When the propellant charge ignites, it burns almost instantly into a hot high-pressure gas trapped in the chamber. That gas then expands behind the projectile, converting thermal/pressure energy into kinetic energy as it pushes the projectile down the bore — while continuously losing some heat through convection to the cool barrel wall.
V(x) = V0 + A·x (gas volume grows with travel x)
P = (γ−1)·U / V (ideal-gas energy↔pressure link)
T = P·V / (mc·Rs) (equation of state, Rs≈300 J/kg·K)
dU/dt = −P·(dV/dt) − h·Awet·(T−Twall) (work done + convective loss)
m·dv/dt = P·A − friction·A (projectile acceleration)
- Propellant charge sets the initial chemical energy U0 = mc·Hc that becomes hot gas.
- Projectile mass resists acceleration for a given pressure force P·A.
- Barrel length sets how much volume the gas can expand into before the projectile exits.
- Wall heat transfer (h) controls how fast the hot gas cools via Newton's law of cooling, robbing energy that would otherwise become velocity.
This is the same energy balance that gun and artillery designers use to shape pressure-time curves: enough peak pressure for performance, without exceeding the barrel's safe working pressure, while heat loss to the barrel steel limits erosion and efficiency (thermal efficiency of a real gun is often only 25–35%).