{
  "id": "projectile-motion",
  "version": "2b10e270676a",
  "status": "published",
  "name": "Projectile Motion Calculator",
  "question": "Projectile motion: how far and high?",
  "summary": "Works out a projectile’s range, time of flight, maximum height and impact speed from its launch speed, angle and height, with no air resistance, and draws its path.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/projectile-motion-calculator",
  "markdown": "https://www.acalculator.org/physics/projectile-motion-calculator.md",
  "kind": "function",
  "method": "vₓ = v₀ cos θ, vᵧ = v₀ sin θ; t = (vᵧ + √(vᵧ² + 2gh₀)) ÷ g; R = vₓ t; h_max = h₀ + vᵧ² ÷ 2g; v_impact = √(v₀² + 2gh₀).",
  "assumptions": [
    "No air resistance, a flat landing level, and the same gravity everywhere along the path.",
    "The launch height is above the landing level; the launch angle is 0° to 90° above the horizontal."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "v": {
        "title": "Launch speed",
        "description": "How fast the projectile leaves the ground or the launcher.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 0.001,
        "maximum": 100000
      },
      "a": {
        "title": "Launch angle",
        "description": "The angle above the horizontal: 0° is flat, 90° is straight up.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "angle",
        "minimum": 0,
        "maximum": 1.5707963267948966
      },
      "h": {
        "title": "Launch height",
        "description": "How high above the landing level it is launched (0 for level ground).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 100000
      },
      "g": {
        "title": "Gravity (m/s²)",
        "description": "The acceleration of gravity in m/s²: 9.80665 on Earth, 9.8 in many textbooks, 1.62 on the Moon.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000
      }
    }
  },
  "outputs": {
    "range": {
      "label": "Range",
      "description": "The horizontal distance from the launch point to where it lands.",
      "format": "quantity",
      "unit": "ft"
    },
    "time": {
      "label": "Time of flight (s)",
      "description": "How long it is in the air, in seconds.",
      "format": "number"
    },
    "peak": {
      "label": "Maximum height",
      "description": "The highest point above the landing level: h₀ + (v₀ sin θ)² ÷ 2g.",
      "format": "quantity",
      "unit": "ft"
    },
    "apexTime": {
      "label": "Time to the top (s)",
      "description": "v₀ sin θ ÷ g, in seconds.",
      "format": "number"
    },
    "impact": {
      "label": "Impact speed",
      "description": "The speed when it lands: √(v₀² + 2 g h₀).",
      "format": "quantity",
      "unit": "fps"
    },
    "impactAngle": {
      "label": "Impact angle (° below horizontal)",
      "description": "How steeply it comes down when it lands, in degrees below the horizontal.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "v": "20 m/s",
      "a": "45°",
      "h": 0,
      "g": 9.80665
    },
    "outputs": {
      "range": 40.78864851911713,
      "time": 2.8841929963302353,
      "peak": 10.197162129779281,
      "apexTime": 1.4420964981651176,
      "impact": 20,
      "impactAngle": 45
    },
    "text": "It lands 133.821 ft away after 2.88419 s, reaching 33.4553 ft at the top."
  },
  "examples": [
    {
      "given": {
        "v": 70,
        "a": 1.3089969389957472,
        "h": 0,
        "g": 9.8
      },
      "expect": {
        "peak": 233.25317547305477,
        "apexTime": 6.899470187779058,
        "range": 249.99999999999997
      },
      "source": "OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion (Example 4.7: h = 233 m, t = 6.90 s to the top); hand calculation in content.mdx: R = 70² × sin 150° ÷ 9.8 = 250 m"
    },
    {
      "given": {
        "v": 20,
        "a": 0.7853981633974483,
        "h": 0,
        "g": 9.80665
      },
      "expect": {
        "range": 40.78864851911713,
        "time": 2.8841929963302353,
        "peak": 10.197162129779281
      },
      "source": "OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion; hand calculation in content.mdx: R = 20² × sin 90° ÷ 9.80665 = 40.7886 m"
    },
    {
      "given": {
        "v": 10,
        "a": 0,
        "h": 20,
        "g": 9.8
      },
      "expect": {
        "time": 2.020305089104421,
        "range": 20.203050891044214,
        "impact": 22.181073012818835
      },
      "source": "OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion; hand calculation in content.mdx: t = √(2 × 20 ÷ 9.8) = 2.02031 s; x = 10 × 2.02031 = 20.2031 m"
    },
    {
      "given": {
        "v": 30,
        "a": 1.5707963267948966,
        "h": 0,
        "g": 9.80665
      },
      "expect": {
        "range": 0,
        "time": 6.11829727786757,
        "impactAngle": 90
      },
      "source": "OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion; hand calculation in content.mdx: t = 2 × 30 ÷ 9.80665 = 6.1183 s; no horizontal speed, so the range is 0"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §4.3 Projectile Motion (equations of motion, maximum height, time of flight, range; Example 4.7). https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion (retrieved 2026-10-03)",
    "NIST SP 811, Appendix B.8 (standard acceleration of gravity 9.80665 m/s²; 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",
    "velocity",
    "acceleration"
  ],
  "changelog": []
}