Components of Acceleration

Since the unit tangent vector T is F'/||F'||, it can also be written velocity/speed. Thus
velocity=speed*T
Using the product and chain rules (which you should verify), this implies
a=speed'*T+speed^2 dT/ds.
But dT/ds is curvature*N(s).

So, we have
a=d(speed)/dt T+(speed^2)(curvature)(N)
d(speed)/dt is the tangential component of acceleration, and (speed^2)(curvature) is the normal component of acceleration.

Let's look at something that should be familiar: circular motion.

Let's look at something that should be familiar: ...

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