Contact-point velocity (${\mathbf{v}}_P={\mathbf{v}}_{\mathrm{CM}}+\mathbf{\Omega }\times {\mathbf{r}}_P$)

The instantaneous velocity of the material point of a rigid body touching another body. In the displayed formula, ${\mathbf{r}}_P$ is its position relative to the centre of mass, and $\mathbf{\Omega }$ is the body's angular velocity. The rotational term is a cross product. This material-point velocity differs from the motion of the geometrical contact location: in rolling without slipping on a stationary plane it vanishes even while the contact location advances. For a sphere touching a horizontal plane, with rightward centre velocity $v$ and anticlockwise spin $\Omega$, its tangential component is $v+a\Omega$.