Add water. Build pressure. Reach the sky.
Water phase: the trapped air expands p Vairγ = const (γ = 1.4) and pushes water out at ve = √(2 (p − pa) / (ρ (1 − (An/Ab)²))) (Bernoulli). Mass flow ṁ = Cd ρ An ve, thrust T = ṁ ve ≈ 2 Cd An (p − pa).
Air phase: when the water is gone the air blows down through the nozzle: choked flow while p/pa > 1.89, subsonic afterwards, T = ṁ ve + (pe − pa) An.
Flight: m dv/dt = T − ½ ρ CD A v² − m g with air density falling with height, integrated with Runge-Kutta 4. The rocket is guided on a 0.42 m rail, then flies along its velocity (stable fins).
Why an optimum fill exists: little water = little mass to throw; too much water = heavy rocket and the air runs out of pressure. Around one third water is best here.
Simplifications: no wind, no tumbling, no heat exchange with the bottle walls (the real air cools during expansion), constant drag coefficient, flat ground.
Safety: this is a virtual toy. Real pressurised bottles can burst and injure people; never build or pressurise a real rocket from the numbers here. Use only a certified kit with adult supervision.