{
  "id": "potential-energy",
  "version": "cf6265528265",
  "status": "published",
  "name": "Potential Energy Calculator",
  "question": "What is the potential energy?",
  "summary": "Computes gravitational potential energy U = m g h from mass and height, or the elastic potential energy of a spring U = ½ k x², in joules, kilojoules and foot-pounds.",
  "category": "physics",
  "subcategory": "energy",
  "url": "https://www.acalculator.org/physics/potential-energy-calculator",
  "markdown": "https://www.acalculator.org/physics/potential-energy-calculator.md",
  "kind": "function",
  "method": "U = m × g × h (gravitational); U = ½ × k × x² (spring).",
  "assumptions": [
    "Gravitational: the height is measured from a chosen zero level, and g is the same over the whole height (near a planet’s surface).",
    "Spring: an ideal spring that follows Hooke’s law, F = k x, over the whole stretch.",
    "Exact unit sizes: 1 lb = 0.45359237 kg; 1 ft = 0.3048 m; 1 lbf = 4.4482216152605 N; 1 ft·lbf = 1.3558179483314004 J."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "kind": {
        "title": "Kind",
        "description": "Gravitational potential energy (a raised mass) or elastic potential energy (a stretched spring).",
        "type": "string",
        "enum": [
          "gravity",
          "spring"
        ]
      },
      "m": {
        "title": "Mass",
        "description": "The mass that is raised.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000
      },
      "h": {
        "title": "Height",
        "description": "How high the mass is above the level you measure from.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 10000000
      },
      "g": {
        "title": "Gravity (m/s²)",
        "description": "The acceleration of gravity in m/s²: 9.80665 on Earth, about 1.62 on the Moon.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000
      },
      "k": {
        "title": "Spring constant",
        "description": "The spring constant k: the force per unit of stretch, in the unit chosen below.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1000000000
      },
      "ku": {
        "title": "Spring constant unit",
        "description": "The unit of the spring constant: newtons per meter, newtons per millimeter, or pounds-force per inch.",
        "type": "string",
        "enum": [
          "Npm",
          "Nmm",
          "lbfin"
        ]
      },
      "x": {
        "title": "Stretch or squeeze",
        "description": "How far the spring is stretched or squeezed from its rest length.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 1000
      }
    }
  },
  "outputs": {
    "joules": {
      "label": "Potential energy (J)",
      "description": "The potential energy in joules.",
      "format": "number"
    },
    "kj": {
      "label": "Potential energy (kJ)",
      "description": "The same energy in kilojoules.",
      "format": "number"
    },
    "ftlbf": {
      "label": "Potential energy (ft·lbf)",
      "description": "The same energy in foot-pounds (1 ft·lbf = 1.3558179483314004 J).",
      "format": "number"
    },
    "speed": {
      "label": "Speed if dropped (m/s)",
      "description": "For a raised mass: the speed it would reach falling the height with no air resistance, √(2gh).",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "kind": "gravity",
      "m": "10 kg",
      "h": "5 m",
      "g": 9.80665,
      "k": 200,
      "ku": "Npm",
      "x": "10 cm"
    },
    "outputs": {
      "joules": 490.3325,
      "kj": 0.4903325,
      "ftlbf": 361.65069256049475,
      "speed": 9.902853124226372
    },
    "text": "The potential energy is 490.333 J (361.651 ft·lbf)."
  },
  "examples": [
    {
      "given": {
        "kind": "gravity",
        "m": 10,
        "h": 5,
        "g": 9.80665
      },
      "expect": {
        "joules": 490.3325,
        "kj": 0.4903325,
        "speed": 9.902853124226372
      },
      "source": "OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system; hand calculation in content.mdx: U = 10 × 9.80665 × 5 = 490.3325 J; v = √(2 × 9.80665 × 5) = 9.90285 m/s"
    },
    {
      "given": {
        "kind": "gravity",
        "m": 0.45359237,
        "h": 0.3048,
        "g": 9.80665
      },
      "expect": {
        "ftlbf": 1,
        "joules": 1.3558179483314003
      },
      "source": "OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system; hand calculation in content.mdx: 0.45359237 × 9.80665 × 0.3048 = 1.3558179483314004 J = 1 ft·lbf"
    },
    {
      "given": {
        "kind": "spring",
        "k": 200,
        "ku": "Npm",
        "x": 0.1
      },
      "expect": {
        "joules": 1
      },
      "source": "OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system; hand calculation in content.mdx: U = ½ × 200 × 0.1² = 1 J"
    },
    {
      "given": {
        "kind": "gravity",
        "m": 70,
        "h": 100,
        "g": 1.62
      },
      "expect": {
        "joules": 11340,
        "kj": 11.34
      },
      "source": "OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system; hand calculation in content.mdx: U = 70 × 1.62 × 100 = 11,340 J"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y near Earth’s surface; U = ½ k x² for a spring). https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system (retrieved 2026-10-03)",
    "NIST SP 811, Appendix B.8 Factors for Units Listed Alphabetically (foot-pound-force, pound-force, standard gravity 9.80665 m/s²). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-03)"
  ],
  "related": [
    "kinetic-energy",
    "free-fall",
    "energy"
  ],
  "changelog": []
}