Vectors and Calculus

From my book.

T = v/|v|, by definition. Rearranging gives:

v = |v| T.

Differentiating both sides of the above equation with respect to t, and using product rule:

a = v' = |v|' T + |v| T'

Also, N = T'/|T'|, by definition of N. Substituting into the above equation:

a = |v|' T + |v||T'| N

So the tangential component of acceleration is equal to |v|' and the normal component is equal to |v||T'|.