Bottle Rocket Calculations Example using 40 psi and 700 ml of water

Select air pressure and water volume Select air pressure i.e. 40 psi Select water volume i.e. 700 ml Find mass of water Mass = density x volume Vol = 700 ml = 700 cm3 or 0.000700 m3 ρ = density of water = 998 kg/m3 which is a constant Mass = 998 kg/m3 x 0.0007 m3 = 0.6986 kg

Calculate average water mass flow rate Average mass flow rate, ṁ, of water out of nozzle: ṁ = A X cd X √(2ρΔP) Find A of nozzle in m2: A = πr2 For diameter of ~21 cm = .021 m Radius = d/2 = .021 m / 2 = 0.0105 m A = π (0.0105)2 = 0.0003462 m2 Cd is given at 0.98

(water mass flow rate cont.) Find average pressure acting on the water ΔP = (Pi + Pf) / 2 or (Pi (1+Vi/Vf)) / 2, since PiVi = PfVf, so Pf = (PiVi)/Vf Pi = 40 psi Vi of air = 2 Liter – 0.7 L = 1.3 L Vf = 2 L Pf = 40 (1.3) / 2 = 26 ΔP = (40 + 26) / 2 = 33psi Convert psi to N/m2 14.7 psi = 101,353.56 N/m2, so 33 psi = 101,353.56 x 33 / 14.7 = 227528.4 N/m2 Note: N = kg m/s2

(water mass flow rate cont.) Back to calculating average mass flow rate, ṁ, of water out of nozzle: ṁ = A x cd x √(2ρΔP) ṁ = A x cd x √(2ρΔP) ṁ = 0.0003462 m2 x 0.98 x (√2 x 998 kg/m3 x 227528.4 N/m2) = 7.2302 kg/s

Water Exit Velocity & Thrust Water exit velocity V = ṁ / ρA = 7.2302 kg/s / (998 kg/m3)(.0003462 m2) = 20.926 m/s Rocket thrust ft = ṁ x V = 7.2302 kg/s x 20.926 m/s = 151.3 kg m/s2 or 151.3 N

Net Force on Rocket Net force f = ft – fd – (mave x g) F = 151.3 kg m/s2 – 0 – ((0.3 + 0.7 kg)/2) x (9.8 m/s2) = 146.4 kg m/s2 or 146.4 N Note: Mave = mass of empty rocket + mass of water selected Mass of rocket was weighted at 300 g, or 0.3 kg Water selected was 0.7 kg fd is the drag coefficient and is very low in this case

Rocket Acceleration The rocket acceleration is a result of the net force acting on the mass f = mave x a, so a = f/ mave A = (146.4 N) / ((0.3 + 0.7 kg)/2) = 292.8 m/s2

Range To find the range you need to find the amount of time it takes for the water to exit and the velocity of the rocket The time to expel the water is the mass of the water divide by the mass flow rate: t = m H2O/ ṁ = 0.7 kg / 7.2302 kg/s = 0.097 s

Velocity of the bottle Vrocket is a x t = (Range cont.) Velocity of the bottle Vrocket is a x t = (292.8 m/s2) x (0.097 s) = 28.4 m/s So the range R is V2rocket x sin 2Ө / g Ө is the launch angle which is 45° in this case. R = (28.4 m/s)2 x sin 2(45) / 9.8 m/s2 = 82.3 m

Final Range Final range is affected by drag factor Drag factor, Dc, for bottle shape is low, i.e. 0.15 Drag force D = 1- Dc = 1 – 0.15 = 0.85 Final range Rf = R x D = 82.3 m x 0.85 = 69.96 m