Using PCB coil for proximity sensing
So you need to measure the speed of a fast moving object.

This requires that you measure its position and the time at that position twice and precisely. Or if you know the weight of the projectile you measure its energy and hence derive speed.

Neither of these things can be done with a plastic projectile through an inductive sensor.

Go back to the classic two position opto sensor. Alternately shoot the projectile into a block on a string and measure it's weight and height rise to get energy.