# UM-StL Pages on Map-Based Motion at Any Speed

## a.k.a. relativity from within your space-time slice

...two puzzles...

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)

Above is a "real-time" animation of two "dropped balls" in a 50[m] diameter rotating "space-wheel",
something like that which held an exercise pod in the Hermes spacecraft
which brought travelers from Earth to Mars (and back?) in the 2015 movie The Martian.

Below find an animation of wall-clocks on a "1-gee" interstellar-shuttle
for the 6.6 traveler-year trip between our solar system and the Vega system, about 25 lightyears away.
Interesting questions about simultaneity come up in this "practical" setting.

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 c2γ in place of ½v2 in the work-energy equation. In other words:

becomes .

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

• Other physics education links that may be of interest include those at: physlink, yahoo, quantum, c3p, & tiptop.
• Please drop us a line if you have other links to suggest here.
• Edwin Taylor's web-page on Scouting Black Holes with the metric equation, and Many-Paths QM with propagators. Prof. Taylor's focus has been on new strategies for "a second course in physics", while much of our focus here is on the first.
• This stuff is Copyright (1970-99) by Phil Fraundorf
• Dept. of Physics & Astronomy, U. Missouri-StL, St. Louis MO 63121-4499
• For source, cite URL at http://www.umsl.edu/~fraundor/a1toc.html
• Version release date: 09 April 2005.
• Mindquilts site page requests ~2000/day approach a million per year.
• Page requests to a stat-counter linked subset since 4/7/2005: .
• Note: Recent rain damage is making links to newton.umsl.edu intermittent.
• Also: The approach here has evolved rapidly, so that nomenclature on older pages may lag.
• At UM-StLouis see also: accel1, cme, i-fzx, progs, si-river, stei-lab, turnovers, & wuzzlers.

### Post-Transform Relativity

Other references from this site to the strategies described in this "one of many side-projects" include:

• P. Fraundorf (2016) "The proper-force 3-vector", hal-01344268 draft, in which (we think now correct) integration of the (3+1)D constant proper-acceleration equations yields an answer to the question: How does rapidity generalize in that case?
• P. Fraundorf (2013) "Metric-first and entropy-first surprises" arxiv:1106.4698, in which threads (and illustrations) from the vantage point of more than one side-project appear, and perhaps even intertwine.
• P. Fraundorf (2012) "A traveler-centered intro to kinematics", arxiv:1206.2877 draft, in which we look at prospects for telling intro-students to define "which clock" along with "which coordinate system".
• P. Fraundorf (2005) "Modernizing Newton, to work at any speed", arxiv:physics/0110020, in which three "tweaks" to make intro-physics "anyspeed-smart" are discussed.
• P. Fraundorf (2001) "An experience model for anyspeed motion", arxiv:physics/0109030, in which an inquiry-based empirical observation exercise (involving data from an "extreme physics" airtrack) is described, to give students a modeling-workshop style introduction to the concepts of frame-dependent time.
• P. Fraundorf, "Modeling motion at any speed, ala Minkowski" (poster), 1999 AAPT Summer Meeting in San Antonio.
• P. Fraundorf (1998) "Teaching Newton with anticipation...", arxiv:physics/9710013. This is a short paper (1,2,3,4), whose first draft was named "Travelers keep your yardsticks covered!", about an "anyspeed-smart" way to introduce studies of motion in high school as well as university. Also citable as an abstract to the American Association of Physics Teachers Winter 1998 Meeting.
• P. Fraundorf (1998) "Some minimally-variant map-based rules of motion at any speed", arxiv:physics/9704018, updated 23 Oct 97 with newer errata here, but it still contains errors!
• P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics", arxiv:physics/9611011. This is a paper for teachers (with a v-w-u notation shift from papers involving the Galilean kinematic). See also PDF, new and old HTML versions as well.
• P. Fraundorf (1996) "Non-coordinate time/velocity pairs in special relativity"arxiv:gr-qc/9607038, (xxx.lanl.gov archive, Los Alamos NM, 1996). This is a paper on the "frame-dependent" or "map-based" non-coordinate time/velocity pair strategy in general. Browser (HTML) version is here.
• P. Fraundorf, "Relativistic one-D acceleration with Galileo's equations", American Physical Society March Meeting 1996, abstract [H33.49], (American Physical Society, College Park MD, 1996). This is a note on ways to use Galileo's acceleration equations in "pre-transform" relativity.
• P. Fraundorf (1995) "Three self-consistent kinematics in (1+1)D special relativity", arxiv:gr-qc/9512012. This is my first note, dealing with (1+1)D SR only, on the non-coordinate time/velocity pair approach.