Free recoil

How fast, how hard and with how much energy a firearm free to move is pushed back by one shot, by the formula SAAMI publishes, and why without a powder charge the figure is a lower bound.

What it computes

The recoil answer takes one or more firearms and a load. For each firearm it returns the free recoil of one shot: the velocity the firearm is pushed back at, the energy it takes and the impulse, with the muzzle velocity the load reaches from that firearm's own barrel. When more than one firearm is compared, each is also set against the one that recoils least.

The catalogue records no powder charge. Without one the answer counts the bullet's momentum alone, and since the powder gas leaves the muzzle in the same direction, the real free recoil is larger: every figure is then a lower bound, and the answer says so. Give the load's powder charge and the gas is counted.

Parameters, defaults and the response are in the reference: GET /v1/firearms/recoil

The formula

A firearm free to move takes the momentum the bullet and the powder gas leave with. The Sporting Arms and Ammunition Manufacturers' Institute (SAAMI) publishes the formula in these three steps.

I = m_{b} v + m_{c} f v

V = \frac{I}{M}

E = \frac{1}{2} M V^{2}

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$m_{b}$: the bullet's mass in kilograms, from the load's record
$m_{c}$: the powder charge in kilograms, as the request gives it; left out when none is given
$v$: the muzzle velocity in metres per second, computed for this firearm's barrel length
$f$: SAAMI's gas velocity factor for the class of firearm, from the table below
$M$: the firearm's mass in kilograms: its recorded loaded weight, or its empty weight on request or where no loaded weight is recorded

The powder gas cannot be weighed as it leaves, so SAAMI takes its mass as the powder charge and its velocity as the muzzle velocity times a factor that depends on the class of firearm.

The muzzle velocity is the one the ballistic figures compute for this firearm's barrel from the load's reference velocity and reference barrel, not the reference velocity itself, so a short barrel's slower bullet gives a smaller recoil.

Gas velocity factors

SAAMI's factors, as its sheet prints them. SAAMI states they were derived from experiments published in the British Text Book of Small Arms (1929); Hatcher (1957) prints the same figures. The last column is the catalogue categories each class is read from when the request names none.

SAAMI class	In the request	$f$	Read from
High powered rifles	`rifle`	1.75	rifle, carbine, assault rifle, battle rifle, sniper rifle, machine gun
Shotguns (average length)	`shotgun`	1.50	shotgun
Shotguns (long barrel)	`shotgun_long_barrel`	1.25	Only when the request names it
Pistol & revolvers	`handgun`	1.50	pistol, revolver

A submachine gun or a personal defence weapon is in none of SAAMI's classes, so it has no factor unless the request names a class. With a powder charge and no class, its answer counts the bullet alone and says why.

Worked example: SAAMI's shotgun

SAAMI's own worked example, in SI units: a 7 lb shotgun firing 589.9 grains of shot and wads at 1275 ft/s over 33.4 grains of powder, at the factor for an average shotgun. Every figure below is the answer's own functions, run as the page loads.

Impulse (1)	$I = 0.03822 \times 388.6 + 0.00216 \times 1.5 \times 388.6 = 16.117 N \cdot s$
Recoil velocity (2)	$V = \frac{16.117}{3.175} = 5.076 m / s$
Recoil energy (3)	$E = \frac{1}{2} \times 3.175 \times {5.076}^{2} = 40.9 J$

That is 30.17 ft-lb, the 30.17 ft-lb SAAMI's formula gives in its own units. The sheet prints 30.22 because it rounds one factor to three places before squaring.

A carbine with no powder charge

An M4 Carbine with a 368 mm barrel, 3320 g loaded on its record, firing a 4.02 g bullet, with no powder charge given.

Muzzle velocity from this barrel	$v = 856 m / s$
Impulse (1)	$I \geq 0.00402 \times 856 = 3.442 N \cdot s$
Recoil energy (2, 3)	$V \geq 1.037 m / s, E \geq 1.78 J$

With the powder gas left out, each figure is the least the recoil can be, and the answer marks it as a lower bound. Adding a charge only raises it.

Limitations

Free recoil is the reaction of a firearm nothing holds. It is not felt recoil, which the stock, a recoil pad, the action and the shooter all change.
SAAMI's formula does not count a muzzle brake or a suppressor redirecting the gas, and the firearm's mass is the catalogue's, without any optic or other attachment.
The gas velocity factor is an average for a class of firearm. SAAMI notes that the real gas velocity varies with the muzzle pressure and the projectile's velocity.
Nothing here says how accurately or how comfortably a firearm can be shot.

Sources

The formula and its factors are quoted from these sources.

Sporting Arms and Ammunition Manufacturers' Institute (SAAMI) (2018). Gun Recoil, Technical: Free Recoil Energy. SAAMI technical sheet, Rev. 7/9/2018.https://saami.org/wp-content/uploads/2025/03/Gun-Recoil-Formulae-2018-07-9.pdf
Hatcher JS (1957). Hatcher's Notebook, second edition, chapter The Theory of Recoil. The Military Service Publishing Company.https://archive.org/details/hatchers-notebook-by-julian-s.-hatcher