{
  "id": "escape-velocity",
  "version": "65fc9ec3f6ab",
  "status": "published",
  "name": "Escape Velocity Calculator",
  "question": "What is the escape velocity?",
  "summary": "Escape velocity calculator: v = √(2GM ÷ r) from the Earth, Moon, Sun, a planet or any mass and radius, at the surface or a height above it, with the orbital speed and gravity there.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/escape-velocity-calculator",
  "markdown": "https://www.acalculator.org/physics/escape-velocity-calculator.md",
  "kind": "function",
  "method": "v_esc = √(2GM ÷ r), r = R + h; circular orbit speed √(GM ÷ r); gravity GM ÷ r²; G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² for a typed mass.",
  "assumptions": [
    "The body is a sphere with no air: no drag, and no help from its spin or its motion around the Sun.",
    "Newton’s gravity: an escape velocity at or above the speed of light (a black hole) has no answer."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "body": {
        "title": "Body",
        "description": "The planet, moon or star to escape from, or your own mass and radius.",
        "type": "string",
        "enum": [
          "earth",
          "moon",
          "sun",
          "mercury",
          "venus",
          "mars",
          "jupiter",
          "saturn",
          "uranus",
          "neptune",
          "pluto",
          "custom"
        ]
      },
      "M": {
        "title": "Mass (M)",
        "description": "The mass of the body.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1e+40
      },
      "R": {
        "title": "Radius (R)",
        "description": "The radius of the body, from its centre to its surface.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.001,
        "maximum": 1000000000000000
      },
      "h": {
        "title": "Height above the surface (h)",
        "description": "Where you start: 0 at the surface.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 1000000000000000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The escape velocity to 6 significant figures, in km/s and mph.",
      "format": "text"
    },
    "escape": {
      "label": "Escape velocity (m/s)",
      "description": "v = √(2GM ÷ r), in metres per second.",
      "format": "number"
    },
    "kms": {
      "label": "Escape velocity (km/s)",
      "description": "v in kilometres per second.",
      "format": "number"
    },
    "kmh": {
      "label": "Escape velocity (km/h)",
      "description": "v in kilometres per hour.",
      "format": "number"
    },
    "mph": {
      "label": "Escape velocity (mph)",
      "description": "v in miles per hour.",
      "format": "number"
    },
    "orbit": {
      "label": "Circular orbit speed (m/s)",
      "description": "√(GM ÷ r): the speed of a circular orbit at that distance, v ÷ √2.",
      "format": "number"
    },
    "gravity": {
      "label": "Gravity there (m/s²)",
      "description": "g = GM ÷ r², the pull of the body at that distance.",
      "format": "number"
    },
    "distance": {
      "label": "Distance from the centre (km)",
      "description": "r = R + h, in kilometres.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "body": "earth",
      "M": "5.96e24 kg",
      "R": "6370 km",
      "h": "0 km"
    },
    "outputs": {
      "result": "v = 11.1861 km/s (25,022.7 mph)",
      "escape": 11186.128228778109,
      "kms": 11.18612822877811,
      "kmh": 40270.06162360119,
      "mph": 25022.65620252798,
      "orbit": 7909.787125791265,
      "gravity": 9.820224436579496,
      "distance": 6371.0084
    },
    "text": "The escape velocity is 11.1861 km/s (25,022.7 mph)."
  },
  "examples": [
    {
      "given": {
        "body": "earth",
        "h": 0
      },
      "expect": {
        "escape": 11186.128228778109,
        "mph": 25022.65620252798,
        "gravity": 9.820224436579496,
        "result": "v = 11.1861 km/s (25,022.7 mph)"
      },
      "source": "NASA JPL Solar System Dynamics: Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440), https://ssd.jpl.nasa.gov/astro_par.html; Planetary Physical Parameters (mean radii; escape velocity of the Earth 11.19 km/s, Jupiter 60.20 km/s), https://ssd.jpl.nasa.gov/planets/phys_par.html (retrieved 2026-10-19) (11.19 km/s); OpenStax, University Physics Volume 1, §13.3 Gravitational Potential Energy and Total Energy (Equation 13.6, v_esc = √(2GM ÷ R); Example 13.7: 1.12 × 10⁴ m/s from Earth with M = 5.96 × 10²⁴ kg and R = 6.37 × 10⁶ m, 4.21 × 10⁴ m/s from the Sun at 1.50 × 10¹¹ m with M = 1.99 × 10³⁰ kg), https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy (retrieved 2026-10-19); hand calculation in content.mdx: √(2 × 3.98600435507 × 10¹⁴ ÷ 6,371,008.4)"
    },
    {
      "given": {
        "body": "custom",
        "M": 5.96e+24,
        "R": 6370000,
        "h": 0
      },
      "expect": {
        "escape": 11175.610919983339
      },
