Bidang, Iban people. Sarawak 20th century, 48 x 107 cm. Nabau motif. From the Teo Family collection, Kuching. Photograph by D Dunlop. |

Consider a particle P described by a repetitive chain of historically ordered space-time events

$\Psi \left( \bar{r}, t \right) ^{\sf{P}} = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} \ldots \, \sf{\Omega}_{\it{i}} \, \ldots \, \sf{\Omega}_{\it{f}} \, \ldots \right)$

which are characterized by their position $\bar{r}$ and time of occurence $\, t \,$. The separation between some arbitrary initial and final pair of events is given by

$\Delta \overline{r} = \overline{r}_{\it{f}} - \overline{r}_{\it{i}}$

and the elapsed time between events is

$\Delta t = t_{\it{f}} - t_{\it{i}}$

Definition: the **velocity** is the ordered set of three numbers

$\begin{align} \overline{\sf{v}} \equiv \frac{ \Delta \overline{r} }{ \Delta t } \end{align}$

Definition: the **speed** of P is the norm of the velocity

$\sf{v} \equiv \left\| \, \overline{\sf{v}} \, \right\|$

Consider measuring this quantity based on observations of length and time. Recall that length is defined only for particles that are at least as big as atoms. And we also presume that P is compared to a frame of reference that includes a calibrated rod and a calibrated clock. To determine the velocity first measure the elapsed time between events $\sf{\Omega}_{\it{i}}$ and $\sf{\Omega}_{ \it{f}}$ as

$\begin{align} \Delta t = \left( k-j \right) \hat{\tau}^{ \mathbf{\Theta}} = \left( f - i \right) \hat{\tau} ^{\sf{P}} = \frac{ f - i }{ \nu } \end{align}$

where $k-j$ is the number of clock cycles between events, $\; \hat{\tau}$ is a period and $\nu$ is the frequency of P. Then make three length measurements $\ell _{x}$ $\ell _{y}$ and $\ell _{ z}$ along the spatial axes. Combine them to obtain the separation vector $\Delta \bar{r} = \left( \ell _{x}, \, \ell _{y}, \, \ell _{z} \right)$ between events. This separation is due to a sum of displacements

$\Delta \bar{r} = \Delta \bar{r}^{ \, \sf{\Omega}}_{i} + \Delta \bar{r}^{ \, \sf{\Omega}}_{i+1} + \, \ldots \, + \Delta \bar{r}^{ \, \sf{\Omega}}_{f}$

where $\Delta \bar{r}^{ \, \sf{\Omega}}$ notes the displacement of P during each orbital cycle $\sf{\Omega}$. As discussed earlier atomic cycles are separated from each other by one wavelength $\, \lambda \,$. And if measurements are not too disruptive so that $\lambda$ is constant, then the distance between initial and final events is

$\Delta r \equiv \left\| \, \Delta \bar{r} \vphantom{\ell_{y}} \, \right\| = \left\| \, \left( \ell _{x}, \, \ell _{y}, \, \ell _{z} \right) \, \right\| = \left( f- i \right) \lambda$

Combining these observations gives the measured speed of P as

$\begin{align} \sf{v} = \left\| \, \frac{ \rm{\Delta} \mathit{\bar{r}}}{ \rm{\Delta} \mathit{t} } \, \right\| = \frac{ \rm{\Delta} \mathit{r} }{ \rm{\Delta} \mathit{t}} = \mathit{ \frac{ \left( f- i \right) \, \lambda}{ \left( f- i \right) \, \hat{\tau}} } = \nu \lambda \end{align}$