How this calculator works

Every formula and constant behind the numbers, with the tests they were checked against. The code itself is not published, but nothing in it is hidden: what is written here is what runs.

From muzzle to target

The calculator follows a single pellet from the muzzle. Each step feeds the next:

Diameter and material give the pellet its mass and frontal area.
Altitude and temperature give the density of the air and the speed of sound.
Air drag slows the pellet, worked out metre by metre.
The velocity at each distance gives energy, time of flight and penetration in ballistic gelatine.
Turning the last step around gives the distance at which a quantity has dropped to a chosen value.

The pellet

A pellet is treated as a perfect sphere of diameter d and density ρ:

m = ρ · π · d³ ÷ 6 A = π · d² ÷ 4

Shot sizes are the US sizes. Numbered birdshot follows the rule d (inches) = (17 − number) ÷ 100, so #6 is 0.110 in. Buckshot diameters are the standard ones, from 0.24 in for #4 buck to 0.375 in for 0000 buck [12]. Any other diameter can be typed in as a custom size.

Each material carries three numbers. Density sets the mass. The form factor multiplies the drag coefficient in flight; it stands for how round and smooth the pellet stays after leaving the barrel. The penetration factor f is used by the birdshot formula below. The form factors and penetration factors were set by the calculator's author.

The air

Air density and the speed of sound come from the International Standard Atmosphere [11]. Pressure follows from altitude h; the temperature you enter is taken as the value at sea level and lowered by 6.5 °C for every 1000 m of altitude.

p = 101 325 · (1 − 0.0065 · h ÷ 288.15)^5.2559 Pa

T = T_entered − 0.0065 · h ρ_air = p ÷ (287.05 · T) c = √(1.4 · 287.05 · T)

At sea level and 15 °C this gives 1.225 kg/m³ and 340 m/s (1116 fps). Thinner air, at altitude or in heat, means less drag and so more velocity downrange.

Drag and velocity

Drag on a sphere grows with the square of its velocity. Over one metre of flight that makes the velocity fall by a fixed fraction:

k = ρ_air · C_d · F · A ÷ (2 · m) v(x + 1 m) = v(x) · e^−k

The drag coefficient C_d is 0.47 while the pellet is slower than sound and 1.0 while it is faster; F is the form factor of the material. Measured drag of spheres runs from about 0.42 at Mach 0.3 to 0.52 at Mach 0.8, then climbs to about 0.95 at Mach 1.3 and stays near 0.98 at Mach 2 [10]. The calculator replaces that curve by two steps, which is why the velocity line shows a kink where the pellet drops below the speed of sound.

Time of flight is the sum of the time spent in each metre, and energy is the usual E = ½ · m · v². Both are for one pellet.

Gel penetration of birdshot

For birdshot, from #12 to FF, penetration in ballistic gelatine is estimated with the formula the calculator has used since its first versions:

P = C · f · m · v^1.6 ÷ d^1.2

with P in cm, the pellet mass m in grams, the velocity on arrival v in m/s and the diameter d in cm. C is for 10% gelatine and for 20% gelatine, and f is the penetration factor of the material from the table above. The formula and its constants are the author's own, set from published test results and experience in the field.

Gel penetration of buckshot

Buckshot has its own formula, new in version 4.0. It is used for the buckshot sizes and for custom diameters of 6 mm and up. The birdshot formula grows with pellet size and velocity faster than gel tests of large pellets do, so for buckshot it gave depths about three times the published ones.

The model

A pellet in gelatine is slowed by two things: drag, which grows with the square of its velocity, and the strength of the gel, which resists whatever the velocity. Written out for a sphere of density ρ_p and diameter d this gives a closed formula for the depth at which it stops:

P = 2 · ρ_p · d ÷ (3 · ρ_gel · C_d) · ln(1 + (v ÷ v_c)²)

Depth is proportional to pellet diameter and density, and grows only with the logarithm of velocity. This drag-plus-strength form goes back to Poncelet. Related models in the literature add a resistance that is linear in velocity instead of, or next to, the strength term [5] [6] [7].

The constants

The calculator uses C_d = , v_c = m/s and ρ_gel = kg/m³. The two free constants were fitted to laboratory tests with steel spheres in 10% gelatine:

225 shots with 4.5 mm steel BBs between 134 and 224 m/s, summarised by the authors as depth = 0.584 · v − 20.12 mm [3];
a 4.8 mm steel sphere at 728 m/s that stopped at 236.7 mm [4].

The FBI calibration shot, a steel BB at 590 fps that must stop at 8.5 ± 1 cm [1] [2], is in the table as well. For the BB the fit assumes a real diameter of 4.4 mm.

