I've marked this as a follow-up to an answer that included a crude calculation.
To do a more accurate calculation, you might do something like this.
1. Call the mass of the bottle m, and its volume V. We want to calculate x, the fraction of the bottle's volume to fill with water. The density ρ of the water is very close to 1.0 gm/cm3
. So the initial mass is m+ρVx. Call the initial pressure p.
2. You have to figure out how much momentum each little increment of expelled water gives to the remaining rocket, whose mass is the rocket mass plus the remaining water mass. The change in momentum of the remaining rocket is equal to the speed s of the ejected mass in the frame of the rocket. I don't quite know how to calculate s because I'm not sure how it depends on the pressure. The pressure drops as the empty volume grows, approximately inversely proportional to volume. (For specialists- I'm using isothermal, although adiabatic may be closer.)
3. The speed change of the rocket is s*(ejected mass) / (remaining mass)
4. Since the pressure and the remaining mass keep changing, you have to do an integral over little increments of ejected mass to get how the rocket speed changes as the mass gets ejected. You also need to use the changing s to calculate how much mass is ejected per time, so that you can convert that to distance traveled.
So that's the outline. Even this calculation involves some approximations, such as neglecting air friction. You see why I recommended just trying a few different fill levels around the 1/2 filled value. That saves doing a tough calculation and avoids any approximations or forgotten influences in the calculation.
p.s. I tried playing around a little with doing that integral, still using some approximations. It looks like in the approximate calculations for a range of guesses about the weight of the bottle somewhere around 30% filled gives about as good a result as you can get.
(published on 05/09/11)