Why there is not initial position term in SUVAT equations?

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Just a quick question.
From calculus I know that $s(t)=s_{0}+v_{0}t+\frac{1}{2}at^{2}$ but the initial position term, $s_{0}$, has disappeared from SUVAT formulas.

Why?

Is it always assumed that $s_{0}=0$?

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Usually you always take s0 = 0 in problems , fore example going from place A to C , but there is a point B , where maybe velocity is changing or something happens ... Mostly you divide the pathing in 2 distances AB and BC always starting from the specifc origin , so s0=0