Plain lead flattens

A steel sphere keeps its shape. Plain lead buckshot flattens when it strikes fast, and a flattened pellet stops sooner. The calculator therefore multiplies the depth of plain lead buckshot by a factor that is 1 below m/s and falls in a straight line to at m/s and above. The factor was set on figures for standard lead buckshot at 7 and 50 yards, compiled on Fr. Frog's Pad from the International Wound Ballistics Association and other published tests [13]. It agrees with the observation, reported with those tests, that hardened or plated buckshot goes 10 to 15% deeper than plain lead [14]. Plated or hardened lead, steel and the tungsten materials are treated as spheres that keep their shape.

How it compares with buckshot tests

The table runs the calculator, flight included, for each published test. The first seven rows are the ones the plain-lead factor was set on, so their close agreement is a fit and not a test. The rows below them were not used to set anything.

Brass Fetcher reported its results the other way around, as the farthest distance at which buckshot still reached 12 in [17]. The loads in the table are assumed, because the summary does not name them.

What to make of it

Against laboratory spheres the model stays within about 6%. Against the popular buckshot tests it is within a few per cent for some and 30% off for others. Those tests rarely state whether the gel was calibrated or how fast the pellets arrived, and they disagree with each other by as much: one found about 9 in for #1 buck at 50 yards [15], another still found 12 in at 51 yards for plated #1 buck [17]. Lucky Gunner quotes 18 to 20 in for 00 buck and 15 to 18 in for #1 buck at 1,100 to 1,200 fps [16], well above the 13 to 15 in and 12 to 14 in compiled from the wound ballistics tests [13]. A fair summary is that the model follows the calibrated laboratory tests closely and sits inside the spread of the published buckshot tests. Two more things to keep in mind. Buckshot loses depth slowly with distance, so a small error in depth becomes a large error in the distance at which a given depth is reached. And the depth in 20% gelatine is simply the 10% depth divided by 1.5; it has not been checked against tests of its own.

10% and 20% gelatine

Two recipes are in common use. The FBI protocol uses 10% gelatine at 4 °C and accepts a block when a steel BB fired at 590 fps stops at 8.5 ± 1 cm [1]. Many European and NATO laboratories use 20% gelatine at 10 °C, which is stiffer. In this calculator the depth in 20% gelatine is always the depth in 10% gelatine divided by 1.5. A direct comparison with a rifle bullet found a ratio of 1.44 [8].

The choice matters when a depth is compared with a published requirement, because the requirement was set in one gel or the other. Depths for ducks and geese are quoted for both: Outdoor Life gives 1.5 in and 2.25 in in 10% gelatine [20], while on waterfowl forums the same figures are usually read against 20% gelatine. The same depth is reached at a clearly longer distance in 10% gelatine than in 20%. The calculator opens in 10%.

Depth needed per game

The game list turns a kind of game into a depth, and the calculator then shows how far each load reaches that depth. Depths are applied as listed, whichever gelatine is selected.

Birds

The bird depths are published guidelines: 1 in for doves, 1.25 in for grouse and partridge, 1.5 in for mallards, 1.75 in for pheasants, 2 in for turkeys and 2.25 in for Canada geese [18]. KPY's site repeats 1.5 in for ducks and 2.25 in for geese and gives 2 in for pheasant [19]; Outdoor Life gives 0.5 in for doves [20]. The 2.5 in for large geese is the value this calculator has used since version 3.2.

Furred game

No published gelatine requirement for foxes, coyotes, deer or wild boar was found. The depths for furred game are therefore estimates, and marked as such. They start from the FBI requirement of 12 in in 10% gelatine [1], which was set for an adult human, taken here as 80 kg, and scale it with body size:

depth = 12 in · (body mass ÷ 80 kg)^1/3 × 1.15 for wild boar

The one check available is a British field trial on foxes [21]. It found BB and AAA shot effective to 40 yards and No. 6 shot poor at 25 yards. For lead shot leaving the muzzle at 400 m/s the calculator gives in of 10% gelatine for BB (4.1 mm) at 40 yards and in for No. 6 (2.6 mm) at 25 yards; the 5 in estimate for a fox lies between the two.

Many countries and states do not allow shot for deer or wild boar at all. The list says nothing about what is legal where you hunt.

What is left out

Pattern density. All figures are for one pellet that hits; whether enough pellets hit is a matter of patterning your gun.
Feathers, hide and bone. Gelatine stands in for muscle only.
Pellets damaged in the barrel, which fly slower and wider than the round ones assumed here.
Drop and wind drift.
Loads where the muzzle velocity on the box differs from what your barrel delivers. A chronograph reading is the better input.

