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Old 02-03-2006, 07:42 PM   #1
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Default Instantaneous output - an extension of output into calculus

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.
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Old 02-06-2006, 05:38 PM   #2
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A few replies from me at other forums:

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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.

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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.

Quote:
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.

Quote:
That is completely correct. However, you are getting the units a little wrong. 4X is the same thing as writing 4 ounces per second (4oz/s), which is 120 ml per second. You can not say that 120 ml per second is shot in 0.01 of a second. You'd have to use a different unit, in this case mL. Something that can get a 20X output could have a total shot time of 1 second, but the first 0.25s could shoot most of the 600 mL needed to get that! That's not a full second shot time!

It would be a tedious task to make output curves for a lot of water guns, but then you can directly compare the water guns side by side in a graph. You may think twice about using that 20X because it might only have a shot time over 50% power for a very short period! You also might think twice about choosing a less constant air pressure water gun, favoring a more constant one such as the Super Soaker XP 150. Hopefully, these output curves will help people make better decisions in choosing water guns or help people design water guns with better output curves. The latter is my main goal.

A graph is what we have been waiting for a long time. People can say "this water gun has 100mL average output, but you must also remember it has drop off." Now you know how much it drops off, where the drop-off starts, and what the initial output or maximum output was. Pictures are worth a thousand words afterall.
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Old 02-12-2006, 01:55 PM   #3
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This is interesting - easy enough to follow (at least, I found it pretty easy to grasp the basic concept) and potentially useful, as you point out. Nice to see simple calculus applied in such an obvious example.
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Old 02-12-2006, 02:34 PM   #4
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Quote:
apeopleied

There is another fault of word filters. If there is a filter for p p l, it will change it even if the 3 letters are part of another word.
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