Think of the railgun setup as an inductor, in which one element is free to move.
I'm not in the habit of giving people verbatim answers for their homework, and you haven't given all the information anyway, but we can do a walkthrough.
You already know the current though the loop (Ohm's law). Use this to find the induced magnetic field. You could form a Biot-Savart integral in 4 parts (both rails, the rod, and the power connection) but this might be too deep. Whichever method you choose, be careful with vector directions.
In real life, resistance wouldn't be zero; you could apply Faraday's law to this function, and get the induced EMF. It's actually a very simple inductor, so the EMF effectively causes a "counter-current" such that the actual current flow through the rod is less than you would expect from V=IR
Anyway, back to the question. You have a function for the induced magnetic field; add this to the existing ('background') magnetic field B to get a function for the total magnetic field. Use Lorentz to find the force that this field creates on the rod as current flows through it.
Once you have the force on the rod, the rest is basic calculus - just use F=ma &c. It's not linear because the force is a function of the rod's position.
If you want to be really
thorough, go through the same solution relativistically - after all, the rails are infinitely long, the rod will eventually be moving pretty fast...