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.

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.

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.

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 cm

The units of measure

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.

The nozzle of our rocket has an inside diameter of 22mm. Remember that

11 mm * 11 mm * pi = 380 mm^{2} (aproximately)

The pressure is in cm^{2} but we have mm^{2}. Quick, how many mm^{2}
are in one cm^{2}. The answer is 100 so divide 380 by 100 and we have 3.80 cm^{2}.

The area of our nozzle is about **3.80cm ^{2}**.

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 3.80cm^{2}.
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/cm^{2} ) * ( 3.80cm^{2} )

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

17.2 newtons * 3.80cm^{2}_______________________ cm^{2}

We see that there is a cm

We had a plug with a known area: 3.80cm

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

In this equation the phrases **newtons** and **cm ^{2}** are our

**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 3.80cm^{2} 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/cm^{2} and
the nozzle is 3.8 cm^{2} .)

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.

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.

F = 2 P A

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

F = 2 * 17.2 newtons/cm

Look closely and make certain you can recognize each of the factors of the equation. Remembering that the cm

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.

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.

- Pressure in the rocket: 172,000 Kpa
- Diameter of the nozzle: 22 mm
- 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