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Projectile Motion Calculator

Calculate an ideal projectile’s velocity components, flight time, maximum height and horizontal range.

Calculate

Use m/s, degrees, metres and m/s². Height is above the landing plane. Angles from −90° to 90° are measured from the positive horizontal direction.

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Formula and variables

vx = v₀ cos θ; vy = v₀ sin θ; t = (vy + √(vy²+2gh))/g; range = vx t

v₀ is launch speed, θ launch angle, h starting height above ground, and g a positive downward acceleration magnitude. Maximum height is measured from the landing plane.

Worked example

Launch at 10 m/s and 30° from ground level with g = 9.81 m/s². vx ≈ 8.6603 m/s and vy = 5 m/s. Flight time = 10/9.81 ≈ 1.01937 s, peak height = 25/19.62 ≈ 1.27421 m, and range ≈ 8.82799 m.

When to use this formula

Use this idealized model when gravity is constant and air resistance is negligible. Launching from a platform is supported by positive initial height; terrain above the launch point is not.

Common mistakes

The shortcut R = v₀² sin(2θ)/g only works when landing and launch heights match. A downward launch reaches its maximum future height immediately, at the launch point.

Why is flight time zero for a downward launch at ground level?

The model’s landing plane is already reached at launch. Give a positive initial height to model a projectile thrown downward from a platform.

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