Escape Velocity Calculator
Calculate the minimum velocity needed for an object to escape a celestial body's gravitational pull without further propulsion. Uses the formula v = √(2GM/r).
Escape velocity is the minimum speed an object needs to break free from a celestial body's gravity and not fall back, assuming no further propulsion. It is one of the most fundamental quantities in space science — every spacecraft mission's energy budget, every interstellar voyage analysis, every black hole calculation relies on it.
The formula v = √(2GM/r) reveals two important things. First, escape velocity depends only on the central body's mass M and the distance r from its center — not on the escaping object's mass. A pebble and a spaceship need the same escape velocity from Earth's surface (about 11.2 km/s). Second, escape velocity scales with the square root of mass and inversely with the square root of distance. A more massive body needs higher escape velocity; farther from the center, escape becomes easier.
Real rocket flights don't simply reach escape velocity instantly — that would require infinite acceleration. Instead, rockets accelerate gradually through staged burns, gaining speed while climbing through the gravitational well. The "delta-v budget" — total velocity change needed — for an Earth-escape trajectory is more than just the surface escape velocity, due to gravity losses and atmospheric drag.
Common applications: spacecraft mission design (lunar, planetary, interplanetary), orbital mechanics, planetary atmospheric retention (light planets lose H, He to space), black hole physics (event horizon at v_escape = c), and conceptual understanding of gravity.
Inputs
Results
Escape Velocity
11.19 km/s
Surface Gravity
9.82 m/s²
In Earth g
1.001 g
Escape Velocity Results
| Parameter | Value |
|---|---|
| Body Mass | 5.9720e+24 kg |
| Distance from Center | 6.3710e+6 m (6371.0 km) |
| Escape Velocity | 11185.98 m/s |
| Escape Velocity | 11.186 km/s |
| Escape Velocity | 25022 mph |
| Surface Gravity | 9.8200 m/s² |
| Surface Gravity (g) | 1.0010 g |
| Formula | v = √(2GM/r) |
Formula
How to use this calculator
- Enter mass of the central body in kg.
- Enter distance from center of the body in meters (surface = body radius).
- Calculator returns escape velocity in m/s.
- Note: escape velocity doesn't depend on the escaping object's mass.
- For real missions, add gravity losses (~1.5 km/s) and atmospheric drag (~0.2 km/s).
- From orbit, escape velocity is lower than from surface.
Worked examples
Earth surface escape
**Scenario:** Minimum speed to escape Earth's gravity from the surface (ignoring atmosphere and Earth's rotation). **Calculation:** v_esc = √(2 × 6.674e-11 × 5.972e24 / 6.371e6) ≈ 11,186 m/s. **Result:** ~11.2 km/s (~25,000 mph). Saturn V rockets achieved this in about 11.5 minutes of burn time. Most missions don't go straight up at this speed; they spiral up through stages, fighting both gravity and atmospheric drag.
Moon escape
**Scenario:** Apollo astronauts launched from the lunar surface to return to Earth. Minimum speed? **Calculation:** v_esc = √(2 × 6.674e-11 × 7.342e22 / 1.737e6) ≈ 2,380 m/s. **Result:** ~2.38 km/s (~5,300 mph). Apollo's Lunar Module ascent stage achieved ~1.7 km/s to reach orbit, then docked with the Command Module which had ~1 km/s reserve for trans-Earth injection. Much easier than launching from Earth.
Black hole event horizon
**Scenario:** Mass of the Sun condensed to make a black hole. What radius? **Calculation:** Schwarzschild radius: r_s = 2GM/c² = 2 × 6.674e-11 × 1.989e30 / (3e8)² ≈ 2,952 m. **Result:** ~2.95 km radius. Anything within 2.95 km of the singularity needs v_esc > c to escape — impossible. The Sun's actual radius is 696,000 km — far from being a black hole. Stellar black holes form when massive stars collapse below their Schwarzschild radius.
When to use this calculator
**Use escape velocity for:**
- **Spacecraft mission planning**: launch energy budgets. - **Orbital mechanics**: distinguishing bound vs unbound trajectories. - **Planetary science**: atmospheric retention analysis. - **Astrophysics**: black hole event horizon, neutron stars. - **Comparative planetology**: ranking gravitational "wells". - **Sci-fi/conceptual**: understanding gravity as energy.
