The two formulas, and where the odd constants come from
Both of these are ordinary physics wearing archery units. Kinetic energy is half the mass times the velocity squared, and momentum is mass times velocity. The awkward divisors exist because arrows are weighed in grains and clocked in feet per second, and neither of those is a coherent unit for the job.
Kinetic energy in foot-pounds = grains × fps² ÷ 450,240
Momentum in slug-feet per second = grains × fps ÷ 225,218
The 450,240 is 7,000 grains per pound multiplied by 32.174 feet per second squared, then doubled to absorb the half in the formula. The 225,218 is the same 7,000 times 32.174 without the doubling. Work the first one through by hand: a 400 grain arrow at 280 fps gives 400 × 78,400 ÷ 450,240, which is 31,360,000 ÷ 450,240, which is 69.65 foot-pounds. Its momentum is 400 × 280 ÷ 225,218, or 0.497 slug-feet per second. Converting the same arrow into metric and running it as half m v squared gives 94.4 joules, which is 69.6 foot-pounds, so the shortcut and the long way agree.
Where a heavy arrow and a light one part company
Put a 400 grain arrow at 280 fps next to a 350 grain arrow at 300 fps. On energy they are 69.65 and 69.96 foot-pounds, a difference of less than half a percent, which is inside the spread you would see between individual shots. On momentum they are 0.4973 and 0.4662, and the lighter arrow is more than six percent behind.
The reason is in the exponents. Energy counts speed twice, so the light arrow gets to apply its speed advantage twice over and very nearly makes up the mass it is missing. Momentum counts speed once, so the same advantage is applied once and it does not close the gap. Mass and speed are interchangeable for momentum, and they are not interchangeable for energy.
Push it further and it gets starker. A 540 grain arrow at 240 fps also lands near 69 foot-pounds, matching both of the others on energy, while carrying 0.575 slug-feet per second of momentum, which is about fifteen percent more than the 350 grain arrow. Three arrows, one energy figure, three quite different momentum figures.
What it costs to go heavier
A common workshop figure is that adding roughly three grains to an arrow costs about one foot per second. It varies with the bow, and if it matters to you the honest way to get the number is to chronograph two arrows of different weights off your own bow. Used as a rough guide, though, it shows the trade cleanly.
| Arrow | Speed | Kinetic energy | Momentum |
|---|---|---|---|
| 400 gr | 280 fps | 69.65 ft-lb | 0.4973 |
| 450 gr | 263.3 fps | 69.31 ft-lb | 0.5262 |
| 350 gr | 296.7 fps | 68.36 ft-lb | 0.4610 |
Fifty grains either way barely moves the energy column, which is the surprise for most people. It moves momentum by about six percent each way, and it moves the trajectory by much more than either. The real price of the heavier arrow is not paid in these two columns at all — it is paid in drop, and it shows up as every sight mark past the middle distances moving, which you can watch happen on the sight tape calculator.
What these numbers do not tell you
Both figures describe the arrow at one instant, as it leaves the string. From then on drag is removing speed from it, and drag does not treat all arrows alike: a light fast arrow sheds a larger share of its speed over the same flight than a heavy one does, so the gap between two setups at forty yards is not the gap at the chronograph.
They also say nothing at all about whether an arrow shoots well. Grouping is set by consistency of the shot, by spine matching, by clearance and by the archer, and an arrow can top the table on both counts and still land somewhere unhelpful. If you are choosing between two builds, work these out because they are cheap to work out, then decide the question on the target face. The mass side of the input comes from the arrow weight and FOC calculator, and the shaft that carries it comes out of the spine band.
Questions people ask
What is the formula for arrow kinetic energy?
Kinetic energy in foot-pounds is the arrow weight in grains multiplied by the speed in feet per second squared, divided by 450,240. A 400 grain arrow at 280 fps gives 400 times 78,400, which is 31,360,000, divided by 450,240, which is 69.65 foot-pounds. The constant is 7,000 grains per pound times 32.174 feet per second squared times two, and it exists purely so that grains and feet per second can go straight into a formula that was written for slugs. Converting the same arrow into metric and using half m v squared returns 94.4 joules, which is the same answer.
How do I calculate arrow momentum?
Momentum in slug-feet per second is arrow weight in grains times speed in feet per second, divided by 225,218. The same 400 grain arrow at 280 fps gives 112,000 divided by 225,218, or 0.497 slug-feet per second. The constant is 7,000 times 32.174, the same as the energy divisor but without the doubling, because momentum has no half in it and speed appears only once. Momentum is quoted to three or four decimal places because the numbers are small; the differences between setups show up in the third decimal, not the first.
Which matters more, kinetic energy or momentum?
Neither is more correct; they answer different questions and the useful move is to look at both. Energy is the work the arrow can do, and because it counts speed twice it rewards a fast arrow disproportionately. Momentum is how hard the arrow is to stop or deflect, and because it counts mass and speed equally it rewards a heavy arrow more. For target shooting the practical significance of both is modest next to trajectory, wind drift and consistency, and the main reason to calculate them is to understand what your last equipment change actually did.
Why can a heavy slow arrow and a light fast one have the same energy?
Because energy is proportional to the square of speed, a small speed advantage counts twice for the lighter arrow and can offset a much larger mass deficit. A 400 grain arrow at 280 fps and a 350 grain arrow at 300 fps are within half a percent of each other on energy. Momentum is proportional to speed only once, so the same trade does not balance there and the heavier arrow comes out more than six percent ahead. If two setups look identical, you are almost certainly looking at only one of the two numbers.
Can I use the speed printed on my bow specification?
Not for this. Published bow speeds are produced under a defined test setup with a specific arrow weight, draw length and draw weight, and every one of those is probably different from yours. Draw length alone changes the figure substantially. What you need is a chronograph reading of your arrow off your bow, which is a five minute job at most clubs and pro shops. Feed a specification number into these formulas and you get an answer that is confidently wrong, and because energy squares the speed, a ten percent speed error becomes a twenty-one percent energy error.