1. The Physics of Bottle Rocketry
Rocket Science
The Physics of Bottle Rocketry

2. Variables to be used: P = pressure (60 psi, 414 kPa) V = volume
v = velocity
m = mass
M* = mass flow rate
a = acceleration
t = time
F = force
Cd = Coefficient of Drag
ρ = density

3. Mass Flow Rate
We’ll begin by finding the mass flow rate of the water as it exits the nozzle. Mass flow rate is literally just that – the rate at which a mass of fluid moves through an opening.
The equation for mass flow rate is:
M * = A x Cd x √(2ρΔP)
First, find the area of the nozzle (in meters-squared).
Find the diameter of the nozzle (it’s 21 millimeters)
Use that to find the radius of the nozzle (it’s m)
The Coefficient of Drag for the exiting water is a constant .98. This was determined experimentally.

4. Average Pressure
Now, find the average pressure acting on the water. Why isn’t the pressure set at a constant 60 psi? Well, as the water exits, the pressure acting on it changes! That’s why we’ll need to find ΔP.
ΔP = .5(Pi + Pf).
Pi = 60 psi, or 414 kPa.
We’ll need SI units for this: 1 Pascal = 6895 psi = 1 N/m2!
Pressure FORCE/AREA…that’s why the unit is Newtons per meters, squared.
Remember Boyle’s Law? PiVi = PfVf, so to find the final pressure in the bottle:
Pf = (PiVi)/Vf
But, what about the volume of the bottle? How will we ever find that?
Well, it’s a 2-Liter bottle, and you put a measured amount of water in. The final volume is 2.0 liters, and the initial volume is (volume of water – 2.0 L).
Use the above info to calculate the average pressure in the bottle, in Newtons per Meters, squared.

5. Mass Flow Rate We now resume calculating the flow rate…
Plug in your values to the flow rate equation. Take a moment to remember the density of water…
Determine the average flow rate, in units of mass/time.

6. Exit Velocity & Thrust
Next up is the exit velocity of the water. It’s not that difficult!
v = M * /(ρA)
Simple, right?
Then find the thrust of the rocket – the force propelling the rocket upward.
Ft = M* x v

7. NET FORCE
Moving on along, we’ll need the net force acting on the rocket. Draw a free-body diagram of the rocket, label all forces, and set up an expression to find the net force acting on the rocket.
Use your expression, along with Newton’s 2nd Law, to find the average acceleration of the rocket during the propulsion stage. Keep in mind that the mass of the rocket will change! You need to use the average mass: .5(mi + mf), where the initial mass is the mass of the bottle PLUS the water, and the final mass is the mass of the bottle.
Note that the drag of the air on the rocket is acting over a fairly short distance, and will be pretty small. We will assume it to be nearly zero.

8. TIME TO ACCELERATE
The amount of time it takes the water to leave the rocket depends on the mass of the water, and the mass flow rate. You must set up your own expression to solve for the time.
Once you have the time, you can find the final velocity of the rocket. This is the velocity it will have when it enters into free-fall.
At this point, kinematics can be used to assess the motion of the rocket.

9. PROJECTILE KINEMATICS
Determine the flight time of the rocket when launched upwards, and when launched at a 45 degree angle. We will try to launch at both angles. Determine the maximum height for the rocket launched upward, and the range of the rocket when launched at 45 degrees. Prepare a “report” that includes all of the items listed on the following page. The report is due on Tuesday, Oct 28th.

10. Necessary Components Components:
The following values, with all calculations shown neatly
Thrust, Net Force, Acceleration, Velocity after Acceleration Stage, Flight Times, Heights and Ranges.
A blueprint of your rocket that includes the height of the rocket, mass (without water) , volume of water used. It does not need to be drawn to scale.
Free body diagrams for the three stages of (upward) flight: Acceleration, Traveling upward, Traveling downward.