arXiv:physics/9611011 (xxx.lanl.gov archive, Los Alamos, NM, 1996).

It follows from above that two velocities will arise as well, namely the
coordinate-velocity **v** = d**x**/d*t*, and
proper-velocity **w** = d**x**/d*tau*. The
first velocity measures map-distance traveled per unit map time, while the
latter measures map-distance traveled per unit traveler time. Each of these
velocities can be calculated from the other by knowing the
velocity-dependence of the ``traveler's speed of map-time'' *gamma* =
d*t*/d*tau*,since it is easy to see from the definitions above that:

Because all displacements d**x** are defined with respect to our map frame,
proper-velocity is not simply a coordinate-velocity measured with respect to
a different map. However, it does have a well-defined home, in fact with
many ``brothers and sisters'' who live there as well. This family is
comprised of the velocities reported by the infinite number of moving
observers who might choose to describe the motion of our traveler, with
their own clock on the map of their common ``home'' frame of reference
{Noncoord}. One might call the
members of this family ``non-coordinate
velocities'', to distinguish them from the coordinate-velocity measured by
an inertial observer who stays put in the frame of the map. The cardinal
rule for all such velocities is: *everyone measures displacements from
the vantage point of the home frame* (e.g. on a copy of a reference-frame
map in their own vehicle's glove compartment). Thus proper velocity **w**
is that particular non-coordinate velocity which reports
the rate at which a given traveler's position on the reference map changes,
per unit time *on the clock of the traveler*.

Because Eqn. 2 allows one to relate velocities to energy, an important part of relativistic dynamics is in hand as well. Another important part of relativistic dynamics, mentioned in the introduction, takes on familiar form since momentum at any speed is

Given these tools to describe the motion of an object with respect to single
map frame, another type of relativistic problem within range is that of time
dilation. From the very definition of *gamma* as a ``traveler's speed of
map-time'', and the velocity relations which show that *gamma* >= 1, it
is easy for a student to see that the traveler's clock will always run
slower than map time. Hence if the traveler holds a fixed speed for a finite
time, one has from Eqn. 2 that traveler time is dilated
(spread out over a larger interval) relative to coordinate time, by the
relation

Convenient units for coordinate-velocity are [lightyears per map-year] or
[c]. Convenient units for proper-velocity, by comparison, are [lightyears
per traveler year] or [ly/tyr]. When proper-velocity reaches 1 [ly/tyr],
coordinate-velocity is 1/Sqrt{1+1}=1/Sqrt[2] = 0.707[c]. Thus
*w*=1[ly/tyr] is a natural dividing line between classical and
relativistic regimes. In the absence of an abbreviation with mnemonic value
for 1 [ly/tyr], students sometimes call it a ``roddenberry'' [rb], perhaps
because in english this name evokes connections to ``hotrodding''
(high-speed), berries (minimal units for fruit), and a science fiction
series which ignores the lightspeed limit to which coordinate-velocity
adheres. It is also worth pointing out to students that, when measuring
times in years, and distances in light years, one earth gravity of
acceleration is conveniently *g* = 1.03[ly/yr^2].

We show here that the major difference between classical and two-clock
relativity involves the dependence of kinetic energy *K* on velocity.
Instead of (1/2)*mv*^2, one has *mc*^2(Sqrt[1+(*w*/*c*) ^2]
-1) which by Taylor expansion in *w*/*c* goes as (1/2)*mw*^2 when
*w* << *c*. Although the relativistic expression is more complicated, it is not
prohibitive for introductory students, especially since they can first
calculate the physically interesting ``speed of map-time'' *gamma*, and
then figure *K*=__mc__^2(*gamma* -1). If they are given
rest-energy equivalents for a number of common masses (e.g. for electrons
*m*_{e}*c*^2 ~ 511[keV]),
this might make calculation of relativistic energies even less painful than
in the classical case!

Concerning momenta, one might imagine from its definition that
proper-velocity *w* is the important speed to a relativistic traveler trying
to get somewhere on a map (say for example to Chicago) with minimum traveler
time. Eqn. 3 shows that it is also a more interesting speed
from the point of view of law enforcement officials wishing to minimize
fatalities on futuristic highways where relativistic speeds are an option.
Proper velocity tells us what is physically important, since it is proportional
to the momentum available in the collision. If we want to ask how long it
will take an ambulance to get to the scene of an accident, then of course
coordinate velocity may be the key.

Given that proper velocity is the most direct link to physically important
quantities like traveler-time and momentum, it is not surprising that a
press unfamiliar with this quantity does not attend excitedly, for example,
to new settings of the ``land speed record'' for fastest accelerated
particle. New progress changes the value of *v*, the only velocity they are
prepared to talk about, in the 7th or 8th decimal place. The story of
increasing proper velocity, thus, goes untold to a public whose imagination
might be captured thereby. Hence proper-velocities for single 50[GeV]
electrons in the LEP2 accelerator at CERN might be approaching
*w*=*gamma* *v*=*E*/*mc*^2 *v* ~
50[GeV]/511[keV] c ~ 10^5[lightyears per traveler year], while the educated
lay public (comprised of
those who have had no more than an introductory physics course) is under a
vague impression that the lightspeed limit rules out major progress along
these lines.

For more on this subject, see our web construction table of contents. Please share your thoughts using our review template, or send comments, answers to problems posed, and/or complaints, to philf@newton.umsl.edu. This page contains original material, so if you choose to echo in your work, in print, or on the web, a citation would be cool.

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