Physics › Projectile

Projectile motion

Throw a ball at any angle. The path is always a parabola — an algebra equation, drawn by gravity.

Projectile launcher

angle45°

speed20 m/s

Range: 40.8 m

Max height: 10.2 m

Flight time: 2.9 s

Two motions at once

A projectile (ignoring air) does two independent things:

Horizontal: constant speed. No force pushes it sideways.
Vertical: constant downward acceleration g.

Combining the two gives the parabolic arc you see above.

Useful formulas

Horizontal velocity: vₓ = v · cos(θ)
Vertical velocity: vᵧ = v · sin(θ)
Flight time: t = 2vᵧ / g
Range: R = v² · sin(2θ) / g
Max height: h = vᵧ² / (2g)

The 45° sweet spot

For a fixed speed, range is maximised at 45°. That's because sin(2θ) hits its peak (1) when 2θ = 90°. Higher angles give more height but shorter range; lower angles give range but less air time.