Consider a moving rigid body and the velocity of a point
P on the body being a function of the position and velocity of a center-point
C and the angular velocity \boldsymbol \omega. The linear velocity vector \mathbf v_P at
P is expressed in terms of the velocity vector \mathbf v_C at
C as: \mathbf v_P = \mathbf v_C + \boldsymbol \omega \times (\mathbf r_P - \mathbf r_C) where \boldsymbol \omega is the
angular velocity vector. The
material acceleration at
P is: \mathbf a_P = \frac{d \mathbf v_P}{dt} = \mathbf a_C + \boldsymbol \alpha \times (\mathbf r_P - \mathbf r_C) + \boldsymbol \omega \times (\mathbf v_P - \mathbf v_C) where \boldsymbol \alpha is the
angular acceleration vector. The spatial acceleration \boldsymbol \psi_P at
P is expressed in terms of the spatial acceleration \boldsymbol \psi_C at
C as: \begin{align} \boldsymbol \psi_P &= \frac{\partial \mathbf v_P}{\partial t} \\[1ex] &= \boldsymbol \psi_{C} + \boldsymbol \alpha \times (\mathbf{r}_{P} - \mathbf{r}_{C}) \end{align} which is similar to the velocity transformation above. In general the spatial acceleration \boldsymbol \psi_P of a particle point
P that is moving with linear velocity \mathbf v_P is derived from the material acceleration \mathbf a_P at
P as: \boldsymbol{\psi}_{P} = \mathbf{a}_{P} - \boldsymbol{\omega} \times \mathbf{v}_{P} ==References==