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Velocity is computed as the time derivative of the position vector, recognizing that as both r and er vary in time, the chain rule must be applied:
(12) V = Ṗ = ṙer + reṙ
Substituting for eṙ from eq. 5↑
(13) V = Ṗ = ṙer + rφ̇eφ + rθ̇sinφeθ
(14)
a
=
V̇ = P̈
=
(d)/(dt)(ṙer + reṙ)
=
(d)/(dt)(ṙer + rφ̇eφ + rθ̇sinφeθ)
Taking the derivatives and substituting for eṙ, eθ̇ and eφ̇, a becomes
(15)
a =
(r̈ − rφ̇2 − rθ̇2sin2φ)er +
(rφ̈ + 2ṙφ̇ − rθ̇2sinφcosφ)eφ +
(rθ̈sinφ + 2ṙθ̇sinφ + 2rφ̇θ̇cosφ)eφ
Further applications of this coordinate system can be found in cross products.