Variable mass calculation of thrust for a water rocket

[Engn_sam]

Homework Statement

I am trying to derive the formula for time varying thrust given that I know the initial mass/volume of water in a water rocket. Knowns will include initial pressure, initial volume of water, and nozzle cross sectional area.

Homework Equations

Bernoullis equation: (p/ρg) + (2V²/g) + z = c

Isentropic expansion: pv^γ = c

The Attempt at a Solution

[/B]
I derived the formula for the exit velocity of the water from the nozzle using the Bernoullis equation,
https://www.dropbox.com/s/gnatjasgb2fjx5i/Photo%202015-02-05%2C%203%2050%2055%20PM.jpg?dl=0
https://www.dropbox.com/s/gnatjasgb2fjx5i/Photo 2015-02-05, 3 50 55 PM.jpg?dl=0

I know that I am supposed to derive a differential equation for the velocity and pressure but I can't seem to setup the equation using the isentropic flow formula.

any help will be appreciated. thank you.

 

[Simon Bridge]

Welcome to PF;
this is something that is normally done by taking a small dm ejected in time dt, and working out the dp that results.