Velocity

Definition

Velocity is defined as the rate of change of the position of the origin of an object frame, \(F_\alpha\) with respect to the origin of a reference frame, \(F_\beta\), resolved around the axes of \(F_\beta\):

\[ \mathbf{v}^{\beta}_{\beta \alpha} \triangleq \dot{\mathbf{r}}^{\beta}_{\beta \alpha}. \label{5.1} \]

This may, in turn, be resolved about the axes of a third frame, \(F_\gamma\) using the rotation matrix:

\[ \mathbf{v}^{\gamma}_{\beta \alpha} = \mathbf{R}^{\gamma}_{\beta} \dot{\mathbf{r}}^{\beta}_{\beta \alpha} = \mathbf{R}^{\gamma}_{\beta} \mathbf{v}^\beta_{\beta \alpha}.\label{5.2} \]

Transport Theorem

Velocity, \(\mathbf{v}^{\gamma}_{\beta \alpha}\), is not equal to the time derivative of \(\mathbf{r}^{\gamma}_{\beta \alpha}\) when there is a rotation of the resolving frame \(F_\gamma\), with respect to the reference frame \(F_\beta\). The time derivative of \(\mathbf{r}^{\gamma}_{\beta \alpha}\) is:

\[ \begin{align} \dot{\mathbf{r}}^{\gamma}_{\beta \alpha} &= \frac{d}{dt} \left( \mathbf{R}^{\gamma}_{\beta} \mathbf{r}^{\beta}_{\beta \alpha} \right) \\ &= \dot{\mathbf{R}}^{\gamma}_{\beta} \mathbf{r}^{\beta}_{\beta \alpha} + \mathbf{R}^{\gamma}_{\beta} \dot{\mathbf{r}}^{\beta}_{\beta \alpha} \\ &= \dot{\mathbf{R}}^{\gamma}_{\beta} \mathbf{r}^{\beta}_{\beta \alpha} + \mathbf{v}^{\gamma}_{\beta \alpha} \\ &= \mathbf{v}^{\gamma}_{\beta \alpha} + \boldsymbol{\Omega}^{\gamma}_{\gamma \beta} \mathbf{R}^\gamma_\beta \mathbf{r}^\beta_{\beta \alpha} \\ &= \mathbf{v}^{\gamma}_{\beta \alpha} + \boldsymbol{\Omega}^{\gamma}_{\gamma \beta} \mathbf{r}^\gamma_{\beta \alpha}. \label{5.3} \end{align} \]

which is called the transport theorem.

Velocity Registry

Figure 1 shows types of motion that can cause a velocity to register.

Figure 1 Motion causing a velocity to register (Groves, p49)

Velocity is registered when:

1. The object frame, \(F_\alpha\), moves with respect to the origin of reference frame, \(F_\beta\)
2. The reference frame, \(F_\beta\), moves with respect to the origin of object frame, \(F_\alpha\)
3. The reference frame, \(F_\beta\), rotates with respect to the origin of object frame, \(F_\alpha\)

Velocity is not registered when:

1. The object frame, \(F_\alpha\), rotates with respect to the origin of reference frame, \(F_\beta\)

Properties

Direction reversal and velocity addition doesn't hold unless there is no rotational motion between the object and the reference frame:

\[ \begin{align} \mathbf{v}^{\gamma}_{\beta \alpha} & \neq \mathbf{v}^{\gamma}_{\beta \delta} + \mathbf{v}^{\gamma}_{\delta \alpha} \\ \mathbf{v}^{\gamma}_{\alpha \beta} & \neq - \mathbf{v}^{\gamma}_{\beta \alpha}. \label{5.4} \end{align} \]

The correct relationship is:

\[ \mathbf{v}^{\gamma}_{\alpha \beta} = -\mathbf{v}^{\gamma}_{\beta \alpha} - \mathbf{R}^{\gamma}_{\alpha} \dot{\mathbf{R}}^{\alpha}_{\beta} \mathbf{r}^{\beta}_{\beta \alpha}, \label{5.5} \]

although:

\[ \left. \mathbf{v}^\gamma_{\alpha \beta} \right|_{\dot{\mathbf{R}}^\alpha_\beta = 0} = - \mathbf{v}^{\gamma}_{\beta \alpha}. \]

Velocity may be transformed from one resolving frame to another using the appropriate coordinate transformation matrix:

\[ \mathbf{v}^{\delta}_{\beta \alpha} = \mathbf{R}^\delta_\gamma \mathbf{v}^{\gamma}_{\beta \alpha}. \]

Speed is the magnitude of the velocity and is independent of the resolving axes, so \(v_{\beta \alpha} = |\mathbf{v}^\gamma_{\beta \alpha}|\). However, the magnitude of the time derivative of velocity, \(|\dot{\mathbf{v}^\gamma_{\beta \alpha}}|\), is dependent on the choice of the resolving frame.