What equation should I used for a water bottle rocket lab?

I am using distance vs volume of water. Of course, the volume of water is my independent variable. What equation should I use for this lab?

According to a Wiki article, the formula for the instantaneous thrust is

F=2PAt
where
F=thrust
P=pressure
At=area of the nozzle

Obviously the mass changes with time as well as the pressure. So both F, mass and P are functions of time, and you will need to integrate the resulting acceleration over time to get the distance.

Do read through the safety precautions before attempting this lab.

Ref:
http://en.wikipedia.org/wiki/Water_rocket

For a water bottle rocket lab where you are investigating the relationship between distance and the volume of water, you can use the equation for projectile motion to analyze your data. This equation relates the range (distance) of a projectile to its initial velocity and launch angle.

However, since your independent variable is the volume of water, you need to make some assumptions and approximations to establish a relationship between volume and initial velocity. Assuming that the pressure and water composition remains constant for each trial, the volume of water can affect the mass of the rocket and, consequently, its initial velocity.

To determine the initial velocity, you can use the ideal gas law to estimate the pressure inside your water bottle rocket as a function of the volume of water. The ideal gas law states:

PV = nRT,

where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. By rearranging the equation, you can solve for pressure:

P = (nRT)/V.

Now, by assuming that the number of moles of gas (n) and the temperature (T) remain constant, you can establish a relationship between the volume of water (V) and the pressure (P) inside the bottle.

Next, you need to consider how the pressure affects the initial velocity of the rocket. Again, making some assumptions, you can use Bernoulli's principle, which states that an increase in velocity results in a decrease in pressure. This principle can help you estimate the initial velocity of the rocket based on the pressure inside the bottle.

Finally, using the initial velocity and the launch angle, you can apply the equations of projectile motion to calculate the range (distance) of your water bottle rocket.

It's important to note that this approach involves making simplifying assumptions and approximations. To get a more accurate and precise relationship between volume and distance, you may need to conduct repeated experiments and consider other factors such as air resistance, rocket design, and launch conditions.