Transcript Bottle Rocket Calculations 2013

Bottle Rocket Calculations
Example using 40 psi and 700 ml of
water
Select air pressure and water volume
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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
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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
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• 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
(Range cont.)
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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
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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