Arrow Kinetic Energy Calculator

A Two-Input Kinetic Energy Calculator for Arrows

The homepage arrow speed calculator needs five IBO inputs because it predicts speed from bow specs. This page is the opposite: you already know your arrow weight and speed (from a chronograph or the homepage tool), and you just want the kinetic energy and momentum those two numbers produce. That is a different search intent — “kinetic energy calculator”, “ft lbs energy calculator”, “KE calculator for arrows” — and it deserves a dedicated two-input tool rather than a five-field form.

The dashboard above computes both energy metrics in real time and tags your build against the widely cited bowhunting game-class floors. Switch the KE unit to joules, calories, kilocalories, BTU or erg without re-entering anything — useful when you are reading international ballistics data or comparing arrow energy to other projectile systems.

The Two Formulas — Side by Side

Kinetic energy and momentum describe the same moving arrow from two different angles. KE scales with the square of velocity, so speed dominates it. Momentum scales linearly with both mass and velocity, so a heavy slow arrow carries more momentum than a light fast arrow at the same KE.

From ½·m·v² to the 450,240 Shortcut

The kinetic energy calculator above uses the SI form KE = ½·m·v² re-arranged for archery units. The derivation: (1) start with KE in joules where m is in kg and v is in m/s; (2) convert arrow grains to slugs by dividing by 225,218 (1 slug = 225,218 grains); (3) convert fps to ft/s (already in the right unit); (4) apply the ½ factor; (5) convert joules to ft·lbf by dividing by 1.3558. Bundling all constants yields KE (ft-lbs) = (grains × v²) / 450,240. Same physics, fewer keystrokes.

How to Calculate Average Kinetic Energy

Average kinetic energy is the arithmetic mean of single-shot KE values: KE_avg = (KE₁ + KE₂ + … + KE_n) / n. Two averaging contexts matter for archers. (1) Arrow-to-arrow averaging — shoot 5–10 arrows through a chronograph at the same distance, compute KE for each shot with the calculator above, and take the mean; a standard deviation under 3% indicates a well-tuned setup with consistent arrow weight and form. (2) Distance averaging — KE decays with range because air drag bleeds speed: a 300 fps arrow loses roughly 5–10% of its KE by 30 yards and 12–18% by 50 yards. For an ethical hunting floor, average KE at your maximum effective range, not at the muzzle. This calculator reports muzzle KE — multiply by (1 − drag_loss) for downrange estimates.

Using This as an Arrow Momentum Calculator

The same two inputs (arrow weight + speed) that produce KE also produce momentum in real time, so this tool doubles as an arrow momentum calculator. The momentum output uses p (slug·ft/s) = (grains × v) / 225,218 and can be switched to newton-seconds (N·s) with the unit dropdown. Because momentum scales linearly with mass and velocity while KE scales with v², a heavy slow arrow (595 gr @ 282 fps) carries 40% more momentum than a light fast arrow (350 gr @ 340 fps) at near-identical KE — which is the technical reason heavy arrows penetrate shoulder bone when light arrows stop on it. For a deeper dive on the momentum vs KE trade-off, read the in-depth guide: How to Calculate Arrow Kinetic Energy.

Kinetic Energy Thresholds by Game Class

These are the minimum ft-lbs widely cited for ethical harvesting of each game class. They are floors, not targets — add a 15% safety margin because real KE drops 5–10% at the target from air drag, and broadhead sharpness varies. A 70 ft-lbs elk setup leaves no margin; a 75 ft-lbs setup does.

Slow & Heavy vs Fast & Light — the Trade-Off

Two builds at opposite ends of the 5-to-10 grains-per-pound range. The speed build is 60 fps faster; the heavy build is only 5 ft-lbs higher in KE — but carries 40% more momentum. That momentum gap is why heavy arrows pass through shoulder bone when light arrows stop on it.

Penetration vs KE — What an Arrow Penetration Calculator Would Tell You

There is no true arrow penetration calculator, because penetration depth is not a single-number output — it depends on broadhead geometry, tissue resistance, bone contact angle, and shaft diameter, none of which a two-input tool can know. What this page computes is the physics proxy pair a penetration model needs: KE (how much total work is available at impact) and momentum (how effectively that work keeps being delivered as the arrow decelerates). The two predict different failure points — KE governs whether the broadhead deploys and opens reliably; momentum governs how deep the arrow drives before it stops.

Two competing depth models explain why. The energy model says depth ≈ KE ÷ average resistance: the arrow stops when its kinetic energy is spent pushing against tissue, so a light fast arrow with the same KE should penetrate as deep as a heavy slow one. The momentum model says depth tracks p: under constant resistance the arrow sheds velocity linearly with time, and the integrated distance scales with momentum. On soft tissue the models roughly agree; on heavy bone they diverge — and decades of field data consistently favor the momentum model, because bone resistance spikes so hard that a low-mass, high-KE arrow sheds its speed in the first inch while a high-momentum arrow keeps driving. That is the physics behind the 8–10 gr/lb heavy builds standard for elk and dangerous game even when their muzzle KE matches a light build’s.

