Where can/should I, as a high schooler, publish a paper regarding the derivation of a formula? For a high school engineering project, a friend and I have found ourselves deriving an equation for how to most efficiently configure a system.

We found an equation that, given a few parameters about the system, equates the amount of material added (and thus weight), to the amount of energy this system will gain. Along the way, we needed to find an equation that, given an amount of material, would describe how to most efficiently use the material.

We were thinking we'd write this up in a nice paper, and publish it somewhere. (For resume building, college applications, etc.).

Where/how could we do this (if anywhere), and how long would it take?



I posted this on Physics StackExchange and it was recommended that I post it here as well.
The big question here is whether what you've done is new to the world - whether it's a genuine new advance that nobody in the world has done before, or whether it's something known to the world that you, and perhaps your teacher, have not seen before. Without meaning any disrespect to you and your friend, the second option here is more likely - but the first is of course possible. 

If it's genuinely new, you can look to publish in a scientific journal. If it's not new (but perhaps still unusually advanced for a high school student), you should perhaps look for other types of publication, which could still be good for your resume. The difficult question is how to tell which category your work falls into.

My suggestion is to speak to your high school teacher about this. They have seen a lot of different students' work. If they think that it may be a novel development, then either write, or get your teacher to write, to a professor in a relevant subject at a local university. See if they will look it over and advise you.