[MadMax]

The gist of my posts that mcguyver is referring to is that rapid fire does not affect the relationship between cylinder cross sectional area, piston travel and spring force profile so the calculations I use are applicable to your case (if my assumptions are valid).

Unfortunately if you don't want to get into differential equations you're going to have to a tough time optimizing your bolt action rifle. The difficulty is that the situation is mathematically pretty complicated. More complicated than just plugging gr. 12 physics energy equations. My previous analysis was simplified because I was plugging actual numbers from diameter, force, and velocity measurements. In a sense, reality did all the calculation and all I did was close the theoretical loop by plugging all the numbers into simple equations.

Figuring out the results in the other direction is much more difficult. At any given point in the piston travel, you can calculate the instantaneous spring force which gives you an idea of how much energy has been transferred to the piston, spring (it's centre of gravity moves about half the speed of the piston at any point), and pellet.

However, you are faced with not knowing the breech pressure at that point. It is definitely higher than atmospheric. You can't assume zero compression and therefore proportional speeds of piston and pellet based on cross sectional areas.

Optimizing your system to maximize spring and piston velocity is not optimal for energy transfer to a pellet. The faster the heavy things move (piston/spring) the more energy they absorb in accellerating to speed.

It's really not worth going the theoretical route. It's interesting to see if reality can be codified, but designing products soley from a theoretical perspective is pretty prone to forgetting about important things which are subtle, but actually manifest themselves in important ways in materials.

In product design I do use mathematical physical relations to arrive to some understanding, but it's very important for me to get physical fast and make proof of concepts in materials. Usually my first model shows me how I am so wrong I am on paper. Product design is more about developing not "know how" but "know how NOT" which is a bitch.