**Real-world rocket delta-v:**
Total delta-v needed includes: - **Burnout altitude penalty**: thrust at low altitude wastes energy. - **Gravity losses**: ~1-1.5 km/s for typical launches. - **Atmospheric drag**: ~0.1-0.5 km/s. - **Steering losses**: ~0.05-0.5 km/s. - **Orbital velocity for parking orbit**: 7.8 km/s for LEO.
To reach LEO from Earth surface: ~9.3-10 km/s total delta-v. To escape Earth from LEO: additional 3.2 km/s. Total Earth-escape from surface: ~12.5 km/s of rocket delta-v (vs theoretical 11.2 km/s).
**Atmospheric retention:**
Whether a planet keeps a gas depends on whether typical molecular thermal speeds reach escape velocity.
RMS speed of gas: v_rms = √(3kT/m).
If v_rms approaches v_esc/6 or so, gases leak away in geological time:
- **Moon (v_esc 2.4 km/s)**: lost essentially all atmosphere. - **Mars (v_esc 5 km/s)**: thin atmosphere, ongoing H₂O escape. - **Earth (v_esc 11.2 km/s)**: retains all gases except H, He. - **Jupiter (v_esc 60 km/s)**: retains H, He (primordial atmosphere).
**Gravity assists (slingshot):**
Spacecraft can effectively "borrow" speed from planets by passing through their gravity wells. Voyager 1 and 2 used Jupiter, Saturn, Uranus, Neptune assists to reach interstellar space without needing massive direct delta-v.
The planet's orbital motion adds energy to the spacecraft (and removes a microscopic amount from the planet).
**Common spacecraft destinations and delta-v:**
| Mission | Total ΔV (km/s) | |---|---| | LEO | 9.4 | | GEO | 13.5 | | Earth escape | 12.5 | | Lunar orbit | 14.5 | | Mars transfer | 15.0 | | Jupiter | 18.5 | | Pluto | 20+ |
**Hyperbolic excess velocity (v_∞):**
For interplanetary missions: v² = v_esc² + v_∞²
If launched at v_esc, the probe just barely escapes (parabolic). If at v > v_esc, has residual speed at infinity: v_∞ = √(v² − v_esc²)
Used to plan interplanetary transfer energies (C3 = v_∞²).
**Common applications:**
- **NASA mission planning** (e.g., New Horizons to Pluto: highest C3 ever). - **SpaceX Mars architecture** (Starship Heavy Lift, multiple refueling for interplanetary). - **CubeSat deployment** (need to escape parking orbit if destined for deep space). - **Cubesat deorbit** (sub-escape trajectories: dive into atmosphere).
**Software:**
- **NASA SPICE Toolkit**: orbital mechanics calculations. - **STK (Systems Tool Kit)**: industry standard for mission design. - **GMAT (General Mission Analysis Tool)**: NASA's open source. - **MATLAB Aerospace Toolbox**: educational and engineering.
**Pitfalls:**
- **Forgetting atmospheric/gravity losses**: real launches need ~10-30% more delta-v. - **Confusing escape velocity with orbital velocity**: escape ≈ 1.41 × orbit. - **Using radius vs altitude**: r is from body center. - **Ignoring rotation**: launching east near equator saves ~0.5 km/s (Earth's rotation). - **Assuming object mass matters**: it cancels out. - **Linear vs nonlinear**: escape velocity drops with √r, not linearly.
Common mistakes to avoid
- Confusing escape velocity with orbital velocity (escape is √2 times orbital).
- Forgetting to add gravity losses and atmospheric drag for real missions.
- Using altitude instead of distance from body center.
- Treating escape velocity as the rocket's burnout speed (it accelerates gradually).
- Thinking heavier objects need higher escape velocity (mass cancels).
- Confusing escape velocity with terminal velocity.
- Forgetting Earth's rotation adds ~0.5 km/s if launching east near equator.
- Applying classical formula to neutron stars or black holes (need general relativity).