity √2 times orbital velocity?
Orbital velocity balances gravity for a circle. Escape requires enough extra kinetic energy to overcome the remaining potential, which works out to a factor of √2.
What is the escape velocity of the Moon?
About 2.38 km/s, much lower than Earth's because the Moon is less massive and smaller.
Ready to run the numbers?
Open Escape Velocity Calculator