How to fade tire tracks? (materials/modeling)

I want to have a tire track follow a curve, and I would like it to fade out in both ends.
Can this be done with one material, or in some way that automatically moves the faded “end pieces” to the respective ends of the curve?

Here’s a blend file of what I’ve achieved so far. I can have one faded “end piece” that follows the curve as I move it, but I cannot figure out how to make the other end. A solution to this, or a better method altogether, would be greatly appreciated.
test_tireTracks.blend (1.11 MB)