Speed
//Acanthophracta (detail),// Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.
Acanthophracta (detail), Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.

An analysis of velocity measurement shows that the speed of a particle can be written in terms of its frequency $\nu$ and wavelength $\lambda$ as

$\begin{align} \sf{v} = \nu \lambda \end{align}$

If the particle contains lots of quarks we can use Planck's postulate $E = h \nu$ to express the frequency in terms of the mechanical energy $E$ as

$\begin{align} \sf{v} = \frac{ \it{E} }{\it{h}} \lambda \end{align}$

And if the frame of reference is inertial then de Broglie's postulate $\lambda = h / p$ can be used to replace the wavelength with the momentum $p$ to obtain

$\begin{align} \sf{v} = \left( \frac{\it{E}}{\it{h}} \right) \left( \frac{\it{h}}{\mathit{p}} \right) = \frac{ \mathit{E} }{ \mathit{p} } \end{align}$

Now if the particle under consideration is a photon then its mass $m$ is zero, and its energy is $E=cp$. So the speed of a photon under these conditions is

$\begin{align} \sf{v} = \frac{ \mathit{E} }{ \mathit{p} }= \frac{ \mathit{cp} }{ \mathit{p} }= \mathit{c} \end{align}$

and for this reason the constant $c$ is usually called the speed of light. But if the particle is Newtonian then it is presumably in dynamic equilibrium where its mechanical energy is $E = 2K$ and $K$ is its kinetic energy. But recall that kinetic energy is defined by $K \equiv p^{ 2} / \,2m$ so that for Newtonian particles

$\begin{align} \sf{v} = \frac{\mathit{E}}{\mathit{p}} = \frac{ \mathrm{2}\mathit{K} }{ \mathit{p} } = \left( \frac{ \mathrm{2} }{ \mathit{p} } \right) \left( \frac{ \mathit{p}^{ \mathrm{2}}}{\mathrm{2}\mathit{m}} \right) = \frac{\mathit{p} }{ \mathit{m} } \end{align}$

The term momentum is the modern English word used for translating the phrase "quantity of motion"1. So the foregoing relationship which can be rearranged as

$p   = m \sf{v}$

was stated by Isaac NewtonXlink.png when he wrote

"Quantity of motion is a measure of motion that arises from the velocity and the quantity of matter jointly."2

This direct proportionality between speed and momentum is traditional and simple. It can be used to eliminate the momentum in some previously defined quantities such as the Lorentz factor which can now be expressed in a way that is closer to Lorentz's original3 formulation

$\begin{align} \gamma \equiv \frac{ 1 }{ \sqrt{ \ 1 - \left( \, p/mc \right)^{2} \ \vphantom{{\left( \, p/mc \right)^{2}}^{2}} } } = \frac{ 1 }{ \sqrt{ \ 1 - \left( \, \sf{v}/\mathit{c} \right)^{2} \ \vphantom{{\left( \, p/mc \right)^{2}}^{2}} } } \end{align}$

And for Newtonian particles the kinetic energy can be written as

$\begin{align} K \equiv \frac{\, p^{ 2} }{ 2m } \end{align} = \small{\frac{1}{2}} \normalsize{ m \sf{v}^{\mathrm{2}} }$

Right.png Next step: acceleration.
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License