Spatial acceleration
# Spatial acceleration

In physics, the study of rigid body motion allows for several ways to define the acceleration of a body. The usual definition of acceleration entails following a single particle/point of a rigid body and observing its changes in velocity. Spatial acceleration entails looking at a fixed (unmoving) point in space and observing the change in velocity of the particles that pass through that point. This is similar to the definition of acceleration in fluid dynamics, where typically one measures velocity and/or acceleration at a fixed point inside a testing apparatus.

## Definition

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:

${\begin{aligned}{\boldsymbol {\psi }}_{P}&={\frac {\partial \mathbf {v} _{P}}{\partial t}}\\[1ex]&={\boldsymbol {\psi }}_{C}+{\boldsymbol {\alpha }}\times (\mathbf {r} _{P}-\mathbf {r} _{C})\end{aligned}}$

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:

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

