Hi..
given that you're trying to emulate a pinball launch, all you need to know is the energy imparted to the ball - which is proportional to the deflection of the spring squared plus a constant - so all you need
to measure is how far the spring is pulled back. Use a (weakly) spring-loaded linear potentiometer attached to the plunger , and avoid all the hassle, and do the rest in software. You only need 2 numbers - the extension value, and the release time.
regards
Allan