{
  "id": "terminal-velocity",
  "version": "f3c275ddc57a",
  "status": "published",
  "name": "Terminal Velocity Calculator",
  "question": "What’s the terminal velocity?",
  "summary": "Computes the terminal velocity of a falling object, v = √(2mg ÷ ρCA), from its mass, frontal area and drag coefficient and the air density, with the drag force at that speed.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/terminal-velocity-calculator",
  "markdown": "https://www.acalculator.org/physics/terminal-velocity-calculator.md",
  "kind": "function",
  "method": "v_T = √(2 m g ÷ (ρ C A)), where the drag ½ C ρ A v² equals the weight m g.",
  "assumptions": [
    "Drag grows with the square of the speed (a fast, turbulent flow), with a fixed drag coefficient and frontal area.",
    "The air density is the same along the fall; real air thins with height, so falls from very high reach higher speeds up top.",
    "Exact unit sizes: 1 lb = 0.45359237 kg; 1 ft² = 0.09290304 m²; 1 in² = 0.00064516 m²; 1 lb/ft³ = 0.45359237 ÷ 0.028316846592 kg/m³; 1 mph = 0.44704 m/s."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "m": {
        "title": "Mass",
        "description": "The mass of the falling object.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "A": {
        "title": "Frontal area",
        "description": "The area the object shows to the air as it falls.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "area",
        "minimum": 1e-9,
        "maximum": 1000000
      },
      "C": {
        "title": "Drag coefficient",
        "description": "The drag coefficient C: about 0.45 for a sphere, 0.7 for a skydiver feet first, 1.0 spread-eagle.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 10
      },
      "rho": {
        "title": "Air density",
        "description": "The density of the air (or other fluid): about 1.225 kg/m³ at sea level and 15 °C.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "density",
        "minimum": 0.0001,
        "maximum": 20000
      },
      "g": {
        "title": "Gravity (m/s²)",
        "description": "The acceleration of gravity in m/s²: 9.80665 on Earth, 9.8 in many textbooks.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000
      }
    }
  },
  "outputs": {
    "speed": {
      "label": "Terminal velocity",
      "description": "The top speed of the fall, where the drag equals the weight.",
      "format": "quantity",
      "unit": "mph"
    },
    "mps": {
      "label": "Terminal velocity (m/s)",
      "description": "v_T in meters per second.",
      "format": "number"
    },
    "mph": {
      "label": "Terminal velocity (mph)",
      "description": "v_T in miles per hour.",
      "format": "number"
    },
    "kmh": {
      "label": "Terminal velocity (km/h)",
      "description": "v_T in kilometers per hour.",
      "format": "number"
    },
    "weight": {
      "label": "Drag at terminal velocity (N)",
      "description": "The drag force at that speed, equal to the weight m g, in newtons.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "m": "85 kg",
      "A": "0.7 m²",
      "C": 1,
      "rho": "1.21 kg/m³",
      "g": 9.8
    },
    "outputs": {
      "speed": 44.35022151872897,
      "mps": 44.35022151872897,
      "mph": 99.20862007589695,
      "kmh": 159.66079746742432,
      "weight": 833
    },
    "text": "The terminal velocity is 99.2086 mph (44.3502 m/s)."
  },
  "examples": [
    {
      "given": {
        "m": 85,
        "A": 0.7,
        "C": 1,
        "rho": 1.21,
        "g": 9.8
      },
      "expect": {
        "mps": 44.35022151872897
      },
      "source": "OpenStax, University Physics Volume 1, §6.4 Drag Force and Terminal Speed (F_D = ½ C ρ A v²; v_T = √(2mg ÷ ρCA); Example 6.17), https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed (Example 6.17: 44 m/s); hand calculation in content.mdx: √(2 × 85 × 9.8 ÷ (1.21 × 1.0 × 0.70)) = 44.3502 m/s"
    },
    {
      "given": {
        "m": 1.6,
        "A": 0.5,
        "C": 0.8,
        "rho": 1.25,
        "g": 10
      },
      "expect": {
        "mps": 8,
        "weight": 16
      },
      "source": "OpenStax, University Physics Volume 1, §6.4 Drag Force and Terminal Speed (F_D = ½ C ρ A v²; v_T = √(2mg ÷ ρCA); Example 6.17), https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed; hand calculation in content.mdx: √(2 × 1.6 × 10 ÷ (1.25 × 0.8 × 0.5)) = √64 = 8 m/s exactly"
    },
    {
      "given": {
        "m": 0.00045,
        "A": 0.0000314,
        "C": 0.45,
        "rho": 1.225,
        "g": 9.80665
      },
      "expect": {
        "mps": 22.580963420726523
      },
      "source": "OpenStax, University Physics Volume 1, §6.4 Drag Force and Terminal Speed (F_D = ½ C ρ A v²; v_T = √(2mg ÷ ρCA); Example 6.17), https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed (sphere C = 0.45); hand calculation in content.mdx"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §6.4 Drag Force and Terminal Speed (F_D = ½ C ρ A v²; v_T = √(2mg ÷ ρCA); Example 6.17; Table 6.2 drag coefficients). https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed (retrieved 2026-10-03)",
    "NIST SP 811, Appendix B.8 (1 lb = 0.45359237 kg; 1 ft = 0.3048 m; 1 mi = 1,609.344 m). 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": [
    "free-fall",
    "force",
    "density"
  ],
  "changelog": []
}