Editing Kinematics (section)

===Acceleration===
The acceleration of a point ''P'' in a moving body ''B'' is obtained as the time derivative of its velocity vector:
<math display="block">\mathbf{A}_P = \frac{d}{dt}\mathbf{v}_P
= \frac{d}{dt}\left([S]\mathbf{P}\right)
= [\dot{S}] \mathbf{P} + [S] \dot{\mathbf{P}}
= [\dot{S}]\mathbf{P} + [S][S]\mathbf{P} .</math>

This equation can be expanded firstly by computing
<math display="block"> [\dot{S}] = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\mathbf{d} -\Omega\dot{\mathbf{d}} + \ddot{\mathbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\mathbf{d} -\Omega\mathbf{v}_O + \mathbf{A}_O \\ 0 & 0 \end{bmatrix}</math>
and
<math display="block"> [S]^2 = \begin{bmatrix} \Omega & -\Omega\mathbf{d} + \mathbf{v}_O \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} \Omega^2 & -\Omega^2\mathbf{d} + \Omega\mathbf{v}_O \\ 0 & 0 \end{bmatrix}.</math>

The formula for the acceleration '''A'''<sub>''P''</sub> can now be obtained as:
<math display="block"> \mathbf{A}_P = \dot{\Omega}(\mathbf{P} - \mathbf{d}) + \mathbf{A}_O + \Omega^2(\mathbf{P}-\mathbf{d}),</math>
or
<math display="block"> \mathbf{A}_P = \alpha\times\mathbf{R}_{P/O} + \omega\times\omega\times\mathbf{R}_{P/O} + \mathbf{A}_O,</math>
where ''α'' is the angular acceleration vector obtained from the derivative of the angular velocity vector;
<math display="block">\mathbf{R}_{P/O}=\mathbf{P}-\mathbf{d},</math>
is the relative position vector (the position of ''P'' relative to the origin ''O'' of the moving frame ''M''); and
<math display="block">\mathbf{A}_O = \ddot{\mathbf{d}}</math>
is the acceleration of the origin of the moving frame ''M''.