For air pressure guns, the D would change. If we plot BPF as D increases (going backwards here), then would it be possible to find the total are below the line?
In other words, given a parabola (BPF = D^2 * k) with domain restrictions at zero and at another point (because of shot time), what is the area below the parabola?
Basic
fundamental theorem of calculus stuff. This problem is actually pretty simple (calculus is not hard because of the calculus, it's hard because of all the algebra and rules you have to memorize). I could go into the details for quite some time, but essentially, the area under the curve of D^2 is (D^3)/3. If you have a graphing calculator, there almost always is a function to approximate the area under a curve by using lots of little rectangles.
For this formula however, I don't really know why you would need the area below the curve to calculate anything. It is difficult to make a curve of distance versus time into the shot, which is what matters here. Maximum range would matter, or perhaps average range over the entire duration of the shot.
But if you wanted to use this formula, I suppose the most accurate version would have the integral in it. Personally, I think this formula might not be completely necessary. Battle practicality is subjective and measuring subjective values numerically is difficult.
Now, I think I have a few things that are pretty close to the "silver bullet" homemade water gun.
There is little I would change on my
CPS homemade water gun given the choice. The PC can hold as much as you want, which is a very big advantage. The power is unmatched, and soon to be greater. In it's older self, it could get about 60 feet of range to the puddle (I do not measure to the last drop any longer because it's simply inaccurate). To the last drop would be like 63 feet. It's current state only has two layers on it's PC and is sadly only about as powerful as a CPS 2000, with only about 52 - 53 feet of range (this is a problem?). I intend to upgrade the PC, valve, and nozzle system this year. The PC and valve will be of a larger diameter to allow for greater flow and greater force. The PC also will be very heavily layered with bike tubes over top it's already thick CPS tubes. The nozzle area will be upgraded with a rotating nozzle selector of my own design (based mainly upon older homemade Nerf barrel rotation systems). I consider this water gun to the best one I have ever made, and it is soon to be even better. This I seriously consider to be my own "silver bullet" homemade water gun.
Of course, some may not prefer this design because it does not have an included reservoir. I prefer and for the most part use only backpack reservoirs.
Lately I also have been working on an improved SuperCAP type design. With this design I intend to correct several failings of the original SuperCAP design, namely, the enormous size, tubing even being able to come off, the lack of ability to read the gauges on the back, riot-blasts so large that you spin when shooting, etc. The gun actually was far too powerful and for the new gun I bought a regulator with half the flow (and you must consider that I never used the other regulator at more than maybe 50% of it's ability). I never measured range, but I know for a fact that SuperCAP easily shot over 65 feet. This new gun will be a much more compact and better designed SuperCAP. I don't have it made yet (I just ordered a few parts 4 days ago!), but when it is made it will replace my CPS water gun.
The advantages of this design are numerous. Once the air chamber is full, you will get many full tanks of shots out before needing to refill, especially on a low pressure such as 30 PSI, which still would have have over 350 pounds of force (which is a lot when compared against other water guns). The tank pressure will be visible on the gun in this design, to make it easy to know when to refill with air. The power will be unmatched by anyone else. The reservoir should hold a smaller, but definitely not unsubstantial 6.1 liters. The air volume, power, and water capacity are the biggest advantages of this design. The biggest disadvantage would be size for most people, because this will be about as big as a frame backpack. Weight will not be a problem however because the water will only weight 13.5 pounds and the frame backpack puts the weight on the hips, which makes carrying weight much easier.
For people who lack a frame backpack, building may be expensive. The frame backpack may be over half the cost, unless you get a used one or get only the frame part. I own two frame backpacks myself, so this isn't a problem for me.
The new CAP design actually would score extremely high on SilentGuy's BPF. The range is unmatched, the weight is comparable to most water guns (and balanced better), and the shot time is unmatched as well. The largest problem in my opinion is the size. Perhaps any BF formula should factor in size along with weight?