Retrieving "Arc Length Formula" from the archives

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1. Distance Traveled

   Linked via "arc length formula"

   $$s = \int{t1}^{t_2} |\mathbf{v}(t)| \, dt$$
   where $\mathbf{v}(t)$ is the instantaneous velocity vector. Because distance traveled only considers the magnitude of the motion, it is fundamentally linked to the concept of path length in differential geometry, often analyzed through the arc length formula.
   However, in non-Euclidean systems, such as those studied in chronogeometry, distance traveled is also significantly affected by l…