Flat 45°
h₀ = 0, α = 45° → Z = v₀²/g.
Full projectile motion
One launch — range, max height, time, and path. Supports h₀ > 0 (tower launch), not only flat ground.
Theory and problem: projectile motion guide. Range or H_max alone on flat ground: keep the range and height long-tail tools.
Result
Enter values — the result shows up here.
How it works
The full tool gathers what the Z/H long-tails do not do together: launch height h₀, flight time from y = 0, velocity components, and a trajectory chart y(x).
With h₀ = 0 and α = 45°, Z = v₀²/g. With h₀ > 0, Z grows vs flat ground. At α = 0 the model reduces to a horizontal throw.
H_max = h₀ + v₀² sin²α / (2g) when the vertical launch component points up. Flight time is the positive root of h₀ + (v₀ sin α)t − ½gt² = 0.
Standalone range/height pages stay for “only Z” / “only H_max” intents. Deep-link here when you want everything.
The chart is an x–y parabola (not bar spikes) — Chart.js trajectory for Omni-level UX.
How to use
- Enter v₀ and angle α in degrees.
- Enter h₀ (≥ 0); zero = launch at landing level.
- Leave g blank for Earth or change it.
- Read Z, H_max, t, vₓ and landing vᵧ.
- Inspect the y(x) path on the chart.
Formula
vx = v0 cosα, vy0 = v0 sinα
Hmax = h0 + vy02/(2g) (when vy0 > 0)
0 = h0 + vy0 t − ½g t2 → t, then Z = vx t
Real-life examples
From a tower
v₀ = 15 m/s, α = 30°, h₀ = 10 m.
Horizontal throw
α = 0, h₀ = 8 m, v₀ = 6 m/s.
Steep angle
v₀ = 25 m/s, α = 60°, h₀ = 0.
Flat 30°
v₀ = 18 m/s, α = 30°, h₀ = 0.
Roof
v₀ = 12 m/s, α = 20°, h₀ = 15 m.
Moon
v₀ = 20 m/s, α = 45°, h₀ = 0, g = 1.62 m/s².
Small v₀
v₀ = 8 m/s, α = 40°, h₀ = 2 m.
Near vertical
v₀ = 16 m/s, α = 80°, h₀ = 0.
Ways to use this calculator
- Problems with h₀ > 0 (tower, roof, cliff).
- Compare with range-only / H_max-only tools.
- Visualise the path on a chart.
- Reduce to a horizontal throw (α = 0).
- Check Z at 45° on flat ground.
- School ballistics (no drag).
- Bridge to Eₖ / Eₚ.
- Learn vₓ, vᵧ components.
- Deep-link from Z/H long-tails.
Frequently asked questions
Different from the range calculator?
Range long-tail focuses on Z at h₀ = 0. Here: h₀, H_max, t, speeds, and path together.
And vs projectile height?
Height long-tail targets H_max. Here H_max is one of several results.
How is flight time computed?
From y(t) = 0: the positive root of h₀ + vᵧ₀ t − ½gt² = 0.
What if α = 0?
Horizontal throw: t = √(2h₀/g), Z = v₀ t when h₀ > 0.
h₀ = 0 and α = 45°?
Z = v₀²/g — the classic flat-ground result.
Air drag?
Not in v1 — vacuum parabola.
H_max at α = 0?
H_max = h₀ (no climb).
Units?
Speed and length follow the header; angle in degrees; g in acceleration units.
Why does Z grow with h₀?
Longer flight time at the same vₓ → larger range.
What does the chart show?
The y(x) path from launch to landing (y = 0).
What next?
Launch Eₖ, apex Eₚ, or the Z/H long-tails for a narrower intent.