Projectile Motion Calculator

Throw something and gravity splits the problem in two. Sideways, nothing pulls on it, so it drifts at a steady speed. Downwards, it accelerates the whole time. Every number this Projectile Motion Calculator reports comes out of holding those two ideas side by side.

Give it a launch speed, an angle and the height it starts from. Back comes the range, the peak, the seconds it spends airborne, how hard it hits and at what angle, plus the arc itself drawn to scale with the apex and the landing point marked.

A Real Example: 20 m/s Off the Ground at 45°

Load Sample opens on the classic: 20 m/s at 45° from a start height of 0, Earth gravity, with the position box asking about t = 1 s.

• Range: 40.77 m, which is just v² ÷ g when the launch and landing heights match
• Apex: 10.19 m, reached at 1.441 s, exactly half of the 2.883 s flight
• Impact: 20 m/s at 45° below horizontal, the launch mirrored
• At t = 1 s: 14.14 m downrange, 9.24 m up, still moving 14.79 m/s

Press it again for a 15 m/s throw off a 25 m cliff, where that neat symmetry breaks: the fall takes longer than the climb, so apex time and half the flight time stop agreeing.

Formula: Splitting the Launch into Two Motions

The speed divides into vₓ = v cos θ and vₔ = v sin θ. Height follows y = h₀ + vₔt − ½gt², and solving that for y = 0 gives the flight time (vₔ + √(vₔ² + 2gh₀)) ÷ g, which the horizontal speed then turns into range. Gravity is a field you can type into, not just the four preset buttons, so 9.81 can become 1.62 for the Moon or anything else you need.

When Not to Use This for a Real Throw

Air resistance is not modelled. These are vacuum numbers, and reality always falls short of them: a golf ball loses roughly a third of its vacuum range, a badminton shuttle far more. Treat the output as physics homework and as an upper bound, never as ballistics. The readout most people overlook sits below the chart, in green. It names the angle that actually maximises range for the speed and height you entered, and only from ground level is that 45°. Start 25 m up at 15 m/s and the best angle drops to about 29°, because the extra drop is buying flight time a steeper climb would spend. Ask for an impossible launch and the tool says so rather than returning zero: “Launched at or below the horizontal from ground level, it never gets airborne — raise the angle above 0, or start it above the ground.”