## Water Rocket Version 0.2 06.06.2017

Construct a simulation model to calculate the height our (or your own)
bottle will achieve.

For instance, the bottle has a volume 0f 1.5 liters and weighs 40 grams,
has a 10.5 cm diameter and a 2.5 cm diameter of the opening.
Through that opening, 1 liter of water is expelled in 0.3 seconds.
The drag coefficient is thought to be 0.6.
Finally, the 5 pumps of Prof. Hoekstra each addeed 0.25 liter of air
to the bottle, giving a net pressure of about 250000 Pa (350000 inside
against 100000 ambient pressure).

Try to compute the height that the bottle will reach.
Also try to investigate the effect of the mass of the water in the bottle:
what is the optimum amount of water?

The differential equations for velocity v and altitude h are solved numerically. A "small" time step of 0.01 sec was arbitrarily chosen, and a simple forward integration scheme was implemented

v1 = v0 + a * dt

h1 = h0 + (v0 + v1) / 2 * dt

The simulation is written in Python, both version 2 and 3 work on my Linux box.
You can find the program here:

download rocket-launch.py here

Some results are shown in the table and the plots.

**Table 1. Results for different parameters**

Pressure [Pa] | Air volume [ml] | Altitude [m] | Velocity [m/s] | Jet time [s] | Jet speed [m/s] | Mass at takeoff [kg] | Mass at landing [kg] | Plot file |
---|

350000 | 400 | 13.2 | 14.1 | 0.480 | 22.4 | 1.140 | 0.140 | pressure 350 kPa, 400 ml air volume |

350000 | 500 | 18.4 | 32.7 | 0.302 | 22.4 | 1.040 | 0.040 | pressure 350 kPa, 500 ml air volume |

350000 | 600 | 20.4 | 42.0 | 0.223 | 22.4 | 0.940 | 0.040 | pressure 350 kPa, 600 ml air volume |

350000 | 750 | 21.9 | 51.4 | 0.158 | 22.4 | 0.790 | 0.040 | pressure 350 kPa, 750 ml air volume |

350000 | 825 | 22.1 | 54.0 | 0.134 | 22.4 | 0.715 | 0.040 | pressure 350 kPa, 825 ml air volume |

350000 | 900 | 22.3 | 56.2 | 0.114 | 22.4 | 0.640 | 0.040 | pressure 350 kPa, 900 ml air volume |

350000 | 1000 | 22.7 | 60.3 | 0.090 | 22.4 | 0.540 | 0.040 | pressure 350 kPa, 1000 ml air volume |

300000 | 400 | 5.2 | 8.0 | 0.400 | 20.0 | 1.140 | 0.340 | pressure 300 kPa, 400 ml air volume |

300000 | 500 | 15.3 | 20.2 | 0.496 | 20.0 | 1.040 | 0.040 | pressure 300 kPa, 500 ml air volume |

300000 | 600 | 17.5 | 32.0 | 0.273 | 20.0 | 0.940 | 0.040 | pressure 300 kPa, 600 ml air volume |

300000 | 750 | 19.6 | 41.9 | 0.184 | 20.0 | 0.790 | 0.040 | pressure 300 kPa, 750 ml air volume |

300000 | 825 | 20.0 | 45.1 | 0.155 | 20.0 | 0.715 | 0.040 | pressure 300 kPa, 825 ml air volume |

300000 | 900 | 20.6 | 48.8 | 0.130 | 20.0 | 0.640 | 0.040 | pressure 300 kPa, 900 ml air volume |

300000 | 1000 | 20.5 | 49.7 | 0.102 | 20.0 | 0.540 | 0.040 | pressure 300 kPa, 1000 ml air volume |

It can be seen that the maximum altitude is not very sensitive to the air volume - the main reason for this is that the air drag reduces the attainable altitude drastically.

For small air volumes (400 ml), there is not enough pressure to expell
all of the water,

**Comparison to the actual experiment:** looking at the video and measuring
the altitude with a ruler on the screen, I arrive at somewhat above
10 meters, but the rocket did not fly exactly vertically.
The agreement with my calculation is therefore nor very good.