Article

Projectile motion

v0 at angle α. On flat ground: Z = v0² sin(2α)/g. From a tower — use the full tool.

Part 6 of Physics without mysteries. Hub: throw comparison.

How it works

Projectile motion splits v₀ into v_x = v₀ cos α and v_y0 = v₀ sin α. On flat ground (h₀ = 0) flight time is 2v_y0/g, range Z = v₀² sin(2α)/g, and H_max = v_y0²/(2g).

Calcboxer long-tail pages deliberately separate “mostly Z” and “mostly H_max”. When you launch from a roof or want the path chart and landing components, open full projectile motion.

At α = 45° and h₀ = 0 the flat-ground maximum range is Z = v₀²/g. With v₀ = 20 m/s and g = 9.81 you get Z ≈ 40.8 m — a classic check.

Which formula when

Goal	Formula (h₀ = 0)	Tool
Range Z	Z = v₀² sin(2α)/g	projectile-range
H_max	H = v₀² sin²α/(2g)	projectile-height
Z + H + t + path	y(t)=0 root, Z=vₓ t	full-projectile-motion
h₀ > 0	quadratic root of y=0	Full model only

Solved problems

Flat 45°

Projectile on flat ground: v0 = 20 m/s, α = 45°, h0 = 0, g = 9.81 m/s². Find range Z.

Steps

On flat ground at 45°: Z = v₀²/g.
Substitute: Z = 400 / 9.81.
Z ≈ 40.77 m ≈ 40.8 m.
H_max = v₀²/(4g) ≈ 10.19 m (sin45° = √2/2).
Fill full projectile motion: v₀=20, α=45, h₀=0, g=9.81.

Answer: Z ≈ 40.8 m

Fill in the calculator

Range only (long-tail)

Same data (v0 = 20 m/s, α = 45°, g = 9.81). Use the “mostly Z” intent.