Building accelerator coils is a pain. For starters, they take forever to build. Most importantly though if you build the firing tube and it's coils too small you won't get the power you need to accelerate but if you build them too large the coils won't fire quickly enough.
Why is that? Most serious coil guns use SCRs. You'll have a capacitor, you want to discharge it into the coil. If you just use a switch, the amperage involved is probably more than the 20A most switches are rated for so the switch will be destroyed (in many cases these machines work at thousands or tens of thousands of amps). In fact, this current is generally too strong to use even normal power mosfets. But never fear, there is a product called an SCR. It's essentially a diode that doesn't conduct in either direction till you turn it on. And it happens to be able to survive extremely amperages compared to other devices. The way it does this is that it only turns on.
The amperage limits of other electrical components often comes from heating. The wattage dissipated by a transistor is the voltage across it times the amperage through it. If either is small, there is no heat dissipation. If both are significant, that's when the heat is on. And predictably, if a transistor is partially conducting, it will be taking on a lot of heat. Now SCRs only turn on. Thus, they go very quickly from zero amperage and high voltage to high amperage and zero voltage. This keeps them from building up much heat and thus you can get megawatt switches for double digit prices.
So now you've got a coil that can fire. However, if it fires too long, the projectile will be leaving out the other side and get pulled back in. Thus, you'll have to design the coils to fire for just long enough to pull the projectile in but to have been expended after that. That is why the coils must be neither too large or too small.
How many times should I wind the wire around the firing tube to make the coils? 10? 10,000? How to know? Should I be using 10 gauge wire? 30 gauge wire? Here's where some seriously shoddy math comes in. Essentially, doing some calculations about the expected inductance, discharge time, etc. I don't want to go into the math because it's complex and most people don't give a damn. So I'll just link you to the spreadsheet that you can plug numbers in to calculate. That said, the spreadsheet won't save you. It's not ballistics where you'll find out within 1% where the projectile will land. This will get you within 2x or 4x in either direction. Plug in numbers and then look at the timings to make sure they're not telling you the rise and fall times are too far off. Then do the same thing with the next coil and so on for ever coil you'll be making. For me, that boiled down to about the settings in the spreadsheet.
When you decide on a set of parameters, build a coil or two and fire it. Get a triggering scope to check if the timing you had is what you expected.
Remember those other firing tubes at the top? Well they're not being used for a reason. Countless hours making parts that can't get up to and surpass the very difficult to beat 60m/s marker.
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