Tangential and Normal Components of Acceleration

Topics: Calculus - Vector-Valued Function - Physics

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(theorem)

Let $r(t)$ be a vector-valued function that describes the position of an object. We know the second derivative of this function $a(t)$ describes the acceleration of this object.

We can describe $a(t)$ as a linear combination of the function’s tangent unit vector $T(t)$ and its normal unit vector $N(t)$:

$$a(t)=\frac{d^{2}s}{d{t}^2}T(t)+\kappa {\left(\frac{ds}{dt}\right)}^{2}N(t)$$

…where $s$ represents the arc length parameter and $\kappa$ the curvature.