Start The Model Instructions Launch

#### The Equations the Simulation Uses

The forces acting on the rocket are:

1. Thrust from water leaving the rocket which acts in the direction of the rocket’s motion
2. Drag force from air which acts opposite to the direction of the rocket’s motion
3. Gravity force which acts down

#### Thrust

The air in the bottle expands so rapidly that it cools because it can not absorb heat from the surroundings.  This process is called “adiabatic expansion”.  The equation that relates the pressure of the air in the bottle to the volume of air in the bottle is:

where

• k = Cp/Cv (the ratio of heat capacity at constant pressure to heat capacity at constant volume, 1.4 for air)
• P = the pressure of air in the bottle after the launch
• Pi = the initial pressure of air in the bottle
• Vi = the initial volume of air in the bottle
• V = the volume of air in the bottle

Water is forced out the nozzle at high speed (called; “fully developed turbulent flow”).  The velocity of the exiting water is calculated from Bernoulli’s equation, which gives:

where

• Patm = the pressure of the atmosphere (101 kPa or 14.7 psi)
• vw  = the velocity of the water relative to the bottle (a vector)
• r = the density of water (1000 kg/m3 or 62.4 lbm/ft3)

The force of the expulsion of water pushes the rocket forward in the direction of the rocket is pointed

Where

• Fw = the thrust from the water
• Where Fw = the thrust from the water
• |vw| = the magnitude of the velocity
• vw = the velocity of the water ( a vector)

The powered phase of the flight continues until either all the water is pushed out of the rocket or the pressure in the bottle drops to atmospheric.  In reality, if water is left in the bottle, it will cause the flight to become unstable because it will move around.  The model also neglects the force of any air left in the bottle when the last water

#### Drag

The drag coefficient is defined as:

where

• A = the cross sectional area of the rocket (we neglect the frontal area of the fins and assume that the area is that of a 2 L bottle
• CD = the drag coefficient
• Fd = the drag force
• vr = the velocity of the rocket

The drag coefficient is related to the aspect ratio of the nose cone (ratio of diameter of the base to height).  This relationship is specified for several aspect ratios (“A Brief Intro to Fluid Mech, 2nd ed.”, Donald Young, Bruce Munson, Theodore H. Okiishi, Wiley p. 393) and was fit to the equation

where r = the ratio of diameter to height

#### Gravity

The force of gravity acts down and is given by:

where

• Fg = the force of gravity
• m = the mass of the rocket and the water in it
• g = the gravitational constant (9.8 m/sec2 or 32.2 ft/sec2)

#### How the equations are solved

The model consists of a system of ordinary differential equations which could be solved analytically.  In this case, since we want to animate the simulation, it is more convenient to simply integrate the equations numerically using Euler integration.

What is Missing from the Model?

Models are only approximations of the real thing.  Sometimes details are left out because they were thought to be unimportant or unknown.  If you fly a rocket from a launcher you notice that it can wobble and shorten the distance the rocket flies.  The program assumes the thrust is always along the direction of flight.  The p[rogram ignores wobble.  Technically, the program assumes that the center of mass is on the axis of the center of pressure.  This simplifies the calculation and tends to give the best (longest) flight.

#### Acknowledgements

Richard DiCristino took the pictures and helped develop the JAVA applet and its layout
Philo's Sky Collection provided the image of the sky
Andrew Davison provided the tree images