Modern .308 Winchester ammunition drives a 150-grain bullet as fast as a .270 Winchester. People hunt moose with the .270 and call the .308 “marginal.” Both can’t be true.

The gun world explains this with a folk taxonomy: this cartridge is a “deep penetrator,” that one is “explosive,” this one has “knockdown power.”

Every quantity in this deck is a mass, a length, a time, or something built from them by that law: momentum p = mv, force, pressure, density. Nothing else is invoked, and no quantity is ever squared for its own sake. Where a v² appears, it is a momentum flux and you will see it derived.

Units: grains (1 gr = 64.8 mg), inches, feet per second, and SI where the arithmetic is cleaner. The reference load throughout is a .308″ 180-grain bonded bullet arriving at 2,700 ft/s.

Note what dropped out: b/a = κρ/R. The argument of the logarithm contains neither mass nor area. Mass multiplies depth; area divides it; velocity only enters through ln(1 + u), and u ≈ 5,452 at 2,700 ft/s. The strength term is small at any hunting speed, but it is the floor of the logarithm: without it the bullet would never stop.

Solve step 4 for w as a function of depth instead of stopping at the end point, and the whole track falls out in closed form:

The force decays exponentially with a length scale m/2b ≈ 2.4″. Half of the bullet’s momentum has been handed to the medium by 3.5″, ninety percent by 11″; the first foot takes about 1.6 ms. The violence is front-loaded, which is exactly where gel shows the stretch cavity.

Write the expanded area as a disk of diameter D = k·d and the retained mass as f·m₀. Expanded sectional density factors into the three things a bullet engineer controls:

Take the logarithm of the closed form and differentiate. The elasticities, the percentage change in depth per percentage change in each input, are exact:

For u ≫ 1, ε ≈ 2/ln u: it is 0.25 at 2,000 ft/s, 0.23 at 2,700 and 0.21 at 4,000, and it keeps falling. A unit of recoil momentum spent on bullet mass buys about 4 times the depth that the same unit spent on velocity buys.

On 11 April 1986, eight agents of the Federal Bureau of Investigation cornered two heavily armed bank robbers on a Miami street. Early in the fight a 9 mm round hit Michael Platt and stopped just short of his heart. A wound that should have ended the fight didn’t: Platt kept shooting, and two agents were killed and five wounded.

The post-mortem verdict wasn’t “wrong caliber.” It was too little penetration; the bullet never reached the vitals. So the Bureau stopped arguing calibers and built one repeatable test: a duty bullet fired into calibrated 10% ordnance gelatin must stop between 12 and 18 inches. That window is the yardstick behind every gel number in this deck.

It is a defensive standard for stopping a human; big game often wants more, which is why bonded hunting loads already sit past it near 21″. Gel is a consistent yardstick, not a 1:1 model of tissue.

Inside the expansion window, two effects fight to a draw:

• faster → deeper, but only through ln(1 + u): elasticity ≈ 0.23
• faster → wider mushroom → more area → less depth, as 1/k²

They cancel. An expanding bullet is self-regulating. The chart runs the closed form with an empirical expansion ratio k(v) and retention f(v); the flat band is not drawn in, it is computed.

The bullet’s momentum at impact is mv₀ = 9.6 N·s for the reference load, and that is the most it can ever deliver to the animal, gel, or anything else: momentum is conserved, and the medium gets it all when the bullet stops inside.

The rifle delivered that same bullet momentum, plus the powder gas, into the shooter’s shoulder: about 13 N·s for a .308 Winchester. By the third law the push on the animal is never larger than the push on the shooter.

In air the strength term vanishes (air has no crush strength) and the momentum-flux term is all that is left, with a shape factor c that varies slowly with speed and the air density ρ_air:

Sectional density sets the retention length, just as it set the penetration depth. The sporting “ballistic coefficient” is σ₀ divided by a form factor: the same quantity in different clothes. For the 7 mm bullet on the next slide, L ≈ 1,883 m.

Hold σ₀ at 0.29 (enough for 24″ with margin) and the impact velocity inside the expansion window. Then bullet mass is σ₀d², and both of the things caliber still controls scale with it, linearly and together:

Every step up in caliber buys crush volume in exact proportion to recoil. There is no optimum in the mathematics; there is a budget. Most people place a 4 kg rifle well up to about 4.5 m/s of recoil velocity, and that budget spans .264 to .308 inches with 7 mm in the middle.

A .30-06 Springfield burns about 17% more powder than a .308 Winchester. At the muzzle that buys only ~4% more velocity. Momentum shows exactly where the rest goes.

• peak pressure is capped by the brass: a bigger case can’t raise it, only stretch the tail
• the bullet is already moving fast, so it crosses the late inches, where the extra powder burns, almost instantly and at falling pressure
• +17% powder adds only ~4% to the area under the curve → ~4% more bullet momentum

The convergence assumed one thing: retained mass barely moves, f ≈ 0.93–0.97. Cheap cup-and-core bullets shatter that premise.

• rapid cup-and-core designs (the Hornady SST and dozens like it) can shed 35–50% of their mass
• the lost mass becomes a snowstorm of lead: hundreds of fragments flung inches past the wound, through meat the bullet never touched
• retained m collapses, so X ∝ m/A craters: the same launch weight now penetrates far less