"Coerce" has two syllables, while "coarse" has only one. So pronouncing it correctly a few times should go a long way toward retaining its spelling. This seems to me to be a lot easier to memorize than the following words: though, through, bough, thorough and so on. Knowing how to pronounce these words only makes them more difficult to spell, I would think!

The noun that goes with ‘coerce’ is ‘coercion’ whose pronunciation tells you that you need a ‘c’. As far as the letters that make up the sounds of ‘coarse’ there are just a lot of possibilities. (coarse, course, horse, force etc) Good luck!

There is no logic to it that I am aware of. The good news is, if you make a spelling mistake, at least people will see what you are saying.