This is a derivation first presented at a UM-StL Physics & Astronomy Journal Club session in March 1996, and then incorporated into our poster for the March 1996 American Physical Society Meeting. The posting here will begin with some scratch GIF files with figure labels missing, and be updated and improved for reference by students, educators, and textbook authors as time permits. For now, better plots

From this single curve in the variable range of interest, one can
thus see the quantitative relationship between *V*/*c*,
*aT*/*c* and *ax*/*c*^{2} for any
unidirectional constant acceleration problem! Co-incidentally,
if the acceleration is *a* = 0.969g (i.e. about one
"earth-gravity"), the "dimensionless" x-axis reads out
directly in lightyears, while times on the y-axis are directly in years.
Regardless of acceleration, velocities on these plots are always
in units of the speed of light.

To quantify this, we capitalize the familiar Galilean variables
time T and velocity V = d*x*/dT, but define coordinate-velocity
v = d*x*/dt in terms of the map time t elapsed on synchronized
clock's in the inertial frame in which distance *x* is
measured, and proper-velocity *w* = d*x*/dτ in terms
of the proper time τ (tau) elapsed on our accelerated
traveler's clock. To keep the 3 velocities conceptually distinct,
"coordinate" velocity in special relativity v is usually
measured in units of [lightyears *per map year*] or
*c*, while traveler (proper) velocity w can be
measured in units of [lightyears *per traveler year*]
or "rodden-berries". In these terms, Einstein's high
speed clock predictions say that
γ = 1/Sqrt[1-(*v*/*c*)^{2}], and
w = γv. From these, it follows also that
γ = Sqrt[1+(w/*c*)^{2}]. For our constant
acceleration problem here, it is now easy to show that
γ = 1 + *ax*/*c*^{2} =
1 + ½(*aT*/*c*)^{2} =
Sqrt[1 + (at/*c*)^{2}] =
Cosh[a τ/*c*] as well.

These equations provide 3 more curves for our original universal
acceleration plot (or 4 if we want to also plot γ =
1 + *ax*/*c*^{2} ). The 3 curves are:

**hyperbolic:**w/c = at/c = Sqrt[γ^{2}- 1] = Sqrt[{ 1 + (ax/c^{2}) }^{2}- 1]**exponential:**aτ/c = ArcCosh[γ] = ArcCosh[1 + ax/c^{2}], and**asymptotic to c:**v/c = Sqrt[1-1/γ^{2}] = Sqrt[1 - 1/{1+ax/c^{2}}^{2}].

However, when we plot these curves for larger velocities, the velocity
and time curves
split up to reveal major differences in the experience of observers
party to high speed
adventures. In this way a single plot can be used to visualize any
relativistic 1D constant
acceleration problem, and particularly the relationship of all
6 variables describing
time and velocity for inertial, accelerated, and "Galilean-kinematic"
observers, as a function of traveler position *x* as measured
in the map-frame of choice. Since all of this is done using
distances and 3 kinematics referred to a single
inertial frame, one does not need multiple inertial frames,
Lorentz transforms, length contraction, or even calculus to
understand and put
these results to quantitative use.

Sections to follow here examine *quantitative solution* of
relativistic unidirectional constant
acceleration problems, using either:
(i) the above graphs,
(ii) Galileo's equations with added help from
conversions to the kinematic variables of interest, or (iii) acceleration equations in inertial, traveler, and mixed
kinematics as well. Analytic solutions for unidirectional acceleration problems are provided for most, but not all,
possible combinations of input variables.

Send comments, possible answers to problems posed, and/or complaints, to philf@newton.umsl.edu.

` (Thanks. /philf :)`