Bottle Rocket Calculations

1
Bottle Rocket Calculations

• Example using 40 psi and 700 ml of water

2
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
• ? density of water 998 kg/m3 which is a
  constant
• Mass 998 kg/m3 x 0.0007 m3 0.6986 kg

3
Calculate average water mass flow rate

• Average mass flow rate, ?, of water out of
  nozzle ? A X cd X v(2??P)
• Find A of nozzle in m2 A pr2
• For diameter of 21 cm .021 m
• Radius d/2 .021 m / 2 0.0105 m
• A p (0.0105)2 0.0003462 m2
• Cd is given at 0.98

4
(water mass flow rate cont.)

• Find average pressure acting on the water
• ?P (Pi Pf) / 2 or (Pi (1Vi/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
• ?P (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

5
(water mass flow rate cont.)

• Back to calculating average mass flow rate, ?, of
  water out of nozzle ? A x cd x v(2??P)
• ? A x cd x v(2??P)
• ? 0.0003462 m2 x 0.98 x (v2 x 998 kg/m3 x
  227528.4 N/m2) 7.2302 kg/s

6
Water Exit Velocity Thrust

• Water exit velocity V ? / ?A 7.2302 kg/s /
  (998 kg/m3)(.0003462 m2) 20.926 m/s
• Rocket thrust ft ? x V 7.2302 kg/s x 20.926
  m/s 151.3 kg m/s2 or 151.3 N

7
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

8
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

9
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/ ? 0.7 kg / 7.2302 kg/s 0.097 s

10
(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 2? / g
• ? 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

11
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