Water and air

In a bottle rocket the air stores the energy, and the water is what gets thrown out. Compressed air above the water pushes on it, and the water squirts out of the nozzle. When the water is gone, the little air that is left rushes out too, but air is so light that this gives only a short, weak push.

Pressure here is measured as gauge pressure: how much it is above the pressure of the air around us. In this lab the unit is the bar: 1 bar = 100 000 pascals, about the pressure of the atmosphere. The pressure difference between inside the bottle and outside is ΔP.

How fast does the water leave? Bernoulli’s principle says the pressure difference pays for the water’s motion energy:

½ ρ v² = ΔP, so v = √(2 ΔP / ρ)

where ρ is the density of water, 1000 kg/m³. The thrust is the momentum thrown out each second, which is (mass per second) × (speed). The mass leaving per second is ρ A v, where A is the area of the nozzle opening. So:

T = ρ A v × v = ρ A v² = 2 ΔP × A

A worked example with simulated numbers: a nozzle 22 mm across has an area A = π × (0.011 m)² ≈ 3.8 × 10⁻⁴ m². At a simulated gauge pressure of 4 bar (4 × 10⁵ Pa), the speed is √(2 × 4 × 10⁵ ÷ 1000) ≈ 28 m/s and the thrust is 2 × 4 × 10⁵ × 3.8 × 10⁻⁴ ≈ 300 N. This is the thrust at the first instant. It drops as the air expands and the pressure falls.

Two rules follow from the formulas. Doubling the pressure multiplies the exit speed by √2 ≈ 1.41, not by 2. Doubling the nozzle area at the same pressure doubles the thrust, but the water also leaves twice as fast in volume, so the pressure drops sooner.

Now the real question: how much of the bottle should be water? It is a trade-off.

• Very little water. The water is gone almost at once. There is not much mass to throw, so little momentum is given to the bottle.
• A lot of water. The bottle is heavy, so the same thrust gives less acceleration. There is also little air left, and a small air volume loses pressure quickly as the water leaves, so the push fades early.
• In between. There is enough water to carry momentum and enough air to keep the pressure up for a while.

A simple model shows this clearly. The model used a 2-litre bottle, a 22 mm nozzle, an empty mass of 0.1 kg, a simulated pressure of 4 bar and air drag, and it ignored the short air-only push at the end. The height reached, compared with the best fill, came out roughly like this:

• 10 percent water: about two thirds of the best height.
• About a third to 40 percent water: the best height.
• 60 percent water: about 80 percent of the best.
• 70 percent water: less than half.
• 90 percent water: it barely leaves the launcher.

This is a result of a simple simulation, not a law of nature. A real bottle, nozzle and pressure move the best fill up or down a little. The lesson is the shape of the curve: too little and too much are both bad, and the best is in the middle.