Projectile Motion Calculator
Work out the range, maximum height and time of flight of a projectile launched from ground level at a chosen speed and angle. Ideal for physics students, ballistics problems and sports-motion questions.
Enter Values
How to use this calculator
- Enter the initial launch velocity v₀ in metres per second.
- Enter the launch angle in degrees (0 to 90).
- Optionally change gravity g (defaults to 9.81 m/s² for Earth) — use 1.62 for the Moon or 3.71 for Mars.
How it works
For a projectile launched from and landing at the same height, the horizontal range is R = v₀²·sin(2θ)/g, the maximum height is H = v₀²·sin²θ/(2g) and the time of flight is T = 2·v₀·sinθ/g. The angle is converted from degrees to radians before the trig functions are applied. Air resistance is ignored, so results describe ideal (vacuum) motion.
Worked example
Worked example. Launch a ball at 20 m/s at 45° under Earth gravity (9.81 m/s²): range = 40.77 m, maximum height = 10.19 m and time of flight = 2.88 s.
Common mistakes
- Using an angle above 90° or below 0° — the launch angle must sit between 0 and 90 degrees.
- Expecting the formulas to hold when the landing height differs from the launch height; these equations assume level ground.
- Forgetting that the range is maximised at 45° for a fixed speed, not at steeper angles.
Frequently asked questions
Does this account for air resistance?
No. Like most textbook projectile problems it assumes a vacuum, so real-world distances will be shorter.
Why does 45° give the greatest range?
Range depends on sin(2θ), which peaks when 2θ = 90°, i.e. θ = 45°, for a given launch speed and level ground.
Related tools
- Circular Motion Calculator
- Triangle Solver (SSS)
- Distance Formula Calculator
- Acceleration Calculator
- Sine Cosine Tangent Calculator
- Speed, Distance & Time Calculator
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Tip: Enter any known values to calculate the remaining results.
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