      "source": "OpenStax, University Physics Volume 1, §13.3 Gravitational Potential Energy and Total Energy (Equation 13.6, v_esc = √(2GM ÷ R); Example 13.7: 1.12 × 10⁴ m/s from Earth with M = 5.96 × 10²⁴ kg and R = 6.37 × 10⁶ m, 4.21 × 10⁴ m/s from the Sun at 1.50 × 10¹¹ m with M = 1.99 × 10³⁰ kg), https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy (retrieved 2026-10-19); hand calculation in content.mdx with G = 6.67430 × 10⁻¹¹ (OpenStax: 1.12 × 10⁴ m/s with 6.67 × 10⁻¹¹)"
    },
    {
      "given": {
        "body": "custom",
        "M": 1.99e+30,
        "R": 150000000000,
        "h": 0
      },
      "expect": {
        "escape": 42082.23219681516
      },
      "source": "OpenStax, University Physics Volume 1, §13.3 Gravitational Potential Energy and Total Energy (Equation 13.6, v_esc = √(2GM ÷ R); Example 13.7: 1.12 × 10⁴ m/s from Earth with M = 5.96 × 10²⁴ kg and R = 6.37 × 10⁶ m, 4.21 × 10⁴ m/s from the Sun at 1.50 × 10¹¹ m with M = 1.99 × 10³⁰ kg), https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy (retrieved 2026-10-19); hand calculation in content.mdx: √(2 × 6.6743 × 10⁻¹¹ × 1.99 × 10³⁰ ÷ 1.5 × 10¹¹) (OpenStax: 4.21 × 10⁴ m/s)"
    },
    {
      "given": {
        "body": "moon",
        "h": 0
      },
      "expect": {
        "kms": 2.375675851683792
      },
      "source": "NASA JPL Solar System Dynamics: Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440), https://ssd.jpl.nasa.gov/astro_par.html; Planetary Physical Parameters (mean radii; escape velocity of the Earth 11.19 km/s, Jupiter 60.20 km/s), https://ssd.jpl.nasa.gov/planets/phys_par.html (retrieved 2026-10-19); hand calculation in content.mdx: √(2 × 4,902.800118 ÷ 1,737.4) km/s"
    },
    {
      "given": {
        "body": "earth",
        "h": 400000
      },
      "expect": {
        "kms": 10.85068625094122,
        "orbit": 7672.593828568173
      },
      "source": "OpenStax, University Physics Volume 1, §13.3 Gravitational Potential Energy and Total Energy (Equation 13.6, v_esc = √(2GM ÷ R); Example 13.7: 1.12 × 10⁴ m/s from Earth with M = 5.96 × 10²⁴ kg and R = 6.37 × 10⁶ m, 4.21 × 10⁴ m/s from the Sun at 1.50 × 10¹¹ m with M = 1.99 × 10³⁰ kg), https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy (retrieved 2026-10-19); OpenStax §13.4 Example 13.9 (the ISS at 400 km orbits at 7.67 × 10³ m/s); hand calculation in content.mdx"
    },
    {
      "given": {
        "body": "jupiter",
        "h": 0
      },
      "expect": {
        "kms": 60.20152106043277
      },
      "source": "NASA JPL Solar System Dynamics: Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440), https://ssd.jpl.nasa.gov/astro_par.html; Planetary Physical Parameters (mean radii; escape velocity of the Earth 11.19 km/s, Jupiter 60.20 km/s), https://ssd.jpl.nasa.gov/planets/phys_par.html (retrieved 2026-10-19) (60.20 km/s); hand calculation in content.mdx: √(2 × 126,686,531.9 ÷ 69,911) km/s"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §13.3 Gravitational Potential Energy and Total Energy: Equation 13.6, v_esc = √(2GM ÷ R); Example 13.7 (from the Earth with M = 5.96 × 10²⁴ kg and R = 6.37 × 10⁶ m: 1.12 × 10⁴ m/s; from the Sun at 1.50 × 10¹¹ m with M = 1.99 × 10³⁰ kg: 4.21 × 10⁴ m/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy",
    "OpenStax, University Physics Volume 1, §13.4 Satellite Orbits and Energy: Equation 13.7, v_orbit = √(GM ÷ r); Example 13.9 (the ISS at 400 km: 7.67 × 10³ m/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy",
    "NASA JPL Solar System Dynamics, Astrodynamic Parameters: GM of the Sun 1.32712440041279419 × 10²⁰ m³/s², of the Earth 398,600.435507 km³/s², of the Moon 4,902.800118 km³/s², of Mercury 22,031.868551 km³/s² and of Venus 324,858.592 km³/s² (DE440), retrieved 2026-10-19. https://ssd.jpl.nasa.gov/astro_par.html",
    "NASA JPL Solar System Dynamics, Planetary Satellite Physical Parameters: planet GM values (Mars 42,828.37362; Jupiter 126,686,531.9; Saturn 37,931,206.23; Uranus 5,793,951.3; Neptune 6,835,099.97; Pluto 869.3 km³/s²) and the Moon’s mean radius 1,737.4 km, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/sats/phys_par/",
    "NASA JPL Solar System Dynamics, Planetary Physical Parameters: mean radii (Mercury 2,439.4; Venus 6,051.8; Earth 6,371.0084; Mars 3,389.50; Jupiter 69,911; Saturn 58,232; Uranus 25,362; Neptune 24,622; Pluto 1,188.3 km) and escape velocities (Earth 11.19, Jupiter 60.20 km/s), retrieved 2026-10-19. https://ssd.jpl.nasa.gov/planets/phys_par.html",
    "NASA NSSDCA, Sun Fact Sheet: volumetric mean radius 695,700 km, retrieved 2026-10-19. https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html",
    "NIST, CODATA 2022: Newtonian constant of gravitation G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻², retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?bg"
  ],
  "related": [
    "orbital-period",
    "gravitational-force",
    "free-fall",
    "velocity"
  ],
  "changelog": []
}