Champagne is bottled with several grams per litre of CO₂ dissolved under pressure. Henry's law says the equilibrium pressure a dissolved gas exerts rises with temperature — colder liquid holds more CO₂ in solution at lower pressure, warmer liquid releases it more readily at higher pressure. Measurements on real bottles (Liger-Belair et al.) fit approximately:
P_bottle(T) ≈ 4.1 + 0.128·T [bar], T in °C
At 6 °C that's ≈4.9 bar; at 18 °C ≈6.4 bar; at 24 °C ≈7.2 bar — a genuinely counter-intuitive result worth noticing: a colder bottle pops with real physical justification, a slower and safer cork.
Shaking doesn't change that equilibrium, but it nucleates bubbles throughout the liquid, letting dissolved CO₂ come out of solution far faster than diffusion alone — right behind the cork this shows up as a real, if transient, pressure surge above the quiet-bottle equilibrium (modelled here as an effective multiplier on ΔP, capped, since it's a kinetic effect, not a new equilibrium).
The neck holds the cork by friction until the net outward force exceeds it. Once it lets go, the cork accelerates over the short final length of neck where the seal is already broken (effective length ℓ ≈ 5 mm) under impulse–momentum:
F = ΔP·A - F_friction (ΔP = P_bottle·shake_factor - P_atm)
v₀ = √(2·F·ℓ / m)
with A the cork's cross-section (radius ≈9.5 mm). After that the cork is a free projectile: gravity plus quadratic air drag (Cd·A·ρ·v²/2) on the way up and down — the same equations as any other launched body, just with an unusually well-defined initial velocity.