Sources

Hornady Law Enforcement. FBI test protocol.
Hornady (2012). Ballistic gelatin mixing procedures practiced by the FBI.
Pullen A. et al. (2022). Ballistic gelatin calibration standardisation. BMJ Military Health 168(2): 124.
Wen Y., Xu C., Wang H., Chen A., Batra R.C. (2013). Impact of steel spheres on ballistic gelatin at moderate velocities. International Journal of Impact Engineering 62: 142–151.
Sturdivan L.M. (1978). A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant. Report ARCSL-TR-78055, Chemical Systems Laboratory, Aberdeen Proving Ground.
Moxnes J.F. et al. (2016). On the study of ricochet and penetration in sand, water and gelatin by spheres, 7.62 mm APM2, and 25 mm projectiles. Defence Technology 12(2): 159–170.
Liu K., Ning J., Wu Z., Li Z., Ren H. (2017). A comparative investigation on motion model of spherical fragments penetration into gelatin. Scientia Sinica Technologica 47(10): 1081–1092.
Mabbott A., Carr D.J., Champion S. et al. (2016). Comparison of porcine thorax to gelatine blocks for wound ballistics studies. International Journal of Legal Medicine 130(5): 1353–1362.
Open dataset of 10% gelatin penetration experiments and simulation code for 4.5 and 6.35 mm caliber. Zenodo. Further sphere and pellet shots in 10% gelatine, not used in the fit.
Krumins M.V. (1972). A review of sphere drag coefficients applicable to atmospheric density sensing. Report NOLTR 72-34, Naval Ordnance Laboratory.
Wikipedia. International Standard Atmosphere, after ISO 2533:1975.
Wikipedia. Shotgun cartridge, tables of shot and buckshot sizes.
Fr. Frog's Pad. Shotgun. Buckshot penetration in 10% ordnance gelatine after the International Wound Ballistics Association.
Fr. Frog's Pad. Terminal ballistics. Wound profiles of shot and buckshot in 10% gelatine.
Lucky Gunner Lounge. What's the maximum effective range of buckshot?
Lucky Gunner Lounge. Buckshot sizes & more shotgun stuff you should know.
Isegoria (2018). Maximum effective range of buckshot. Summary of a Brass Fetcher test series in 10% gelatine.
Wakeman R. Shotshell selection basics.
KPY Shotshell Ballistics.
Charney N. (2022). Tungsten Super Shot: an inside look at how the deadliest birdshot of all time is made. Outdoor Life.
Fox N.C., Blay N., Greenwood A.G., Wise D., Potapov E. (2005). Wounding rates in shooting foxes (Vulpes vulpes). Animal Welfare 14: 93–102.

What's new in version 4.0

Version 4.0 replaces version 3.2. The flight model and the birdshot results are the same; almost everything around them is new.

Made for phone and for desktop

The page picks a phone or a desktop layout from your screen. The phone layout has four tabs and shows one chart at a time; the desktop layout shows everything at once. You can override the choice in Settings.
Nothing floats over the charts any more. You drag a distance line across the chart, and the values at that distance stand beside the chart title.
Results update as you type. The Calculate button is gone.
Light and dark appearance, and it keeps working without a connection once the page is loaded.

Lookup in both directions

Pick a distance and read velocity, penetration, energy and time of flight for both loads.
Pick a value, for example 1.5 in of penetration, and get the distance at which each load drops to it. The answer is in a table and marked on the chart.

Reach by game

A list of game, from dove to large wild boar, each with the gel depth it takes. One table shows how far each load reaches that depth.
Choose a kind of game and its depth becomes the line on the penetration chart.
Bird depths are published guidelines. Depths for furred game are estimates from body mass and are marked as estimates. The Method page gives the sources and the rule.

Buckshot

Buckshot sizes from #4 buck to 0000 buck, each with its diameter in mm and inches.
Buckshot penetration has its own formula: a sphere model fitted to laboratory tests in 10% gelatine and compared with published buckshot tests.
A choice between plain lead and plated or hardened pellets, because plain lead flattens and stops sooner.

More sizes and more output

Birdshot from #12 to FF, and any other diameter as a custom size.
Time of flight, a range table in steps of 5 or 10, pellet weight in grams and grains, and muzzle energy per pellet.
The speed of sound is drawn on the velocity chart.

Units and languages

One switch between metric and US units. Values you have typed are converted instead of kept as they were.
Altitude in feet and temperature in °F when you want them.
All six languages cover the whole calculator. The Method page and this page are in English.

Corrected

#7 shot is now 2.54 mm (0.100 in). Version 3.2 used 2.59 mm, which made #7 look slightly better than it is.
Switching from m/s to fps no longer turns 380 m/s into 380 fps.

Kept as it was

Velocity, energy and birdshot penetration give the same numbers as version 3.2 for every size except #7.
Loads you saved in version 3.2 are still there, and so is your language.

Open about the method

The Method page lists every formula and constant, the tests the buckshot model was checked against, where it agrees and where it does not, and links to the sources.