From charge to velocity and pressure

Squeeze the whole event into your imagination and slow it down. The firing pin dents the primer; a jet of flame lights the powder; the powder — a solid — begins turning into a great deal of very hot gas in the sealed space behind the bullet. Pressure climbs from nothing to tens of thousands of pounds per square inch in about a millisecond. That pressure shoves the bullet into motion, the bullet accelerates down the bore, and as it moves it enlarges the volume behind it, which relieves the pressure. By the time the bullet reaches the muzzle the powder is mostly spent and the pressure has fallen well off its peak. Interior ballistics is the physics of that millisecond, and its job is to predict the two numbers you care about: how fast the bullet leaves, and how hard it pushed at the worst moment.

A tug-of-war that peaks early

The key intuition is a competition between two effects happening at once:

• Powder is making gas, which raises pressure.
• The bullet is moving, enlarging the volume behind it, which lowers pressure.

Early on, the powder wins easily: the bullet has barely moved, the space is small, and gas is pouring in — so pressure shoots up. But the bullet accelerates, the volume grows quickly, and soon the volume is winning. So the pressure–time curve has a characteristic shape: a sharp rise to a peak early in the bullet’s travel, then a decline as the bullet runs out the barrel. Peak pressure happens when the bullet has moved only a short distance — not at the muzzle. This is why a longer barrel adds velocity (the bullet is pushed for longer) without adding to peak pressure (which already happened, back near the chamber).

The model class: a lumped model, integrated in time

You cannot solve this with a single formula. The pressure, the burnt fraction of powder, the bullet’s position and speed are all changing together, each affecting the others, moment to moment. The standard way to handle that is a lumped-parameter (zero-dimensional) thermodynamic model integrated as a system of ordinary differential equations in time — you start at the instant of ignition and step forward through the millisecond, updating every quantity as you go.

“Lumped” means the model treats the gas behind the bullet as a single well-mixed reservoir with one pressure and one temperature, rather than resolving how conditions vary from the breech to the base of the bullet. That is a deliberate, well-established simplification — the same modeling family used across the classical interior-ballistics literature (Corner; the U.S. Army’s AMCP 706-150; the BRL IBHVG2 code). It captures the physical story faithfully while staying fast enough to run instantly as you sweep a charge.

The model rests on a few textbook relations, each carrying a distinct piece of the physics:

How fast the powder burns — Vieille’s law

Propellant burns faster at higher pressure. The linear rate at which the burning surface eats into a grain follows a power law in pressure:

$r = \beta\, P^{\,n}$

Here $r$ is the burn rate (how fast the flame front advances into the solid), $P$ is the gas pressure, and $\beta$ and $n$ are constants describing the propellant — $n$ is the pressure exponent. Combined with the grain’s geometry (a form function that tracks how much burning surface remains as a grain is consumed), this tells the model how quickly gas is being generated at every instant. Grain shape matters here: a solid sphere loses surface area as it shrinks and burns degressively, while a perforated stick can hold its surface roughly constant and burn neutrally.

What the gas is doing — the Noble–Abel equation of state

At these pressures the combustion gas is nowhere near an ideal gas — the molecules are packed tightly enough that their own size matters. Interior ballistics uses the Noble–Abel (covolume) equation of state:

$P\,(V - b\,m_{\text{gas}}) = m_{\text{gas}}\, R\, T$

$P$ is pressure, $V$ the volume behind the bullet, $T$ the gas temperature, $m_{\text{gas}}$ the mass of gas present, and $R$ the gas constant. The term $b\,m_{\text{gas}}$ — the covolume correction — subtracts the space the gas molecules themselves occupy, which the ideal-gas law ignores. It is the standard equation of state for this regime.

Where the energy goes — the energy balance

The chemical energy released by the burnt powder does not vanish; it is split between heating the gas and driving the ejecta forward:

$f\, m_{\text{burnt}} \;=\; (\text{gas thermal energy}) \;+\; \tfrac{1}{2}\,\varphi\, m_p v^2$

The left side is the energy released so far: $f$ is the propellant’s impetus (its specific energy — how much push per unit mass) and $m_{\text{burnt}}$ is the mass burnt. The right side accounts for it: some becomes heat in the gas, and the rest becomes kinetic energy of the bullet, $\tfrac{1}{2} m_p v^2$, where $m_p$ is the bullet mass and $v$ its speed. The factor $\varphi$ — the Lagrange gradient correction — accounts for the fact that the gas column itself has mass and is being flung forward along with the bullet, so a little more energy is spent than the bullet’s motion alone would suggest.

What moves the bullet — Newton’s second law

Finally, the bullet obeys $F = ma$:

$m_p \frac{dv}{dt} = A\,\big(P_{\text{base}} - P_{\text{resist}}\big)$

The bullet’s mass $m_p$ times its acceleration $\tfrac{dv}{dt}$ equals the net force: the bore cross-sectional area $A$ times the difference between the gas pressure pushing on the bullet’s base, $P_{\text{base}}$, and the resistance opposing it, $P_{\text{resist}}$. Two subtleties live in that resistance term. There is a shot-start threshold — the bullet does not move at all until pressure exceeds the grip of the case and the bullet’s engagement with the rifling — and a bore-resistance term for the friction and engraving as the bullet is forced through the lands.

What comes out, and how much to trust it

Step this system forward through the millisecond and you get muzzle velocity, the pressure–time and velocity–travel curves, barrel time, the fraction of powder burned, and how full the case is — each carried with an interval, never as a bare number. The velocity predictions land within roughly ±2.5% of published values across many cartridges and powders, and — the stronger test — a powder calibrated on one cartridge predicts a different cartridge to the same tolerance without re-tuning, which is the sign the parameters are physical rather than curve-fits.

The one thing interior ballistics cannot guess from geometry is a powder’s burn behavior — that has to be learned from data, and how well it has been learned is exactly what a powder’s grade (estimated, provisional, calibrated) tells you. A separate article covers that pipeline.

Why pressure is only ever a percentage of SAAMI maximum

This is the sharpest edge of the honesty boundary, so it is worth being blunt about it.

The model produces a modeled peak pressure, and modeled pressure is not measured pressure. Against published piezo data it lands within roughly ±10% for firearm — good enough to be genuinely useful, nowhere near good enough to certify a round. So pressure is reported one way only: as a percentage of the SAAMI Maximum Average Pressure (MAP) for the cartridge, carried with its interval.

• Under MAP is shown neutrally — never green. “Under maximum in a model” is not “safe in your firearm.” Your brass, your chamber, your lot of powder, and the temperature on the day all move real pressure in ways a generic model cannot see.
• Over MAP is a caution, never a prohibition — and the caution keys to the modeled high end of the interval, not the midpoint, because the honest question is not “is the best guess under the line” but “could it plausibly be over it.”
• The app will never tell you a charge is safe to shoot. That judgment is yours, working up from current published load data. The model genuinely cannot know your firearm — and a tool that pretended otherwise would be lying to you at the one moment it matters most.

Where the model is least certain. It is a lumped, zero-dimensional model, so it does not resolve pressure waves along the case or fine details of ignition. Peak pressure is intrinsically harder to pin down than velocity — hence the wider tolerance — and vented pistol test barrels are harder still, because the test geometry itself bleeds pressure before it can be measured. All of this is why the pressure number wears an interval and a grade, and why it stays on the honest side of a line it will not cross.