Instantaneous output - an extension of output into Calculus

[SSCBen]

(posted at all forums because I believe this is important and should get fair notice)

Recently my thoughts have been dedicated to making a system to accurately predict a water gun's stream distance and design efficiency (efficiency being a calculated value). While I had made some considerable progress, what I had been working on would take much more time to explain and to calculate than my new idea.

For a long time I had wanted to calculate the output at a specific instant. I could obtain a somewhat accurate number with the method described in my old "XP Physics" article which is no longer online. This method would completely replace that method for two reasons: it is far more accurate and it can describe output that is not constant or constantly changing. The inspiration for my XP Physics article was the Aqua-Nexus pages describing the "drop-off" of pressure with what I assumed to be approximate graphs.

Instantaneous velocity is an easy and common idea. It also isn't very useful except in comparisons. Instantaneous output on the other hand is very useful for several reasons. People have been taking the "average output" for years, which gives air pressure water guns a raw deal because they have about half of the output above that figure and half of the output below that figure. You can not tell how much the figure decreases simply by the average. Instantaneous output can also be used to determine when exactly the shot ends, as I had previously calculated in my old XP Physics article. Output would be best given with several methods: output when t=0, average output, and a graph of the output over the period of the shot.

We all know that water guns' water output, especially the output of non-regulated air pressure water guns, changes. For example, in a non-regulated air pressure water gun's output is constantly dropping. We now can see exactly how the output is changing by calculating the instantaneous output curve.

To calculate the output curve, you first must find a curve that displays how much water is shot over a period of time. The graph below is an example of a possible air pressure system's water. The x-axis represents time (t). The y-axis represents the amount of water shot thusfar in the shot. This test will be hard to make, though a small machine that records mass at certain fractions of a second as the mass is increasing would be the best way to generate this information. This is not the output curve.



The output curve is the derivative of this function. For those who are not familiar with basic Calculus, the derivative essentially is the graph of the slopes of the function. The derivative of the position function is the velocity function. The first graph can be said to be a "position" function for the output, though I am sure that is not the best way to describe the graph.

The first graph can and likely will be represented as a quadratic equation. That would not be the best way to represent the graph in my opinion. The graph will be best drawn with the data points visible as well as the equation made to fit. We do not know yet how the output curves will appear, and to prevent them from being shown as lines, which is not likely how they will appear, we should instead use regressions in the style of more detailed equations as seen fit in each situation.



This graph represents a potential output curve (or line), likely better known as a graph of the drop-off. Air pressure is known to reduce it's output, distance, and pressure as time goes on. You can see that easily in this graph.

This is my basic theory at the moment. I will be writing a much more in depth article eventually with everything including real data generated from a real water gun (likely my CPS 1000 and XP 150 because they are very good water). Output curves should be extremely useful for future water gun designers and those interested in water gun statistics.

[isoaker]

To calculate the output curve, you first must find a curve that displays how much water is shot over a period of time. The graph below is an example of a possible air pressure system's water. The x-axis represents time (t). The y-axis represents the amount of water shot thusfar in the shot. This test will be hard to make, though a small machine that records mass at certain fractions of a second as the mass is increasing would be the best way to generate this information. This is not the output curve.

Um, from what I'm gathering, your 'output curve' is basically a curve representing the deccelleration of the stream over time. However, if one is already measuring (tough to do as you noted) the different output as a function of time, I'm not sure how useful having a theoretical curve would be. Output is affected by many factors including level of water remaining the reservoir, nozzle size, lamination, etc. While one may be able to fit the measured curve to a line, it does not necessarily allow one to extrapolate from that curve alone what needs to be done to improve things. As well, I have a feeling that if it could be measured, soaker output over time would vary in a more complex manner, especially as the pressure gets low for air-pressure systems. While potentially useful, I'd be curious on what actual soaker measured curves would be.

[SSCBen]

No, the curve does not represent the decelleration of the water stream over time. That would be the second derivative of the position function. This is completely different.

As I said earlier, these types of graphs will be useful in several things, namely: seeing really where the output of air pressure water guns is at certain times, seeing how constant a CPS system really is, etc. It is not really here to improve things, rather, the output curve would be used mainly as a statistics. Take the output curve like a distance measurement. The output curve is not here to directly improve your water gun, it is here to see where it stands, and you can improve the output curve if you wish with modifications or a new homemade water gun. The output curve is just another statistic.

I can understand that people may think these curves would in reality be essentially random, but that is not true. I have looked at many water gun systems and the output does not fluctuate randomly. The output always will drop, unless you've managed to make a system where pressure increases, which would be unusual. Output drops in different patterns, and these patterns determine whether or not we say the water gun has drop-off or is CPS.

I also should have said earlier that the output curves for certain water guns on any nozzle size should be similar, but not the same. A longer shot time curve will be easier to collect data from. The integral (area under the curve) from when t=0 to when t=the shot time always will equal the PC size (simply because the function is the derivative of a function that shows how much has been shot over time). A longer shot time graph can, at least in my mind, be compressed and stretched into a correct curve for other larger nozzles, given that the integral stays the same.

