Yesterday’s post highlighted one gun cartridge (the .357 SIG) that, in small pistols, delivers energy disproportionate to its recoil. Today I will describe more generally the physics and practical considerations that go into optimizing a gun for a particular purpose.
The purpose of a gun is generally to project some combination of Energy, Range, and Accuracy.
Today this is done with firearms, which are subject to practical constraints on Length, Cartridge Size, Chamber Pressure, Rifling, and Recoil. Cartridge Size is a function of propellant (gun powder) capacity and projectile (bullet) size. To understand the physics that relate all these variables we will actually start with a Cartridge and work backwards, because:
A. Propellant volume puts an upper limit on the Kinetic Energy a gun can generate.
Solid propellant produces energy through burning, which converts it to a gas, which generates the pressure that accelerates the bullet within the gun barrel. Gun powder is engineered and loaded to avoid exceeding the maximum Chamber Pressure of the gun firing it. For modern rifles the Chamber Pressure limit is typically around 60kpsi. For pistols it is usually no more than 40kpsi. (Yes, that’s in thousands of pounds per square inch!)
In practice guns are under peak pressure right about the moment the bullet leaves the cartridge and enters the rifled barrel. After that, although the propellant is still burning, the bullet is accelerating down the barrel faster than the propellant can create gas to fill the space, so the chamber pressure drops off. In terms of converting the chemical energy of solid propellant to the kinetic energy of a speeding bullet:
1. Longer barrels are more efficient — i.e., they convert more propellant energy into kinetic energy — because they give the gas more time to accelerate the bullet.
2. In practice bullets don’t experience much acceleration after pressure has dropped below about 5kpsi, so the most efficient barrels are no longer than that. However, for rifle cartridges this would still be a very long barrel so to make them portable most barrels are “shorter than optimal.”
3. Shorter-than-optimal barrels “waste” propellant energy in the form of muzzle blast.
4. Waste propellant contributes to recoil if it discharges straight out the front of the barrel. This is called “rocket-effect” recoil, and it can be reduced with muzzle brakes (which redirect the force of waste propellant) or with suppressors (which capture the waste propellant and convert it to heat).
Propellant is not quite half the story. The projectile’s mass and shape has a lot to say about energy, recoil, and range:
B. Energy is proportional to mass times velocity squared.
C. Recoil is proportional to momentum, which is proportional to mass times velocity.
Note that “felt recoil” is not the same as the recoil impulse we quantify here. Felt recoil is mitigated by the mass of the gun and the contact area with the shooter, and the impulse can be further dampened by pads and other devices that spread it out over time. However, holding all these factors constant, felt recoil varies proportionally with momentum.
Since shooters generally want to minimize recoil one might expect to see an arms race for speed: After all, you can project the same amount of energy with less recoil if you drive a lighter bullet faster. Unfortunately, due to Chamber Pressure limits and the burning characteristics that can be achieved with modern gun powder, man-portable firearms can’t discharge bullets much faster than 4000fps. In fact, due to the metallurgy of bullets and barrels 3500fps is a more practical limit since above that speed barrel fouling and erosion become excessive. One more observation explains why shooters are often aiming for even lower speeds:
D. Range is proportional to projectile mass. In particular, for a given bullet diameter and profile: the heavier the bullet the further it will retain its energy. (Another way of stating this is: “Holding all else equal: the denser a bullet, the higher its ballistic coefficient.”)
While investigating the relationship between range and mass I discovered the following key invariant:
E. For a given cartridge and barrel length, muzzle energy is virtually constant across peak-pressure loads for a range of bullet masses.
F. Therefore, a rifle shooter can enhance his range by increasing bullet weight, but this comes at the expense of increased recoil.
The range of bullet weights for which this observation holds true depends on the cartridge. For example, .308 Winchester can produce the same muzzle energy with bullets ranging from 140gr all the way up to 220gr. On the upper end this is limited by the cartridge dimensions: Heavier bullets are longer and have to sit deeper in the case to fit in the chamber, which takes away volume for powder, which reduces the available energy. On the lower end this is limited by the progressivity of modern powders: Lighter bullets accelerate too easily, and no current powder can maintain as high a pressure curve behind them to extract the same energy as heavier bullets that move down the barrel more slowly. But given a particular barrel length and peak pressure, within that bullet range muzzle energy is virtually constant. For example, a 16″ barrel loaded to 55kpsi produces 2300ft-lbs of energy: If it’s a 150gr bullet it will leave the barrel at 2630fps; a 220gr bullet will shoot 2170fps. Extend the barrel to 26″ and use a slower powder and you can get 2900 ft-lbs of energy, whether from a 150gr bullet at 2950fps or a 220gr bullet at 2440fps.
