If we measure all distances *x* from the vantage point of a
mapframe with interlinked yardsticks and
synchronized clocks, along with chase-plane time T
and GK-velocity V≡d*x*/dT
it is traditional to define map-time *t* and coordinate-velocity
*v*≡d*x*/d*t*,
as well as traveler (proper) time *τ* and proper-velocity
*w*≡d*x*/d*τ*. The Galilean
kinematic provides the simplest expressions at low speed,
but the traveler-kinematic (proper-time/velocity/acceleration) is
easier to calculate other stuff with at high speeds.
Proper-velocity *w* is proportional to the momentum of a moving object,
and can be determined from GK-velocity V using the relation
*w* = V Sqrt[1+¼(V/*c*)^{2}].
Coordinate-velocity *v* in turn relates to proper-velocity via
*v* = *w*/Sqrt[1+(*w*/*c*)^{2}].
```
(Q1)
Can you show from this last expression (which follows directly
from the metric equation) that finite proper-velocities imply
coordinate-velocities which are always less than lightspeed
```

*c*?

We thus have three familiar ways to
describe a traveler's velocity at any speed, with respect
to a chosen reference or map frame. To minimize
confusion when talking about inter-convertable
velocities, defined with
reference to distances measured in a single inertial frame
but differing based on whose time
is being considered, we find it convenient to report
coordinate-velocity *v* (distance
traveled per unit map time) in units of [lightyears per coordinate year]
or [*c*], GK-velocities *V* are reported simply
in [lightyears per chase-plane year] or [ly/gy], and
proper-velocity *w* (distance traveled per unit
proper time) in [lightyears per traveler year] or [*rb**].
Units of years and lightyears are used here since
a typical acceleration involving
people, namely the acceleration due to gravity on earth,
is 1.03 [ly/yr^2]. For example, a proper-velocity of
1 [lightyear per traveler year] marks the transition between
relativistic and non-relativistic behaviors. ```
(Q2)
With a bit of manipulation of the equations above, can you show that
1[rb] = 0.7071[c] = 0.9102[ly/gy]?
```

```
(Q3)
Given this, how much traveler time Δ
```*τ* elapses
during constant 1-gee
acceleration for a million lightyears distance?

Also, what's the final proper
velocity

```
(Q4) What
is the proper-velocity "land-speed record" for objects accelerated by
man?
```

This was probably attained in November 1995 by electrons
in the LEP2 accelerator
at CERN in Geneva, which were accelerated through an electric potential
of 70 billion
volts and hence had a kinetic energy of
There is a simple way to keep track of these effects. Note from above
that proper-velocity *w* can be written in terms of coordinate
velocity as
*w* = γ*v* =
*v*/Sqrt[1-(*v*/*c*)^{2}].
If one
object is moving rightward with coordinate speed
*v*_{1} in the frame in which you measure
distances, and a second object is moving leftward toward the first
with speed *v*_{2} in that
same frame, then the proper-velocity of the first object
in the frame of the second is *w*_{12} =
γ_{1}γ_{2}(*v*_{1}+*v*_{2}).
In other words, when calculating relative proper velocities, the coordinate
velocities add while the gamma values have to be multiplied.
This expression then allows
one to calculate the relative speeds and energies attainable when
throwing objects (like
elementary particles) at each other at relativistic speeds from
opposite directions. ```
(Q5)
What is the relative proper-velocity, in lightyears per traveler year,
of two 70 GeV electrons in head-on collision trajectories?
```

This may be
the "relative-speed record" for objects accelerated by man.

Send comments, your 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.

` (Thanks. /philf :)`