{
  "id": "velocity",
  "version": "0be65ab79d79",
  "status": "published",
  "name": "Velocity Calculator",
  "question": "What is the final velocity?",
  "summary": "Solves the constant-acceleration equations of motion: the final velocity, initial velocity, acceleration, time or displacement from any three of them, with the average velocity.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/velocity-calculator",
  "markdown": "https://www.acalculator.org/physics/velocity-calculator.md",
  "kind": "formulas",
  "method": "v = v₀ + a t; s = ½ (v₀ + v) t; s = v₀ t + ½ a t²; s = v t − ½ a t²; v² = v₀² + 2 a s. Average velocity = s ÷ t = (v₀ + v) ÷ 2.",
  "assumptions": [
    "The acceleration is constant over the whole time, and the motion is along one straight line; a negative value points the other way.",
    "When two answers fit (a velocity from v² = v₀² + 2as, or a time from a quadratic), the page shows both."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "u": {
        "title": "Initial velocity (v₀)",
        "description": "The velocity at the start; negative means moving the other way.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": -100000,
        "maximum": 100000
      },
      "v": {
        "title": "Final velocity (v)",
        "description": "The velocity at the end; negative means moving the other way.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": -100000,
        "maximum": 100000
      },
      "a": {
        "title": "Acceleration (m/s²)",
        "description": "The constant acceleration in metres per second squared; negative slows a forward motion.",
        "type": "number",
        "minimum": -100,
        "maximum": 100
      },
      "t": {
        "title": "Time",
        "description": "How long the motion lasts.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "exclusiveMinimum": 0,
        "maximum": 1000
      },
      "s": {
        "title": "Displacement",
        "description": "The change in position over the time; negative means it ended behind the start.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": -10000000,
        "maximum": 10000000
      }
    }
  },
  "solve": {
    "fillAny": 3,
    "variables": [
      "u",
      "v",
      "a",
      "t",
      "s"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "u": {
      "label": "Initial velocity",
      "description": "The velocity at the start.",
      "format": "quantity",
      "unit": "mps"
    },
    "v": {
      "label": "Final velocity",
      "description": "The velocity at the end of the time: v = v₀ + a × t.",
      "format": "quantity",
      "unit": "mps"
    },
    "a": {
      "label": "Acceleration (m/s²)",
      "description": "The constant acceleration in metres per second squared; negative slows a forward motion.",
      "format": "number"
    },
    "t": {
      "label": "Time",
      "description": "How long the motion lasts.",
      "format": "quantity"
    },
    "s": {
      "label": "Displacement",
      "description": "The change in position over the time; negative means it ended behind the start.",
      "format": "quantity"
    },
    "average": {
      "label": "Average velocity",
      "description": "Displacement ÷ time, which under constant acceleration is (v₀ + v) ÷ 2.",
      "format": "quantity",
      "unit": "mps"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "u": 70,
      "a": -1.5,
      "t": 40
    },
    "outputs": {
      "v": 10,
      "s": 1600,
      "average": 40
    },
    "text": "Starting at 70 m/s with an acceleration of -1.5 m/s² for 40 s, the final velocity is 10 m/s."
  },
  "examples": [
    {
      "given": {
        "u": 70,
        "a": -1.5,
        "t": 40
      },
      "expect": {
        "v": 10,
        "s": 1600,
        "average": 40
      },
      "source": "OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.7: v = 70.0 m/s + (−1.50 m/s²)(40.0 s) = 10.0 m/s); hand calculation in content.mdx: s = ½ × (70 + 10) × 40 = 1,600 m",
      "tolerance": 1e-12
    },
    {
      "given": {
        "u": 0,
        "a": 26,
        "t": 5.56
      },
      "expect": {
        "s": 401.8768,
        "v": 144.56
      },
      "source": "OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.8: x = ½ (26.0 m/s²)(5.56 s)² = 402 m); hand calculation in content.mdx: s = 13 × 30.9136 = 401.8768 m, v = 26 × 5.56 = 144.56 m/s",
      "tolerance": 1e-12
    },
    {
      "given": {
        "u": 0,
        "a": 26,
        "s": 402
      },
      "expect": {
        "v": 144.58215657542254,
        "t": 5.56085217597779
      },
      "source": "OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.9: v = √(2 × 26.0 m/s² × 402 m) = 145 m/s); hand calculation in content.mdx",
      "tolerance": 1e-12
    },
    {
      "given": {
        "u": 20,
        "v": 5,
        "s": 50
      },
      "expect": {
        "a": -3.75,
        "t": 4,
        "average": 12.5
      },
      "source": "OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Equation 3.14); hand calculation in content.mdx: a = (5² − 20²) ÷ (2 × 50) = −3.75 m/s², t = 2 × 50 ÷ (20 + 5) = 4 s",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration: Equations 3.10 to 3.14, Examples 3.7 (landing jet), 3.8 and 3.9 (dragster). https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (retrieved 2026-10-01)",
    "NIST Special Publication 811, Appendix B: 1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 nautical mile = 1,852 m exactly. https://www.nist.gov/pml/special-publication-811 (retrieved 2026-10-01)"
  ],
  "related": [
    "speed",
    "pace"
  ],
  "changelog": []
}