The **Results Report** will then read:

*Galilean (Chase-Plane Kinematic) Results*:

These values are obtained from the usual
Galilean constant acceleration equations, in particular
V_{f}=±Sqrt[V_{o}^{2}+2α Δx]
and ΔT=[V_{f}-V_{o}]/α.
The actual constant acceleration problem is now solved,
and only routine conversion of velocities and elapsed time
to the two relativistic kinematics remains.

*Accelerated Traveler (Proper Time) Results*:

These traveler-frame results follow using
a routine
protocol for converting from values for the
Galilean kinematic, via w=V*Sqrt[1+0.25*(V/c)^{2}] and
Δτ=[ArcSinh(w_{f}/c)-ArcSinh(w_{o}/c)]*c/α
where c=3x10^8 m/s is the speed of light. In this
particular problem, the traveler values are quite
different from the Newtonian values.

*Relativistic Inertial-Frame (Map Kinematic) Results*:```
```

`Starting Coordinate Velocity: v`_{o} = 0 [c or lightyears/iyear]

`Final Coordinate Velocity: v`_{f} = {0.981, -0.981} [c or lightyears/iyear]

`Map Time Elapsed: dt = {4.874, -4.874} [years]`

These inertial-frame results follow by routine conversion from the
traveler-frame values, using v=w/SQRT[1+(w/c)^2], and
dt=[w_{f}-w_{o}]/α. In this
particular problem, the map kinematic values are quite different
from the Galilean values.

To consider the clocks of the driver, note that the traveler frame (or proper) time elapsed is 2.247 traveler years! Moreover, her final velocity is 5.028 roddenberries -- meaning that she is now traveling more than 5 lightyears of distance (as measured in the rest frame of our solar system) per year of her own time. Speaking of the spectator or solar-system point of view, however, the time elapsed on earth during this race is 4.874 inertial years. Thus her friends back home are now almost 5 years older at the end of the race, even though fewer than 27 months have elapsed for her. Moreover, from their point of view, she has a coordinate velocity of only 0.981 c, since to inertial observers the interconnected fabric of spacetime does indeed limit travelers to lightspeed or less.

**Picturing It**: An approximate *graphical* solution
using distance-time
relationships between the variables (and letting
α = 1[ly/y^2] exactly) is shown below. Actually,
the solution is exact if the acceleration is 0.97 g = 1[ly/y^2],
instead of 1 g exactly. A
blowup
near the origin of this plot can be used to
graphically
solve acceleration problems at lower everyday speeds.

**Making Quadruplets**: This graphical strategy also works if
we *combine* constant acceleration segments. For example,
one of the famous "twin paradox" problems might involve a 4 ly
acceleration leg like that discussed in this problem, followed
by a second 4 ly leg for decelerating to a stop, a 3rd 4 ly leg
for accelerating in the return direction, and a 4th deceleration
leg bringing an inertial twin's traveling sister back home.
After such an adventure, our traveling twin will have aged 4
times 2.247 or about 9 years, while the twin brother who stayed
home will have aged nearly 20 years. Some cool things about
this approach to calculating and visualizing the adventure are:
**(i)** only one inertial frame is needed, so confusions built
into our language about things like length contraction and
simultaneity aren't involved ; **(ii)** the Galilean solution
of this problem remains part of the overall solution, and in fact
can serve as the means to actually do the calculation,
followed by conversion to velocities and elapsed times
associated with the other kinematics, and **(iii)** all
variables can be shown on a single graph, allowing
visualization and estimation without total dependence on
equations that some find abstract at best.

The resulting twin-adventure graph is also (I think) pleasant to
look at! To keep the details of the plot from obscuring its
symmetry, to keep the onus on the reader to fill in the blanks,
and of course to conserve bandwidth on a very busy web, we
provide a reduced version of this twin plot for your enjoyment
below. Can **you** guess what the curves represent *without
downloading* the full sized image?

Mindquilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter linked subset of pages" since 4/7/2005: .