Projectile Motion Calculator
Vacuum ballistics in one card: range, flight time and apex for a launch speed and angle, with the 45° question answered live and the complement's tie shown where you can check it.
Projectile Motion Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Vacuum model — no drag, spin or wind
- Uniform gravity; angles 0–90 from the ground
In short: Launch 20 m/s at 30° from level ground: the motion splits into vx = 17.320508 m/s (constant, no drag) and vy₀ = 10.000000 m/s (decaying at g). Flight time T = 2 × 10/9.80665 = 2.039432 s, range R = vx · T = 35.324006 m, apex H = vy₀²/2g = 5.098581 m. The 45° card answers the classic question: the same arm at 45° would send it 40.788649 m — and 60° would tie your 30° range exactly, because complementary angles trade height for distance at the same rate. Raise the tee 10 m and the fall keeps working after the symmetrical part ends: T grows to 2.774496 s. All of it is vacuum arithmetic; the air is the unmodelled tax.
Formula
R = v₀²·sin(2θ)/g (flat) · T = 2v₀·sinθ/g · H = (v₀·sinθ)²/2g · complements tie: θ and 90°−θ
Vacuum ballistics is two straight-line problems wearing one curve: constant horizontal velocity, vertical free fall. Range on flat ground peaks at 45° where the sin(2θ) term tops out; from a height the optimum drops BELOW 45° because the extra fall time rewards the flatter shot.
Worked Example
- Enter the launch speed and angle; set the height of the release above the landing.
- Read range, flight time and apex — all three from the same split.
- Read the 45° card before blaming the arm: the angle may be the limiting reagent.
- Moon and Mars chips re-price the whole arc under weaker gravity.
Defaults: 35.324006 m at 30°; 45° gives 40.788649 m; 60° ties 30°. Same launch from 10 m up: 2.774496 s of flight. On the Moon the same 30° throw flies 20/1.62 × ... six times as long.
Strengths & Limits Of This Model
Where this engine is strong
- Full quadratic solved for launch height, not just flat ground
- 45° and complement cards computed from your launch
Where it stops
- No air density or drag laws
- No bounces or impacts — one arc, one landing
Practical Use Cases
Sport
throws and kicks, honestly idealised
Ballistics
first-order reach before drag models
Teaching
independence of components, made live
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Projectile Motion Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does 45° give the longest range?
Because R = v₀²·sin(2θ)/g on flat ground, and sin(2θ) peaks when 2θ = 180° — i.e. θ = 45°. Below 45° you spend too much on speed along the ground; above it, on height. The card prices both neighbours of your angle so the optimum is a number, not a slogan.
Why do 30° and 60° land the same place?
sin(2·30°) = sin(60°) and sin(2·60°) = sin(120°) = sin(60°) — the complementary angles share the range but split the flight differently: the steep one hangs, the flat one skims. The card computes the tie from your own launch so you can watch it hold.
Where is air resistance in all this?
Deliberately absent. Drag grows roughly with the SQUARE of speed and bends every one of these laws — real baseballs fly far shorter than the vacuum card. This page is the honest ideal: the ceiling the air taxes. Modelling the tax is a different page with different inputs.
Does a heavier ball fly farther?
Not here. Mass appears nowhere in the equations — gravity accelerates all masses alike, which is the same hammer-and-feather honesty as the free-fall page. Mass matters only once air and spin arrive, and this page refuses to pretend it prices those.
Why does the launch height change the range at all?
Because the vertical motion keeps falling AFTER the symmetric up-down part finishes: from a tee, the ball lands below its launch line and the extra drop time carries it forward. The page solves the full quadratic rather than the flat-ground shortcut, so a 10 m tee is a real answer, not an extrapolation.
Can I launch downward, below horizontal?
Not on this page — the angle box is the ground game, 0° to 90°. A downward launch is the same arithmetic with a negative angle, but the cards (apex above launch, the 45° tie) stop meaning what they say; honesty beat coverage.
What happens exactly at 0° or 90°?
Honest degeneracies: 0° from ground level never leaves the ground (range 0), 90° goes straight up and comes straight down (range 0, apex maximal). The page prints the zero rather than refusing — the boundary cases ARE the physics teaching.
How does this connect to the free-fall page?
The vertical half IS that page: once launched, the up-down motion obeys the same ½gt² law with a head start. Two pages, one law — the split is your reading order, not the physics's.