Projectile Motion Calculator

Formula and method

Resolve initial speed into horizontal and vertical components: vₓ = v₀ cos(θ) and vᵧ = v₀ sin(θ). Position at time t is x(t) = vₓt and y(t) = h₀ + vᵧt − ½gt². For an upward launch, apex time is vᵧ/g and maximum height is h₀ + vᵧ²/(2g). The positive y = 0 root gives flight time; horizontal range is vₓ multiplied by that time.

If vᵧ is zero, the highest position is the entered initial height at t = 0. At a 90° launch, horizontal range is zero in this model.

Worked example

At 20 m/s, 45°, zero initial height, and 9.80665 m/s² gravity, both velocity components are about 14.1421 m/s. The model gives about 1.4421 s to apex, 10.1972 m maximum height, 2.8842 s flight time, and 40.7886 m horizontal range. Values are rounded for reading after calculation.

Assumptions and limits

This is an ideal, two-dimensional, flat-coordinate, point-mass model with constant downward gravity and landing at y = 0. It excludes air resistance, wind, drag, lift, terrain, and changing gravity. Input angles are limited to 0–90°. Results are educational calculations, not measured trajectories, engineering or safety guidance, or real-world targeting predictions.

All entered values stay in this browser tab. This calculator does not upload them, add them to URLs, or save them to browser storage.

Common question

What does the initial height change? It changes the time available before the ideal path reaches y = 0, and therefore can change the horizontal range even when the launch speed and angle stay the same.

Further reading

OpenStax: Projectile Motion explains horizontal and vertical components and the idealized motion model. This interface and its calculations are independently implemented.

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