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

• 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 mass flow rate cont.)

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

(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