The trade-off, stated for penetration specifically (the build-comparison cards above frame it for build choice): KE wins deployment — mechanical broadheads need an energy threshold to open, and insufficient KE at the target leaves blades partially deployed, capping depth regardless of momentum. Momentum wins depth — through heavy bone, penetration tracks p, so momentum ≥ 0.55 slug·ft/s is the large-game band worth checking alongside the KE floor. Use the dashboard above as the penetration calculator’s front end: check KE against your game-class floor for deployment, then check momentum against the ≥ 0.55 slug·ft/s band for depth. For the related but distinct impact-force question (F = Δp/Δt — how hard the arrow strikes at the instant of impact, not how deep it goes), see the kinetic energy guide’s impact-force answer.

Bow Potential Energy & Arrow Kinetic Energy — the Energy Bow Chain

The kinetic energy number this calculator reports is the last link in what archers call the energy bow chain — the sequence of energy conversions that begins the moment you start drawing the string. Understanding each link tells you why your real arrow KE is always lower than the theoretical maximum.

The bow potential energy at step 1 is the energy stored in the limbs at full draw, calculated as PE = ½ × draw_force × draw_length where the ½ accounts for the roughly triangular force-draw curve (force ramps up linearly from 0 to peak as you draw). For a 70 lb compound bow drawn to 30 inches (2.5 ft), PE ≈ ½ × 70 × 2.5 ≈ 87.5 ft-lbs. Only about 70–85% of that PE reaches the arrow as KE — the rest is lost to limb vibration, string mass acceleration, friction in the cam system, and the audible “twang” of the shot. This is why the same 70 lb bow shooting a 350 gr arrow at 350 fps produces ~67 ft-lbs of KE, not ~87 ft-lbs.

The energy bow concept matters when you are choosing a bow or tuning a build: a more efficient cam system (or a recurve with lighter limbs and a cleaner release) extracts more arrow KE from the same stored PE. To compute the stored PE step yourself, pair this KE calculator with the Bow Draw Length Calculator to get your draw length in feet, then multiply by ½ × your peak draw weight.

Kinetic Energy Unit Conversion Table

Arrow KE is most commonly quoted in ft-lbs, but international data, scientific writing, and heating-energy analogies use other units. The dashboard converts in real time; this table is the reference for hand-converting a number you read elsewhere.

ft-lb, ft-lbs, foot pounds — All the Same Unit

Searchers type “ft lbs energy calculator”, “foot pound energy calculator”, “ft lb energy calculator”, “foot lbs energy calculator” and “ballistic energy calculator” — and they all mean the same thing: a tool that returns kinetic energy in foot-pounds. A foot-pound (symbol ft·lbf, written ft-lbs, ft lbs, foot pounds or foot pound) is the energy to lift one pound of mass through one foot against Earth's gravity, equal to ≈ 1.3558 joules. “Ballistic energy” is the broader term for the kinetic energy of any projectile (bullet, arrow, bolt), so a ballistic energy calculator for archery is identical to this arrow kinetic energy calculator.

Arrow vs Bullet Kinetic Energy — Same Calculator, Different Scale

This same two-input tool works as a bullet kinetic energy calculator with no formula change — KE = (grains × v²) / 450,240 applies to any projectile, whether it leaves a bowstring or a rifle barrel. The difference between arrows and bullets is purely one of scale and what dominates the result: arrows carry 50–100 ft-lbs of KE and rely on momentum for penetration, while bullets carry 25–50× more KE and rely on shock and expansion.

Notice that the heavy hunting arrow (595 gr) carries about the same momentum as a 9mm bullet (0.75 vs 0.59 slug·ft/s) despite having only a third of the KE — that is the physical reason arrows are lethal at low KE. To use this calculator for bullets, enter the bullet weight in grains and muzzle velocity in fps; the KE result is identical to what a dedicated bullet kinetic energy calculator would report.

Your build recipe — copy to your notes

# Arrow KE build card
arrow_weight    = 450.0 grains  (450 grain)
arrow_speed     = 300.0 fps  (300 fps)
---
kinetic_energy  = 89.95 ft-lbs  (121.96 J)
momentum        = 0.5994 slug·ft/s  (2.666 N·s)
game_class      = Dangerous Game  (Cape buffalo, African lion (fixed-blade only))
recommendation  = within ethical band — verify with a chronograph

Frequently Asked Questions

From KE Number to Ethical Build

Kinetic energy is the headline number, but momentum is what carries an arrow through bone — read both together. If your build lands below the floor for your target game, the fix is usually a heavier arrow (more momentum per fps lost) rather than a faster one. If you do not yet know your arrow speed, run the homepage arrow speed calculator first with your bow's IBO rating and real draw parameters, then bring that fps back here.

To understand the physics behind these formulas in depth, read the kinetic energy calculation guide. For the full arrow-build workflow, the Arrow Spine Chart and Bow Draw Length Calculator cover the other two numbers you need.