AkCalculators

Escape Velocity — meaning, formula and practical implications

Escape velocity is the minimum initial speed required for an object to leave the gravitational influence of a massive body without further propulsion, neglecting atmospheric drag and assuming energy-conserving motion. For a spherically symmetric mass distribution, the classical escape velocity from distance r from the center is

v = √(2 G M / r)

Where the formula comes from

The expression follows from conserving mechanical energy. The object needs initial kinetic energy equal to the change in gravitational potential energy required to move from r to infinity: (1/2) m v² = G M m / r. Cancel m and rearrange to get v = √(2GM/r). This derivation assumes Newtonian gravity and neglects other forces.

Typical values and presets

Common surface escape speeds: Earth ≈ 11.186 km/s, Moon ≈ 2.38 km/s, Sun (photosphere) ≈ 617.5 km/s. These are computed using accepted mass and radius values for the bodies. The calculator includes presets for convenience but allows custom M and r for satellites, exoplanets or hypothetical bodies.

Using altitude and surface vs center

Escape speed decreases with altitude because r increases. To compute escape speed from altitude h above the surface: use r = R_body + h. For spacecraft launches the required delta-v is higher than the surface escape speed due to atmospheric drag, gravity losses during ascent, and the need to achieve certain trajectories, so escape velocity is a theoretical baseline rather than a direct launch target.

Limits and assumptions

The formula assumes a point mass or spherically symmetric mass distribution and classical mechanics. It ignores atmospheric drag, rotational energy of the body (which can assist or hinder escape depending on launch azimuth and latitude), and relativistic effects. For relativistic or high-precision astronomical work, general relativity and detailed mass distribution models must be used.

Worked examples

Example 1 — Earth surface: Using M_earth = 5.972e24 kg and R_earth = 6.371e6 m, v = √(2 × 6.67430e-11 × 5.972e24 / 6.371e6) ≈ 11,186 m/s ≈ 11.186 km/s.

Example 2 — From altitude: At 400 km altitude (ISS orbit approximate), r ≈ 6.371e6 + 4e5 = 6.771e6 m. Escape speed ≈ √(2GM / r) ≈ 10.97 km/s (slightly lower than from surface).

Practical implications for missions

Real launch vehicles do not simply accelerate to escape speed; they follow powered trajectories that trade gravity losses and atmospheric drag. For interplanetary missions the spacecraft often first reach orbit and then apply burns (Oberth effect) to raise energy. Additionally, missions use gravity assists, which rely on relative motion rather than pure escape speed calculations.

Units and precision

Use SI for internal calculations. This tool converts common units (km, km for radius, Earth-mass, Solar-mass) into SI and provides outputs in m/s and km/s. For high-precision research, use the latest measured planetary parameters from authoritative sources.

Using this calculator

Select a preset or enter custom mass (M) and radius (r). Optionally enter an altitude to compute from above the surface. Choose precision and enable steps to show the algebraic derivation and intermediate conversions. Export results to CSV or copy them to the clipboard for reports.