Caveat: You can’t just shoot any bullet in any barrel. Barrels are rifled to impart spin to stabilize bullets:
G. Gyroscopic stability is a function of bullet diameter, weight, and length (as well as atmospheric density).
H. If a bullet doesn’t spin fast enough to stabilize it will tumble erratically instead of travelling nose-forward.
I. If a bullet is spun significantly faster than necessary to stabilize then any asymmetries or defects are exacerbated and can degrade accuracy. (However this is not considered a problem with top-grade bullets in realistic barrel twist rates.) In extreme cases bullets can be spun so fast that the centrifugal force disintegrates them.
J. Therefore, a barrel’s rifle rate must be reasonably matched to the bullet’s dimensions and muzzle velocity to ensure that it is spin stabilized on leaving the muzzle.
Since a lot of this discussion centered on rifles, one may be left wondering why handguns tend to use such uniquely stubby cartridges and large calibers. The reasons are rooted in the need for compactness. For pistols that load from a magazine in the grip, cartridges can’t be longer than the smallest grip can accommodate. (Note that revolvers, which don’t have this constraint do tend to use longer cartridges than autoloaders.) But if we prefer light, fast bullets for short-range shooting why aren’t there more extreme bottle-nosed cartridges than the .357 SIG? The problem goes back to the matter of barrel length and peak pressure. We know that short barrels are inefficient at converting powder to kinetic energy. There are only two ways to increase this efficiency: (1) Increase peak pressure, which is already at its practical limit, and (2) increase the amount of time the powder can act on the bullet. The latter can only be done by increasing the mass of the bullet so that it spends longer in the tube. So handguns necessarily stick to heavier bullets to maintain reasonable levels efficiency. Efficiency in isolation is difficult to calculate, but we can say
K. Firearm efficiency is a function of powder burn rate and “barrel time,” which is the interval between powder ignition and the bullet leaving the barrel.
L. Holding powder burn rate constant, efficiency increases with barrel time, which increases with both barrel length and bullet mass.
To illustrate these points I computed the following examples using QuickLOAD, an interior ballistics program often used by advanced reloaders:
A .308 shooting a 168gr bullet out of a 24″ barrel can attain 35% efficiency (e.g., using 40gr H322), although the maximum velocity in that configuration is obtained with 49.4gr RL-17, a slower powder producing 31% efficiency due to shorter barrel time. Obviously the slower powder keeps a higher average pressure behind the round in the barrel, but in the end it’s also going to spit more pressure out the muzzle. Extend the barrel to 30″ and that same RL-17 load reaches 34% efficiency. To take it to an extreme, in simulation a 5-foot barrel boosts bullet velocity another 300fps and brings efficiency to 42%. Note that if we step down to an absurdly light-for-caliber 90gr bullet the .308 can only achieve 29% efficiency using a wide range of intermediate-rate powders. At the upper limit, a 24″ .308 loaded with a 225gr bullet extracts 36% efficiency from RL-17, since barrel time is extended to 1.34ms (but the powder charge has to drop to 41.6gr to keep the pressure peak under 55kpsi).
9mm Luger, shooting a 125gr bullet from a 3.5″ barrel to a safe +P pressure of 35kpsi is also able to extract nearly 35% efficiency from one of the fastest powders, Red Dot. However it gains another 100fps if loaded with 5.7gr of the slightly slower Unique powder, at which point efficiency is 31%. Note that if we look at potential loads using a 90gr bullet the most efficient load, this time with 5.7gr Red Dot, is only 30% efficient. 7.7gr of Unique is 25% efficient but produces a remarkable 420 ft-lbs of muzzle energy. A popular subsonic loading using a 147gr bullet over 4.3gr Universal produces 1000fps and 326ft-lbs. Efficiency is up to 36% because of the heavier bullet and extended barrel time (.43ms), but note that energy is down and recoil is up.
Aside from its academic interest, perhaps the most practical effect of efficiency is on muzzle blast: Less efficient loads result in higher muzzle pressures. Muzzle pressure times powder mass creates both “rocket effect” recoil and potentially deafening noise. The former can be mitigated through muzzle brakes or porting. The latter can only be mitigated with suppressors.
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Just found some excellent work here on empirical measurement of recoil curves.
I just want to note that there is a mistake in B, which reads: “B. Energy equals mass times velocity squared.”
The formula for kinetic energy is in fact: half mass times velocity squared = (mv^2)/2
Otherwise, greate article!
Thanks, and good catch! I just corrected that to note the less precise “proportional” relationship, which is what I meant to focus on.