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Three school throws in one table: when to compute H_max, when Z, and when the full trajectory from a launch height.
Everything in one tool: full projectile motion. Step-by-step theory: Physics without mysteries.
In school physics a “throw” means motion under constant gravity with no air drag. A vertical throw is pure up–down: you want H_max and the rise time. A horizontal throw leaves a height h with speed only in x — flight time comes from free fall, range from vx·t.
An angled (projectile) throw mixes both components: v0 at angle α. On flat ground the classic range is Z = v02 sin(2α)/g and H_max = v02 sin2α/(2g). From a tower (h0 > 0), use the full tool instead of the Z-only long-tail page.
Galileo already split motion into components; Newton closed the F = ma story with g. On Calcboxer the three intents stay separate on purpose — plus a deep link to the full trajectory calculator.
| Model | H_max | Z (range) | t (flight) | Key formula | When to use |
|---|---|---|---|---|---|
| Vertical throw | v₀²/(2g) | — | 2v₀/g (up and back) | H = v₀²/(2g) | Ball up, fountain, jump |
| Horizontal throw | ≈ h₀ (start) | v₀·√(2h/g) | √(2h/g) | x = v₀ t, y = h − ½gt² | From edge, roof, bridge |
| Projectile (flat) | v₀² sin²α/(2g) | v₀² sin(2α)/g | 2v₀ sinα/g | Z = v₀² sin(2α)/g | Sport, α and v₀ homework |
| Full projectile | h₀ + vᵧ₀²/(2g) | vₓ·t (from y=0) | root of y=0 | h₀ + vᵧ₀ t − ½gt² = 0 | h₀ > 0, path, vₓ/vᵧ |
You throw a ball straight up at v₀ = 10 m/s. Take g = 9.81 m/s². What is the maximum height H_max?
Steps
Answer: H_max ≈ 5.10 m
