Creating a simulator for an electromagnetic mass driver

I am trying to build a simulator for designing a mass driver. What I am essentially doing is given an iron rod and a coil of wire, calculate the force on the iron rod at various positions along the coil axis. Ideally, I want to use the various forces to calculate the final velocity of the rod. The overall goal is to compare different windings of an coils to find the one the best accelerates the iron rod. I can then feed that final velocity back into the simulator to test accelerating the rod while its already moving (for using multiple coils in a row). I might be way out of my depth here, but what would be the formulas for these calculations? As long as there is a semi accurate way to do this, even if it is really roundabout and intensive, I can write code to do most of the work for me. For givens, I have the size and weight of the iron rod, and the inner and outer diameter, number of windings and current through the coil.

I've seen a simulator like this done before, but the person used a testing rig to find the forces rather than calculating them, and I want to avoid having to build a similar testing device as I would have to build each coil that I want to test.
I am also new here, sorry in advance if I didn't include something important. I'll be checking here often if I need to post more information.

I think you will need to decide how realistic you want your model to be. There are analytical formulas for the force on an iron rod inside a coil. The numerical calculations are then not difficult.
However, this will always be an idealized case; if you want a more realistic model that takes into account multiple cells etc you will need something like a FEM solver, these are far from trivial to create from scratch and you would be better off using existing software such as COMSOL or Ansys.
It is very possible that there are free software libraries around that could be used;; but they tend to have a very steep learning curve. COMSOL is -relatively speaking- easy to us (which does not mean easy)

if you

Thanks for the reply! I'll have to look into that software to see if it will do what I need it to. I'd like to use the analytical formulas, but since I'm trying to find force outside of the coil, I doubt they would work.