Why can't you hit a feather in mid-air with a force of 200 N?

I would not say it could not be done. The feather could first be charged and then an electrical field could suddenly be applied.

The acceleration would be F/m. A feather's mass is about 0.001 kg, so the acceleration would be about 2*10^5 m/s^2

Any object you are trying to hit it with at that acceleration might break. Air resistance would also be hard to overcome.