Water Rocket Thrust

What makes a water rocket go up?

How fast will it go up?

How far will it go up?

These are all dependent upon the thrust and the mass. We know the mass so now we work on the thrust.

The thrust is the force that pushes the water rocket up. If we know the thrust, and if we know the mass of the rocket, we can calculate how fast it accelerates.
If we know how time it spends accelerating, we can calculate how fast it goes. From that we can calculate how high it goes. Lets begin.

Trying to calculate all the variables is quite difficult and is suitable for college level mathematics. We will use a simplified simple equation for water rockets.

F = 2 P At

This says, Force, or thrust, is equal to twice the pressure times the area of the nozzle.

The 2 is a simplification of some rather complex calculations. For a water rocket all those calculations work out to about 2. We wil return to that number 2 later.

Pressure

Start with the Pressure, the P in the equation. The pressure in the bottle was about 172 KPa.

Recall from our discussion of mass that KPa means one thousand newtons per square meter. Lets make that a bit easier to deal with. We will work mostly with centimeters so it will help to convert that to newtons per centimeter.

Remove the K of the KPa and the number becomes 172 000 Pa.

Exercise

Determine the number of square centimeters in a square meter. Remember that one pascal is the force of one newton on one square meter. One centi means 100 and there are 100 centimeters in a meter. Do your calculations now.

Done? If you were to draw a square, one meter on each side, there would be 100 centimeters on each side. Multiply 100 * 100 to get 10 000 cm2 in 1 m2. All the pressure on that one m2 is evenly shared amongst those 10 000 cm2 so divide out 172 000 by 10 000 to get 17.2 n/cm2. Read that as:

17.2 newtons per centimeter squared.

The units of measure pascal is a shorthand name for newton per m2. We are using cm2 so we revert to the full expression.

Exercise

Lets look closely at a cm2. Get a ruler, pencil, and paper. Draw a square inch on on the paper. Hopefully your ruler is marked in centimeters also. Mark all the sides of that square inch in centimeters then connect the lines on opposite sides. When done correctly there will be four complete square centimeters inside the square inch, with two smaller rectangle along two of the sides and one small square in one corner. One square centimeter is fairly small. Press the end of your finger down on something and the area where it touches is about 1 cm2.

Each of the rectangles along the sides is a little more than 1/2 square cm. There are four of them so that is a little more than 2 square cm. Add in the tiny square in the corner and the little spaces add up to 2.4516 square cm. All together there are 6.5416 square cm in the square inch.

Nozzle Area

The equation we are working on is: F = 2 P At We have the pressure in scientific notation, now we work on the area part.

The nozzle of our rocket has an inside diameter of 22mm. Remember that area = pi * r2. The radius is half the diameter and is 11 mm. The area is:

11 mm * 11 mm * pi = 380 mm2 (aproximately)

The pressure is in cm2 but we have mm2. Quick, how many mm2 are in one cm2. The answer is 100 so divide 380 by 100 and we have 3.80 cm2.

The area of our nozzle is about 3.80cm2.

Keeping Our Terms

Before our rocket launched, there was a plug in the nozzle to hold the pressure while we pumped in air. We just discovered that the area of the nozzle was 3.80cm2. We also discovered that the pressure inside the rocket, and on the plug, was about 17.2 newtons per square centimeter. To find the total pressure on the plug, multiple its area by the pressure. That equation looks like this:

( 17.2 newtons/cm2 ) * ( 3.80cm2 )

We can put everything into a single complex fraction to make is easier to visualize.

			17.2 newtons * 3.80cm2
			_______________________
			       cm2
			

We see that there is a cm2 in the numerator and in the denomonator. Anything divided by itself is one so the equation simplifies to:

17.2 newtons * 3.80 = 65.36 newtons

Very Important Observation

We started with a pressure measured in force per unit of area. In this case: cm2.
We had a plug with a known area: 3.80cm2. That plug was exposed to the pressure per unit of area.

