Abstract
<jats:p>Standard treatments of curvilinear motion decompose acceleration into tangential and normal components, but the resulting components are seldom assembled into an explicit expression for the global direction of the acceleration vector in coordinate-free terms. This tutorial collects, organizes, and works through such an expression for both planar curves and space curves in R³, with worked exercises intended for classroom use. Using the Frenet–Serret frame, curvature κ, and torsion τ, we present four results. First, for a smooth planar curve, the absolute direction of acceleration is assembled into the exact, coordinate-free formula θa(s) = Θ₀ + ∫κ dσ + atan2(κṡ², s̈), whose two terms separate cleanly by what they require of the observer: the tilt φ = atan2(a_N, a_T) is read directly from an onboard accelerometer triad and needs no external input at all, whereas the path orientation Θ(s) requires both an external axis Θ₀ and an external initial speed ṡ₀. Following Galileo and Newton, speed itself is not an intrinsic quantity even in Newtonian mechanics, and we make the resulting three-level hierarchy explicit throughout. Second, for any C² space curve, the acceleration vector is confined to the osculating plane: the binormal component a_B = a·B vanishes identically. Third, the acceleration tilt φ is τ-independent and remains onboard-measurable in three dimensions; the absolute spatial direction of acceleration is â = cos φ T(s) + sin φ N(s), where T, N are determined by integrating both κ and τ. Fourth, setting τ ≡ 0 reduces the three-dimensional formula to the planar formula exactly — the planar result is a strict special case, not an independent result. Worked examples on a circular path and a circular helix confirm consistency with direct Cartesian computation.</jats:p>