Arrow & Bullet Energy Calculator

The standard sporting formulas: KE = grains × fps² ÷ 450,240 (verified against published references: a 350 gr arrow at 300 fps carries ~70 ft·lb; a 150 gr .308 at 2,800 fps ~2,612 ft·lb). Energy favors speed; momentum favors mass — which is why heavy slow arrows out-penetrate light fast ones with identical KE, and why archery states set minimum draw weights rather than energy floors.

The energy-against-momentum argument, checked

Bowhunting forums argue about kinetic energy against momentum, usually with the claim that a heavy arrow beats a fast bullet on momentum even though it looks feeble on energy. I set out to show that with this page's own presets, and it is false.

The .223 has 19× the hunting arrow's energy and also 1.73× its momentum. It wins on both; the arrow is not sneaking ahead on a second metric.

Across all 15 pairings of the six presets, energy and momentum rank them the same way 14 times. The Spearman correlation between the two orderings is 0.94. As a way of separating projectiles they are nearly the same measurement.

The one pair that does flip, and why it is the only one

The exception is .223 55 gr against .45 subsonic. Energy picks .223 55 gr by 1.29×; momentum picks .45 subsonic by 1.84×.

The condition is simple once written down. Comparing a heavy slow projectile against a light fast one, call the weight ratio w and the speed ratio s. Momentum goes as w·s and energy as w·s², so the heavy one wins momentum and loses energy exactly when s falls between 1/w and 1/√w. Here w is 4.4, which puts the window at 0.23 to 0.48 — and the .45 subsonic sits inside it at 0.42. The disagreement is real, narrow, and predictable rather than a general warning.

The metric that actually inverts is sectional density

Sectional density is weight over frontal area — the standard proxy for how well something pushes through rather than how hard it arrives. Rank the six by it and the order nearly reverses:

Spearman between energy and density is -0.43, against 0.94 for energy and momentum. The two arrows are last and second-last on energy and first and second on density. That is the real "which metric" story, and it is not the one the forums have: energy and momentum are near-duplicates, and density is the one that changes the answer.

Which the arrow presets could not show you, because they left diameter blank

The diameter field is optional and only sectional density uses it. The four bullet presets fill it in; the two arrow presets used to leave it empty, so density read as a dash for exactly the two projectiles it would flatter. They now carry an assumed shaft and mark the figure with a star, because a supplied number and a measured one are not the same kind of fact. The starred figures here and in the first table assume a 0.246″ carbon shaft — a number I had to supply rather than read off the page, so it is worth testing the conclusion against it.

The arrow only falls behind the .45-70 500 gr once the shaft exceeds 0.383″. Hunting shafts run about 0.204 to 0.300, so the conclusion holds across the whole plausible range and then some — it does not depend on the diameter I picked.

Two small things the numbers give away

The two published sporting formulas are grains×fps²/450240 for energy and grains×fps/225218 for momentum. Those imply different values of gravity: 32.160 and 32.174 ft/s², a disagreement of 0.044%. The consequence is that energy divided by momentum is not exactly half the velocity as the algebra says — for the arrow it comes out 145.06 against 145.00, off by that same 0.044%. Nothing turns on it; it is a reminder that these are conventional constants with a history rather than derivations.

And the two thresholds this page cites for the same animal are 40× apart: 25 ft·lb for a bow and 1,000 for a rifle. The hunting arrow makes 65 — comfortably over the first and 7% of the second. Two accepted numbers that far apart are the clue that energy is not what is doing the work in the bow case.