# Water rocket and water velocity

• bkelly
In summary, The conversation is about trying to understand the equation for exit velocity of water in a water rocket. The equation, V = sqrt( (2 * pressure * nozzle_area) / water_density ), has incorrect dimensions to be a velocity. The units are simplified to (pascal * m^5 / kg), which does not represent velocity (distance/time). The OP is looking for a reliable source for the correct equation to use in a spreadsheet.
bkelly
I found a few questions on water rockets but am not able to understand the equations. The guide I am using has this for the exit velocity of the water:
V = sqrt( ( 2 * pressure * nozzle_area ) / water_density )
I plug in the values for my water rocket to get
V = sqrt( ( 2 * 17.24Newtons/cm2 * 3.80 cm2 ) / 1 gram/cm3 )
After some simplification:
V = sqrt( 131 Newtons * cm2 / 1 gram )
That does not look anything like a velocity. How do I get the terms to work our correctly?

Firstly, I'd suggest convert all units to SI units. It'll be easier.
Secondly, Newton can be further broken down into two basic units. Force = mass x acceleration
Try it

sophiecentaur
You are correct, the expression does not have correct dimensions to be a velocity (it has dimension length^2/time), although you did a minor mistake in the units that does not make it better either.

Are you certain the area is supposed to be there?

Re siddharth: The Newton is the SI unit for force. The nozzle area is 3.8 square cm. Converting that to 0.000308 meters2 does not help much. But OK, I’ll do that.
Re Orodruin: No, I am not sure at all. That is what the document says. It can be found with a search for “water rocket predictors.pdf” It is the simplest equation I could find for the water flow . Reasonably accurate is good for me. If you have one I should use instead, Please show it and describe each of the terms.

( 2 * pressure * nozzle_area ) / water_density
filling in my values

( 2 * 17 Newtons * 0.000308 meters2 ) / ( 1,000 kgrams/m3 )

The denominator is a fraction so invert and multiply, and consolidate the constants

( 0.010472 Newtons meters2 * meters3 ) / 1,000 kgrams

Consolidate the meters to get

0.000 10472 Newtons meters5 / kgrams

F = MA
or
Newtons = kgrams * ( meters / second2)

yielding

( 0.000,10472 kgrams * meters * meters5 ) / ( kgrams * sec2 )
yielding

0.000 10472 meters6 / sec2

It becomes obvious to the casual observer that I have totally blown this. Please advise.

FYI, the critical values are: Pressure = about 17 Newtons/ square cm and a nozzle area of 3.80 square cm. It is a water rocket and I need to find the flow rate of the water. The velocity combined with the area of the nozzle will yield the flow rate.

Last edited:
Get the units right first, then concern yourself with the values.I agree with Orodruin. You get length^2/time as the final units. Check the initial formula again.

Pressure is not measured in Newton in SI units, it is measured in Pascal = N/m^2 ...

bkelly said:
That is what the document says. It can be found with a search for “water rocket predictors.pdf” It is the simplest equation I could find for the water flow .

"It can be found with a search for" is not a good reference. Even if it does appear we have no way of verifying we actually found the document you refer to. If you want to refer to it, you need to provide a link.

Point taken. The document is here: http://www.schulphysik.de/strutz/predictors.pdf
On the second page, right side, is the equation for flow rate. Is that a valid equation? Do you recommend something different?

I am still working the equation and the units are not working out. I will continue for a while then post my results.

Edit: I suspect I am miss-using the equation from this page, or just flat not understanding what that author is trying to tell me. As I understand the equation, the velocity of the water exiting the bottle is the square root of:
( 2 * pressure * nozzle_area ) / water_density

Ignoring the constants for now, and just trying to work out the units I start with

( pascal * m^^2 ) / ( kilograms / m^^3 )
invert the denominator and multiply to get

(pascal * m^^2 * m^^3 ) / kilograms
which simplifies to
pascal * m^^5 / kilograms

Pascal must be expanded to what is represents, Newtons / m^^2 so the equation becomes

( Newtons * m^^5 ) / ( kilograms * m^^2)

The meters simplifies resulting in
( Newtons * m^^3 ) / kilograms

This does not look like a velocity. Velocity is composed of ( distance / time ). If I use F = MA re-arranged to M = F/A and replace kilograms then I introduce seconds^^2. That will not work because velocity is seconds^^1, not ^^2.

