Imagine someone is standing on the surface of the Earth (let's assume it is completely flat) and they start going exactly in the direction their shadow is cast (that is, exactly opposite of where the Sun is in the sky). What kind of path that would produce after a certain time, say a few days? I assume they stop when the Sun sets and wait for the sunrise to continue with their journey. I also assume they just embark a boat should they encounter water.
I realize this question is a bit under-defined as the precise answer depends at least on 3 parameters (the exact time and location they start the journey from, and their speed) but I'm interested in an answer that takes these into account as parameters.
How to approach this problem? In particular, what kind of formula would give the position of the Sun in the sky as seen from the Earth at any given moment?