We continue with the Water Rocket analysis
with a summary of the four equations developed so far on **Page 1**. Note that all of this analysis
is relevant to the upwards thrust phase of the rocket flight,
while the water is being expelled through the nozzle by the compressed
air.

The basic rocket Force equation:

The thrust force in terms of the water expelled through the nozzle:

Bernoulli's equation, relating the pressure applied by the compressd air to the velocity of the exhausting water:

Note that the resulting thrust force is

twicethe pressure difference times the nozzle area:

Unfortunately the compressed air pressure P is not a constant during the thrust phase, but varies in a nonlinear manner with the expanding volume of the compressed air. This is the main reason for the extremely complex relations resulting from this analysis.

As the water escapes, the air volume increases, causing a decrease in pressure and a corresponding decrease in thrust. We consider this process to be adiabatic (no transfer of heat during the split-second expansion process), which allows us to relate the time variation of the pressure to that of the volume.

The adiabatic expansion process is derived
from the energy equation applied to an ideal gas, and is developed
in the section on **Adiabatic Expansion
Analysis** leading to the following equation:

where:

P0 is the initial absolute pressure at liftoff [Pa]

V0 is the initial volume of the compressed air [m^{}3]

k is the ratio of specific heat capacities [k = 1.4 for air]

P, V are the respective time varying pressure and volume of the compressed air during the thrust phase.

The volume variation of the compressed air due to the water escaping through the nozzle is given by:

Substituting equations 3 and 5 into equation 6 and simplifying, we obtain:

Equation 7 is the differential equation for the volume variation of the compressed air as a function of time t. It cannot be solved explicity since the volume V is deeply embedded in a nonlinear manner in the equation, hence we resort to a numerical solution.

The numerical solution of ordinary differential equations (ODEs) is an important generic problem in engineering, and you will learn various methods of solving them (such as the Runge-Kutta methods) when you study Math 344. The approach adopted by Dr. Nielsen uses an approximate numerical integration method by replacing the derivative by a first order difference method, as follows:

where:

t is the elapsed time [s], thus V(t) refers to the volume at elapsed time t

is the time step increment

and P is obtained from equation 5 as:

This leads to the following solution for V(t):

In the forthcoming exercises we will use this
method as well as a similar technique called the Trapezoidal method
for numerical integration in order to evaluate the upward velocity
and height of the rocket. This entire project will be developed
over six programming exercises, and in this first exercise we
wish to set up the basic **class** structure which will allow
us to define specific rocket objects.