Momentum. With no external horizontal force, M v = m V: the barrel group (mass M) recoils with speed v = mV/M while the ball (mass m) leaves at V. The recoil energy ½Mv² = (m/M)·½mV² is a small fraction of the ball’s energy because M ≫ m. The ledger above checks ΣP = ∫Fground dt at every step, to rounding error.
Mount. The sliding barrel is held by a spring (k) and a damper (c) against the carriage: M s̈ = −(k s + c ṡ). For a fixed carriage s̈ + 2ζω ṡ + ω² s = 0 with ω = √(k/M), ζ = c / (2√(kM)). On a free carriage, replace M by the reduced mass μ = M·Mc/(M+Mc) — the same equation holds for the relative stroke s.
Initial value problem. s(0)=0, ṡ(0)=v. For ζ<1: s = (v/ωd) e^{−ζωt} sin ωdt (the barrel swings back through battery); ζ=1: s = v t e^{−ωt}, peak v/(eω) at t = 1/ω; ζ>1: sum of two decaying exponentials, the slow one has rate ω(ζ − √(ζ²−1)).
Why ζ = 1. If the barrel must return to battery without ever passing it (it would strike the stop), ζ must be ≥ 1, and among those the return time is shortest at exactly ζ = 1; further damping only slows the creeping tail. “Back to battery” means within 2 % of the peak stroke; while the carriage is still rolling, friction holds the stroke at a small offset μMg/k, which is added to that band. Underdamped settings reach battery sooner but overshoot. If a tiny overshoot were acceptable the optimum would move slightly below 1; the sweep measures this honestly. The sweep’s criterion: overshoot ≤ 10⁻¹² of the peak stroke.
Ground. Rolling resistance on the carriage is μ(M+Mc)g (sticks when the spring force is smaller). A locked or anchored carriage lets the ground push back: the peak force equals the peak spring+damper force. Gravity only changes this friction and the ball’s fall.
Assumptions. 1-D motion, impulsive launch (muzzle speed is an input), linear spring and viscous damper, no air drag. Real hydro-pneumatic mounts use velocity-dependent orifices; this is the textbook reduction. This is a physics teaching model of momentum and damped oscillation; it contains no propellant, pressure, ammunition or targeting model.