Instantaneous output - an extension of output into calculus
Posted: Sat Feb 04, 2006 1:42 am
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.
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.