Please reveal my fundamental error to me.

Last edited:
The equation is wrong. As the dimensions are not correct, the equation simply cannot be correct. I would not trust a source making this error.

Orodruin said:
(it has dimension length^2/time)
Sorry for being pedantic :s but this may be my only opportunity to ever correct you. :) It's length^(3/2)/time

Nathanael said:
Sorry for being pedantic :s but this may be my only opportunity to ever correct you. :) It's length^(3/2)/time

The OP has made an error in the simplification of units. The dimensional analysis goes:
pressure x area / density = force / density = (mass x length / time^2)/(mass / length^3) = length^4 / time^2
Taking the square root gives length^2 / time.

Nathanael said:
Sorry for being pedantic :s but this may be my only opportunity to ever correct you. :) It's length^(3/2)/time
It's m^2/sec

I just looked at the final equation in the OP sorry hehe

Nathanael said:
I just looked at the final equation in the OP sorry hehe
I guess today is not the day you get the better of him

Well then. I will shift my reference. I will start here: http://tbpmindset.org/modules/SECME/Rockets/Rockets_Calculations_Manual.pdf
and search for a few more. If you have a known good reference, please post.

My goal is to enter instantaneous formulas into an Excel spread sheet and iterate my way through the flight, maybe 100 steps per second, more if needed.

I thank each of you for your time.

Last edited:
The document referenced in my previous post provided a formula for mass flow per seconds. After working it for a while the result was 6.98 kg per seconds. With a water load of 0.5 kg, the water rocket is out of water at about the 0.07 second mark. That agrees with our rather imprecise observation of “whooose and the water is gone.” It also agrees with an earlier calculation I made that the rocket would be something less than 2 meters high when out of water. Unless someone asks I won't show the calculations because I am sure everyone reading this can already do the math.

Thanks for confirming that the first equation was bogus (or maybe I was unable to understand the author's intent).

Is there a way to mark this answered?

Did your iterations adjust the pressure value as the water flows out of the rocket?

I am going rather slow, spending too much time on Quora. But yes, and I will add in a phrase to consider temperature drop and the resultant further pressure drop. I am doing this because I am a volunteer at an elementary school STEM class, one hour a week. We launched a water rocket a few times and the kids all thought that was just great. One of them noticed the fog remaining in the bottle and I was impressed at his taking notice. I am putting my lesson plans on my personal website and that takes me a rather long time.

mic*

## 1. What is a water rocket?

A water rocket is a type of rocket that uses water as its propellant. It is typically made up of a plastic bottle, which is partially filled with water and pressurized with air. When the pressure is released, the water is forced out of the bottle, creating thrust and propelling the rocket upwards.

## 2. How does a water rocket work?

A water rocket works based on the principle of Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. When the water is forced out of the bottle, it creates a downward force on the water rocket, which in turn creates an upward force (thrust) on the rocket, propelling it upwards.

## 3. What factors affect the velocity of a water rocket?

The velocity of a water rocket is affected by several factors, including the amount of water and air in the bottle, the size and shape of the bottle, the surface area of the fins, and the launch angle. The more water and air in the bottle, the greater the thrust and velocity of the rocket. A larger and more streamlined bottle can also increase the velocity, as well as fins that provide stability and reduce drag.

## 4. Can you measure the velocity of a water rocket?

Yes, the velocity of a water rocket can be measured using a variety of methods, such as a speedometer or by tracking its flight using a camera and measuring the distance traveled over time. It is important to note that the velocity may vary based on the launch angle and environmental factors.

## 5. Are water rockets safe to launch?

Yes, water rockets are generally safe to launch as long as proper precautions are taken. It is important to always launch in a wide open area away from people and buildings, and to use protective gear such as goggles and gloves. It is also important to never aim the rocket at anyone and to carefully follow instructions for pressurizing and launching the rocket.

Replies
17
Views
1K
Replies
26
Views
2K
Replies
4
Views
2K
Replies
13
Views
1K
Replies
1
Views
6K
Replies
8
Views
2K
Replies
14
Views
1K
Replies
9
Views
1K
Replies
32
Views
4K
Replies
3
Views
1K