# Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of velocity of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus.

## Definition

Consider a moving rigid body and the velocity of a particle/point P along the body being a function of the position and velocity of a center particle/point C and the angular velocity ${\vec {\omega }}$ .

The linear velocity vector ${\vec {v}}_{P}$ at P is expressed in terms of the velocity vector ${\vec {v}}_{C}$ at C as:

${\vec {v}}_{P}={\vec {v}}_{C}+{\vec {\omega }}\times ({\vec {r}}_{P}-{\vec {r}}_{C})$ where ${\vec {\omega }}$ is the angular velocity vector.

The material acceleration at P is:

${\vec {a}}_{P}={\frac {{\rm {d}}{\vec {v}}_{P}}{{\rm {d}}t}}$ ${\vec {a}}_{P}={\vec {a}}_{C}+{\vec {\alpha }}\times ({\vec {r}}_{P}-{\vec {r}}_{C})+{\vec {\omega }}\times ({\vec {v}}_{P}-{\vec {v}}_{C})$ where ${\vec {\alpha }}$ is the angular acceleration vector.

The spatial acceleration ${\vec {\psi }}_{P}$ at P is expressed in terms of the spatial acceleration ${\vec {\psi }}_{C}$ at C as:

${\vec {\psi }}_{P}={\frac {\partial {\vec {v}}_{P}}{\partial t}}$ ${\vec {\psi }}_{P}={\vec {\psi }}_{C}+{\vec {\alpha }}\times ({\vec {r}}_{P}-{\vec {r}}_{C})$ which is similar to the velocity transformation above.

In general the spatial acceleration ${\vec {\psi }}_{P}$ of a particle point P that is moving with linear velocity ${\vec {v}}_{P}$ is derived from the material acceleration ${\vec {a}}_{P}$ at P as:

${\vec {\psi }}_{P}={\vec {a}}_{P}-{\vec {\omega }}\times {\vec {v}}_{P}$ 