[isoaker]

What are the units on the axes on your graphs?

Also, I never stated random changes in output over time, just more complex ones. A quadratic equation may do ok to approximate a measured curve, but may a higher order equation may fit better, but that's a moot point currently.

In terms of measuring things, as noted at SSCentral, you can get an approximation of stream speed if firing a level shot and seeing how far it travels. Doing a little backwards calculations, you can determine how fast the stream must have travelled for it to reach a distance since acceleration due to gravity is assumed constrant. If one knows the diameter of the nozzle, together with stream speed, you can approximate output for any given distance. Then one just needs to measure how range changes over time to get the first graph (not that simple, but can be done to some extend without the need for more expensive equipment).

I realize it could be a nice graph to see. I'm just not sure whether it's worth it for general soaker reviewers to attempt a more complex statistic since the benefit to the common user is not so significant.

[SSCBen]

A quadratic equation may do ok to approximate a measured curve, but may a higher order equation may fit better, but that's a moot point currently.

Yep, and I mentioned that. Some situations will work best with a simple quadratic equation, while in other situations other types and orders of equations will be more accurate. Fractional orders likely will be the most accurate for most situations, though I have no way of knowing until the tests are made! We'll know soon, at least for the water guns I will be testing.

I do not believe that this is for general use. I sure wouldn't want to put this in general reviews. Most people won't have any idea what this curve means (and by the lack of interest in this thread, I can easily see that!). A lot of effort will be required to make these curves as well, and effort could be much better spent in other areas. Output curves are really for people like me who are interested in these sorts of things. I mainly hope that fairly general use of this concept by those interested in it will result in more constant performance because the output can be seen as a function of time. As I said before, this is only another statistic to be improved upon. "Perfect" would have a flat output curve in my opinion.

I suppose I won't be posting things like this much more due to the lack of interest. Rather, I'll just go ahead and post about the concept later.

[Soaker Master]

This is to confusing. :O ???

[SSCBen]

Not really. This is the most basic Calculus around. However, I suppose this is the most advanced thing we have seen in water gunning. If you are interested, ask your math teacher about differentiation and you should learn enough to make these graphs (the power rule is plenty for these purposes).

[isoaker]

No, the curve does not represent the decelleration of the water stream over time. That would be the second derivative of the position function. This is completely different.

Actually, I've been doing a little thinking on this. Output is more cloasely related to stream velocity than it is related to stream position (output is defined as volume pushed per unit time). If you're taking the derivative with respect to time on an output vs. time graph, you're basically getting an accelleration curve of the stream (but a 'velocity' curve if you're assuming that output is similar to 'position').

I still stand by the statement that the end graph is basically a deccelleration graph when it comes to stream speed since, to me, output is related to stream velocity (though, in this case, it's more of a water volume deccelleration graph to be more accurate).

[SSCBen]

No, that's not what I meant. In what you are saying, the position function tells where the stream head is located. The velocity function (the first derivative of the position function) tells how fast and in what direction the stream is going. The acceleration function (the second derivative of the position function or the first derivative of the velocity function) describes how the velocity is changing.

We all know that stream velocity and output are not related very. You can have an extremely high velocity and a very low output (small nozzles). You can have a low velocity and high output (the pour effect). If you divide output by nozzle orifice size squared and use that as opposed to output, you should find a little more correlation, but they will not be perfectly correlated.

The first function is how much water in total has been shot. That is not an output vs. time graph, rather, it is a total water shot vs. time graph. The output vs. time graph is the derivative with respect to time of the total water shot vs. time graph. The "decelleration" of the output graph would be the derivative of the output vs. time graph. Decelleration would be a poor term in this situation because acceleration implies relation to the velocity. I believe that is where my confusion was.

Perhaps it would be best to describe the graph like this: The first graph is a graph of the total water shot in units of mL. The second graph (the output curve) is the graph of the output in units mL/s. The derivative of that graph with respect to time would be in units of mL/s^2. The last one would be the "decelleration of the output" graph.

Let me know if this better explains what I was saying. I would like to make this concept as easy to understand as possible.

[isoaker]

So the first graph is simply volume shot plotted versus time? If so, that definitely changes things and thus the derivative of the plot would be the output at any particular moment in time. Ok, that makes sense, now. Somehow, I was under the initial impression you were plotting output versus time, not volume versus time. Silly me.

As an aside, I completely agree that no one should not confuse output and actual stream velocity in some arbitrary situation. However, if one is dealing with a fixed (or given) nozzle size, output is very much related to stream velocity since, well, the opening restricts how fast water must be flowing for a certain output to be measured.

[SSCBen]

Good that I finally said what I meant correctly! Now there should be more people who understand this when I write the article.

I also agree that higher velocity means higher output on a fixed nozzle orifice size. Never considered that's what you meant, but I see now. I should list a few "laws" like that in my planned section on water gun physics because there really are quite a few of them and no one really considers them too much.