Topic of the Month: August
2001
What is Recoil and How is it
Calculated?
The same forces that govern the performance
of rockets in space also cause a firearm to recoil when it is discharged. In the 17th
century, Sir Isaac Newton introduced the concept that for every action there is an equal
and opposite reaction. This concept is now known as Newton’s third law of motion, and
is the basis for the rearward push of a gun when it is fired or the thrust to send a
rocket streaking into the heavens. When a gun is fired the gas from the burning powder
forces the bullet out of the case and down the barrel and exerts an equal and opposite
force on the breech and stock. If the gun weighed the same as the bullet it would recoil
back just as fast as the bullet goes forward. However, the gun weighs much more than the
bullet and powder charge so the bullet is moved much faster and farther than the gun.
Three elements enter in producing recoil.
The first is the reaction to the acceleration of the bullet as it moves down the barrel
until it leaves the barrel at it’s maximum velocity, commonly referred to as muzzle
velocity. The second element is the reaction to the acceleration of the expanding gas
generated by the burning powder. The third element is the reaction of muzzle blast when
the bullet leaves the barrel and the escaping gas gives a reactive push to the muzzle. As
a general rule of thumb, the escaping gas velocity is about one and a half times the
bullet velocity. For example, the escaping gas velocity would be about 4,000 f.p.s. for a
projectile with a muzzle velocity of 2,700 f.p.s. A value of 4,000 is sufficiently
accurate for practical purposes with small arms having muzzle velocities between 1,000 and
4,200 f.p.s.
The factors that enter the recoil can be
written in terms of momentum where momentum is the product of mass and velocity:
|
Mg*Vg
= Mb*Vb + Mc*Vc
|
Where:
|
Mg,Vg
= mass and velocity of the gun
Mb,Vb = mass and velocity of the bullet
Mc,Vc = mass and velocity of the powder charge |
Based on the information above and
correcting for the proper units we can write the following:
|
(1)
I = (Wb*Vb + Wc*Vc)/225400
|
Where: |
I =
recoil impulse, lb.-sec.
Wb = bullet weight, grains
Vb = muzzle velocity of bullet, f.p.s.
Wc = weight of powder charge, grains
Vc = velocity of powder gases, f.p.s. (4,000 in these calcs.)
225400 = unit correction factor (7000 gr./lb.*32.2 f.p.s.2) |
Note: To calculate the recoil of shotguns
it is necessary to add the weight of the wads to the weight of the shot to obtain a value
for Wb.
The free recoil velocity of the gun is
easily found from the recoil impulse and weight of the gun:
|
I =
Wg/32.2*Vg or rearranging terms: (2)
Vg = 32.2*I/Wg
|
Where: |
Vg =
free-recoil velocity of the gun, f.p.s.
32.2 = acceleration of gravity, f.p.s.2
I = recoil impulse, lb.-sec.
Wg = weight of gun, lb. |
The recoil energy of the gun can then be
calculated from the kinetic energy equation:
|
E =
½*Mg*(Vg)2 or simplifying: (3) E = Wg*(Vg)2/64.4
|
Where: |
E =
free-recoil energy, ft.-lbs.
Mg = mass of the gun (Wg/32.2)
Wg = weight of gun, lb.
64.4 = acceleration of gravity*2, f.p.s.2
Vg = free-recoil velocity of the gun, f.p.s |
Recoil energy is found in three simple
steps. First find the recoil impulse (I) from equation 1, then using the recoil impulse
calculated, calculate the free-recoil velocity from equation 2. Finally plug the
calculated free-recoil velocity into equation 3 and solve for the recoil energy (E).
To illustrate these calculations, consider
the .308 NATO cartridge in a rifle weighing 8 lbs., firing a 150gr. bullet at 2800 f.p.s.
with 46 gr. of powder.
I = (150*2800 + 46*4000)/225400 (equation
1)
I = 2.68 lb.-sec.
Vg = 32.2*2.68/8 (equation 2)
Vg = 10.8 f.p.s.
E = 8*10.8*10.8/64.4 (equation 3)
E = 14.5 ft.-lbs.
To recap, the items that affect recoil are
gun weight, bullet weight, powder charge weight and muzzle velocity. Using a heavier gun,
shooting a lighter weight bullet, cutting the powder charge or reducing the muzzle
velocity or a combination of these items can reduce recoil.
A table of recoil values for some typical
firearms is given in the following Table:
RECOIL TABLE
Cartridge
(HG =
handgun) |
Bullet
Weight
(gr.) |
Charge
Weight
(gr.) |
Muzzle
Velocity
(f.p.s.) |
Gun
Weight
(lb.) |
Recoil
Impulse
(lb.-sec.) |
Recoil
Velocity
(f.p.s.) |
Recoil
Energy
(ft.-lb.) |
| .22 Hornet |
45 |
11.5 |
2690 |
7 |
.74 |
3.4 |
1.3 |
| .223 Rem. |
55 |
27 |
3240 |
7 |
1.27 |
5.8 |
3.7 |
| .243 Win. |
80 |
48 |
3350 |
8 |
2.04 |
8.2 |
8.4 |
| .30-30 Win. |
170 |
32 |
2200 |
7 |
2.23 |
10.2 |
11.4 |
| .30-06 |
180 |
56 |
2700 |
8 |
3.15 |
12.7 |
20 |
| .300 Wby. M |
180 |
85 |
3245 |
9 |
4.10 |
14.7 |
30.1 |
| .375 H&H M |
300 |
76 |
2530 |
9 |
4.72 |
16.9 |
39.8 |
| .45-70 |
405 |
30 |
1330 |
8 |
2.92 |
11.8 |
17.2 |
| .458 Win. M |
500 |
66 |
2040 |
9 |
5.70 |
20.4 |
58.1 |
| .460 Wby. M |
500 |
130 |
2700 |
10 |
8.3 |
26.7 |
110.8 |
| .22 L.R. HG |
40 |
1.1 |
800 |
.5 |
.16 |
10.3 |
0.8 |
| .357 Mag. HG |
158 |
16 |
1235 |
2.1 |
1.15 |
17.6 |
10.1 |
| .44 Mag. HG |
240 |
22.5 |
1180 |
3 |
1.66 |
17.8 |
14.7 |
In our ballistics program, Load From A
Disk, we have rated recoil energy into categories that give an indication of felt recoil
or "kick." In the table above the recoil of the .22 Hornet would be rated as
"mild" while the .460 Wby. Mag. would be rated "severe- OUCH!"
Watch our web site for the next topic of
interest "What is Ballistic Coefficient and how is it Calculated." Until then,
shoot safely and know where your bullets are going.
Sincerely,
The Ballistician
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