PACS: 03.30.+p, 01.40.Gm, 01.55.+b

This is in part due to the facts that: **(i)** Newton's laws work so well in
routine application, and **(ii)** introductions to special relativity (often
patterned after Einstein's introductions to the subject) focus on
discovery-philosophy rather than applications. Caution, and inertia
associated with old habits, may play a role as well. It is nonetheless
unfortunate because teaching Newtonian solutions without relativistic ones
at best leaves the student's education out-of-date, and deprives them of
experiences which might spark an interest in further physics education. At
worst, partial treatments may replace what is missing with misconceptions
about the complexity, irrelevance, and/or the limitations of relativity in
the study, for example, of simple things like uniform acceleration.

Efforts to link relativistic concepts to classical ones have been with us
from the beginning. For example, the observation that relativistic objects
behave at high speed as though their inertial mass increases in the
**p** = *m***v** expression, led to the definition
(used in many early textbooks {e.g. French})
of relativistic mass *m*' = *m gamma*.
Such efforts are worthwhile because they can: **
(A)** potentially allow the introduction of relativity concepts at an earlier
stage in the education process by building upon already-mastered classical
relationships, and **
(B)** find what is fundamentally true in both classical
*and* relativistic approaches. The concepts of transverse (*m*'
) and longitudinal (*m*'' = *m gamma*^3) masses have
similarly been used {e.g. Blatt} to preserve relations of the form *F*_{x}=*ma*_{x}
for forces perpendicular and parallel, respectively, to the velocity
direction.

Unfortunately for these relativistic masses, no deeper sense has emerged in which the mass of a traveling object either changes, or has directional dependence. Such masses allow familiar relationships to be used in keeping track of non-classical behaviors (item A above), but do not (item B above) provide frame-invariant insights or make other relationships simpler as well. Hence majority acceptance of their use seems further away now {e.g. Adler} than it did several decades ago {Goldstein}.

A more subtle trend in the literature has been toward the definition of a
quantity called proper velocity {cf. SearsBrehme,
Shurcliff}, which can be
written as **w** = *gamma* **v**. We use the
symbol *w* here because it is not in common use elsewhere in relativity
texts, and because *w* resembles *gamma v* from a distance. This quantity
also allows the momentum expression above to be written in classical form as
a mass times a velocity, i.e. as **p** = *m***w**.
Hence it serves one of the ``item A''
goals served by *m*' above.
However, it remains an interesting but ``homeless'' quantity in the present
literature. In other words, proper-velocity differs fundamentally from the
familiar coordinate-velocity, and unlike the latter has not in textbooks
been linked to a particular reference frame. After all, it uses distance
measured in an inertial frame but time measured on the clocks of a moving
and possibly accelerated observer. However, there is also something deeply
physical about proper-velocity. Unlike coordinate-velocity, it is a
synchrony-free (i.e. local-clock only) {cf. Winnie,
Ungar1} means of
quantifying motion. Moreover, A. Ungar has recently made the case
{Ungar2} that proper velocities, and not coordinate
velocities, make up the
gyrogroup analog to the velocity group in classical physics. If so, then, is
it possible that introductory students might gain deeper physical insight
via its use?

The answer is yes. We show here that proper-velocity, when introduced as
part of a ``one-map two-clock'' set of time/velocity variables, allows us to
introduce relativistic momentum, time-dilation, *and* frame-invariant
relativistic acceleration/force into the classroom without invoking
discussion of multiple inertial frames or the abstract mathematics of
Lorentz transformation (item A above). Moreover, through use of
proper-velocity many relationships (including those like velocity-addition,
which require multiple frames or more than one ``map'') are made simpler and
sometimes more useful. When one simplification brings with it many others,
this suggests that ``item B''
insights may be involved as well. The three
sections to follow deal with the basic, acceleration-related, and multi-map
applications of this ``two-clock'' approach by first developing the
equations, and then discussing classroom applications.

We show further that a *frame-invariant* proper acceleration 3-vector
has three simple integrals of the motion in terms of these variables. Hence
students can speak of the proper acceleration and force 3-vectors for an
object in map-independent terms, and solve relativistic constant
acceleration problems much as they now do for non-relativistic problems in
introductory courses.

We have provided some examples of the use of these equations for high school and college introductory physics classes, as well as summaries of equations for the simple unidirectional motion case (Tables I, II, and III). In the process, one can see that the approach does more than ``superficially preserve classical forms''. Not just one, but many, classical expressions take on relativistic form with only minor change. In addition, interesting physics is accessible to students more quickly with the equations that result. The relativistic addition rule for proper velocities is a special case of the latter in point. Hence we argue that the trend in the pedagogical literature, away from relativistic masses and toward use of proper time and velocity in combination, may be a robust one which provides: (B) deeper insight, as well as (A) more value from lessons first-taught.

**Table II.** Unidirectional motion equations involving constant acceleration from rest, and ``2nd law'' dynamics, in classical and two-clock relativistic form. The full (3+1)D version, with equalities valid for any map-frame, differs in that proper force & acceleration are **divided by** *gamma*_{perp} relative to momentum & velocity, but **multiplied by** *gamma*_{perp} relative to energy & d**v**/d*t*.

**Table III.** Unidirectional motion equations involving distances measured in more than one map-frame, in classical and two-clock relativistic form.

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