In general a (acceleration) is not tangent to the path. So, we conveniently resolve the acceleration into tangent and normal to the path of the particle.
Let el = unit vector tangent to path at point pat time t
or de / dθ is a unit vector and is perpendicular to et . Let us call it en which is normal to de the path of the particle.
Since V is tangent to the path, so
Now, -de / dt = de / dθ , dθ /ds , ds /dt , [ do /ds = 1/p ,p is the radius of curvature of the path of particle
de /dt = en . (1/p ) v = v /p en
Substituting the value of de t /dt from (2.10) in (2.9), we get,
Thus the scalar components of a are
at = dv /dt ,an= v2 p