Map-Based Motion at Any Speed

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**Balancing Apples & Oranges**

"We go through life on earth experiencing

a downward *geometric-acceleration* on each ounce of
our being,

whose effects may be cancelled by an upward *proper-acceleration*

on the soles of our feet, if we wish to stand and not fall."

/Onana Namuh (1998)

**Staying in Our Own Seat**

"We go around curves experiencing

an outward *geometric-acceleration* on each ounce of our being,

whose effects may be cancelled by an inward *proper-acceleration*

from someone/thing nearby, if we wish to sit and not slide."

/Orcim Namuh (2001)

**What's New:** These July 2012 notes on
a modernized
intro to unidirectional motion, these July 2011
voice-thread powered interactive
Kahn-academy style tutorials on
one map + two clocks
and constant acceleration
round-trips,
as well as this
eprint on metric-first & entropy-first
approaches. Also check out this
working draft
of a paper about *proper velocity and F≤mα from the metric*,
plus a couple of summer 2009
x-ct versus
ρ-cτ plots of accelerated motion inspired by Dolby and Gull.
Also check out this
earlier
plot of
Lorentz factor γ-1 versus proper velocity *w* that inspired the dispersion
(KE vs momentum) plot at right. Here are some related images put together for
Wikimedia Commons:
[1,
2,
3,
4,
5,
6,
7,
8
].
Check out these 20 Nov 2007 notes for
intro physics students on what their course might NOT be telling them
about cool stuff you can do with
vector
products, and this
plot of
kinetic
energy versus momentum that includes everything?
Here are some 9 Apr 2007 notes on
natural units
for magnetism, inspired by the concept of
proper-force as a
tool for visualizing the anyspeed connection between m**a** and
d**p**/dt. These 20 Jan 2007 modern physics
notes for a how
things work course illustrate the
circular tradeoff between
motions through time and space, and link to anyspeed dynamics
plots and
calculators.
Here's the 14 June 2005
upgrade
of our note on making the most of ``one-frame concepts'' first in teaching
kinematics, and a ``map-based mechanics''
derivation of
Biot-Savart (illustrated using the frame-invariant proper-force) for
comparison to the
multiframe
derivation here.
A "live
remote platform" for empirical studies of anyspeed spacetime (and
nanoworlds
too) for our April 2005 SLAPT
workshop at SIU-E.
One
path from the
metric equation to Lorentz transforms. See if you can find
some mistakes in this
equation
appendix describing motion in terms of physical coordinates
referenced to a local map-frame. Did you catch "*Anyspeed Acceleration* -
The Movie"?
How about a modern story line
for the
intro-physics transition between kinematics and dynamics? A
javascript calculator and
equation summary for one-frame views of unidirectional motion, plus
beginnings
for a system of classroom-ready "web transparencies" on emerging science.
A draft paper on
subtle ways to modernize introductory dynamics.
An *extreme physics*
motion simulator using Adobe
Atmosphere, along with
some notes on discovering patterns in high speed motion on your own.
Minkowski as a *classic*
pioneer of deep simplification, in the web-version
of our presentation for a symposium on
"The Legacy of Edwin T. Jaynes". An interpretive
cartooning festival with inital focus on the problem sets
above. "Teaching Newton with anticipation..." is our latest adaptation for
introductory physicists. Answers to the puzzlers above can be
found here
and here.
Check out the new discover it
yourself page, and see how far you can take it. A Pythagorean
infomercial, a "vaccine" to empower intro-students
now and minimize pain later, a Java (JDK1.1) applet
version of our anyspeed solver,
and an updated note on map-based
rules. Also find AAPT Conference abstracts (one
for summer 1998, three for
winter), and one way to solve for everything
in an anyspeed acceleration problem.

**What's Young:** A non-trivial set of web puzzlers to
solve with these
equations , a light-meme
on the relation between circular and
hyperbolic angles, and some map-based
rules of motion, patterned after those of Newton but good at any speed.
For more on this cf. physics/9704018.
As to *curved* space-time on earth, here's how Newton-like gravity arises from one non-flat metric and it's derivatives.
An any-speed primer is excerpted here.