We wanted to know the force on that plug. So we multiplied the pressure by the area. The important part is that we kept the units of measure in the equation. Look back up at that equation to see the pressure (newtons / cm2) and see the area: cm2. Those two sets of uints become a factor of one. When dividing and multiplying we can always remove factors of one. So the cm2 drops out leaving us with just a number and the units of measure: newtons. That is just what we wanted. We were calculating the total force on the plug rather than the force per unit of area.

In this equation the phrases newtons and cm2 are our units of measure. They tell us what the numbers are describing and are extremely important. We must pay close attention to our units of measure in order to do the math correctly and to make sense of the numbers we get. You will see more of this as we continue to explore the physics of our water rocket.

To Continue
When the plug was in the nozzle of our rocket, and just before it launched, there was about 65 newtons of force on it. The fricton between the sides of the plug and the inside of the nozzle of the bottle caused it to remain in the neck until the force on the plug exceeded about 65 newtons, then the plug popped out.

At the instant the plug popped out, we can say that there was force of 65 newtons on 3.80cm2 at the top of the bottle that did not have a matching force of 65 newtons at the bottom. (Remember, the pressure inside the bottle was about 17.2 newtons/cm2 and the nozzle is 3.8 cm2 .)

When the plug popped out, the forces were unbalanced. Because the forces were unbalanced the bottle started moving in the direction that had the most force. In this case, that is opposite the direction the nozzle faced and was up. Our rocket went up.

Remember the 2?

Our equation for water rocket thrust is F = 2 P At. We have ignored the "2" until now.

We have just accounted for the inbalance of pressure when the plug popped out. As the rocket takes off, it is forcing water out the nozzle. Pushing that water out means that the water accelerates in the opposite direction that the rocket moves. That acceleration is a force pushing in two directions, pushing the water down and pushing the rocket up.

That force from accelerating the water is about the same as the force from the open hole in the rocket (the nozzle). That pretty much doubles the thrust. Therefore, we have that 2 in our equation. This helps explain why a rocket with water goes faster and further than one without water. There is more to it, but we will get there soon enough.

As noted earlier, all those calculations get rather complex. For now we will just use that constant 2.

Complete the Force Equation

Now we are ready to complete the operations on our force equation. We have already done almost all the work so this is a sumation and review.

F = 2 P At

Substitute in the values that we have so laboriously claculated to get:

F = 2 * 17.2 newtons/cm2 * 3.80cm2

Look closely and make certain you can recognize each of the factors of the equation. Remembering that the cm2 are a factor of one we wind up with:

F = 2 * 17.2 newtons * 3.80 which can be rearanged to: F = 2 * 17.2 * 3.80 newtons.

Do the arithmetic, remove the factors of one, and it works out to: 130.72 newtons.

Rocket Thrust

The thrust pushing our rocket up is about 130.72 newtons.

Remember that one newton is about the same force as one medium sized apples. Stack up 130 apples and you have quite a lot of force. That is why our rocket takes off so fast.

Think About It

It is taken us a long time to get that number, the thrust. We have had to stop and discuss many things along the way. The first time is always the hardest. Now you need to practice. Go over these steps up to here with pencil and paper and work out the numbers for your self. Here are the starting values that you need.

  1. Pressure in the rocket: 172,000 Kpa
  2. Diameter of the nozzle: 22 mm
  3. Equation for thrust: F = 2 * P * A

After you do that once or twice, put this explanation aside, get a fresh piece of paper, get your rocket (your soda bottle that is), make the measurements and do the calculations by your self. Its OK to peek a few times, but keep doing it until you can do it all by your self.

And if your really want to be sure you understand, explain it to someone else. When you do that, you will understand what I mean.

Now that we know our thrust, the next phase will be: acceleration. If you are ready, here is the next web page:
Water Rocket Acceleration

13 Jan 2015
Bryan Kelly
send comments to: on line at bkelly dot ws