For the coil and yoke, I definitely used the TLAR method (though I did run it through FEMM to see if it would work). For voltage, I just went with the biggest supply I had, and for timing, I simply used a program that estimated the extra time needed after the projectile passed the photo-gate.
Keep in mind, I wasn’t very experienced at the time. If I were doing it now, I’d use a random search algorithm to find a coil design that generates the highest force while still switching quickly. For the power supply, I’d use a capacitor bank, and probably stick with similar timing — or possibly use a dedicated capacitor for each stage, charging each differently to deliver longer or shorter pulses as needed