**What's Mature:** We have MathCAD worksheets (snapshot of
one here)
on constant acceleration in two-clock
and three-clock
relativity, readable with MathSoft's free browser.
These allow one to ``play with'' the equations discussed here in
both uni-directional and 3D applications.

**What's Ancient:** An upgrade of physics/9611011
for teachers starting relativity with the metric
equation, rather than with Lorentz transforms. It uses a
``synchrony-free'' speed with no limit, a way to add velocities
fast, and a frame-invariant ``oriented-scalar'' form for Newton's
2nd law. The revision adds clearer language, discussion of
classroom applications, and 3 tables.

Check out the browser-readable (and most up-to-date) version here . Share your thoughts here. A copy optimized for printing with Adobe's free Acrobat PDF reader is here. April APS/AAPT conference abstract.

**Hot Lists:** Our Accel-1D Solver
won a National Academies Press ``coolest science
site'' award in November 1996.

**Soon:** The high (over 30%) fixed "annual percentage
rate" on fuel mass during 1 "gee" accelerations. These
commentable tutorials below might provide some clues in this context...

Check out a Synopsis of the Science These pages contain resources to empower students familiar with only classical kinematics, in the solution of relativistic acceleration problems with variables defined in context of a single inertial frame. They include web-interactive solvers, a variety of examples and derivations, the self-contained Andromeda problem, and a list of yet unanswered questions. The key to these is a more careful operational definition of time (to bypass Newtonian misconceptions from square one), and use of the metric equation as a spacetime extension of Pythagoras' theorem with traveler-time the invariant. These naturally lead to use of multiple kinematics (time/velocity pairs) defined in terms of distances measured with respect to a single inertial reference frame. Such "non-coordinate" variables allow us to see things more simply, and in the process to find surprising uses for Newton's and Galileo's equations.

Something you might discover herein, for example, is that you
can describe accelerated uni-directional motion at any speed by
replacing "coordinate", by "proper",
acceleration *and* velocity, and then using *c*^{2}γ
in place of ½*v*^{2} in the work-energy equation.
In other words:

To list some current outside web-links that *point
in* to the map-based relativity stuff at this site...

- Accel-1D Browser-Interactive Solver - solving anyspeed acceleration problems since mid-1995 w/3393 visits in the first year.
- Newer java applet and java script calculators for unidirectional acceleration at any speed.
- With a few tweaks, your intro-physics class can take Newton and Galileo along into spacetime.
- Your class may prefer to first explore high speed effects first hand on this platform by themselves.
- Defining time carefully lets one introduce Newtonian and anyspeed mechanics.
- A potentially exhaustive set of mini-puzzlers, with
hints and
*some*solutions. - Slow example - A
non-relativistic Solver example.

- Fast example & twin adventure plots - A relativisitic Solver example.

Relativity Rap-Sheet - pedagogical sprinkles & equations dressed up (1887 visits before Aug '96).

- A brief synopsis of map-based relativity science.
- Our 2-clock relativity paper for teachers starting from the metric equation.
- Some laws of motion, like Newton's, but good at any speed.
- A pre-transform primer for students and educators - With exercises!
- Regular derivations -
Single-frame view of constant proper acceleration.

- Conceptual physics derivation - Toward a graphical derivation.

- Inter-kinematic equations - Three kinematics from velocity &
acceleration.

- Unsolved problems -
Unexplored analytical solutions.

The Andromeda Problem - empowering students with a stand-alone problem.

- Debriefing & more
- Overview for the problem.

x-tv Plots - Universal 1D constant acceleration plot applications.

- The map-based (3+1)D universal acceleration plot (JavaView interactive)

- x-ct Plots -
multiframe plots of position & time.

- Time dilation - A two
event problem.

- Length contraction - A
three event problem.

- Magnets - Length
contraction on your refrigerator.

- Barn/Pole adventure -
The farmer's dilemma.

- Twin adventure - Two
inertial views of differentially aging twins.
- 4-vectors - Thinking
in 4 dimensions.

**Other references**
to the strategies described here **include:**