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

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

...two puzzles...

[What's New?] [OverView] [BackLinks] [Table of Contents (Pre/Post)] [References]

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

dv=adt; d(v^2)/2=adx becomes dw=Adt; c^2dG=Adx.

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
  • Send comments/suggestions to
  • For source, cite URL at
  • 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 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.

    AnySpeed Engineering Complex ColorMath Information Physics NanoWorld Explorations Reciprocal World Silicon River StarDust in the Lab Web Puzzlers
    Atomic Physics Lab Center for Molecular Electronics Center for NeuroDynamics Physics & Astronomy Scanned Tip and Electron Image Lab

  • Table of Contents

    AnySpeed Engineering

    Post-Transform Relativity

    Other references to the strategies described here include:

  • P. Fraundorf, "Modernizing Newton, to work at any speed", physics/0110020 ( e-print archive).
  • P. Fraundorf, "An experience model for anyspeed motion", physics/0109030 ( e-print archive).
  • P. Fraundorf, "Modeling motion at any speed, ala Minkowski" (poster), 1999 AAPT Summer Meeting in San Antonio.
  • P. Fraundorf, "Teaching Newton with anticipation...", physics/9710013 ( archive, Los Alamos NM, 1997). 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, "Some minimally-variant map-based rules of motion at any speed", physics/9704018 ( archive, Los Alamos NM, 1997), updated 23 Oct 97 with newer errata here! Also pages 123456789tf in GIF format.
  • P. Fraundorf, "A one-map two-clock approach to teaching relativity in introductory physics", physics/9611011 ( archive, Los Alamos NM, 1996). 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, "Non-coordinate time/velocity pairs in special relativity", General Relativity & Quantum Cosmology gr-qc/9607038, ( 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, "Three self-consistent kinematics in (1+1)D special relativity", General Relativity & Quantum Cosmology, gr-qc/9512012, ( archive, Los Alamos NM, 1995). This is my first note, dealing with (1+1)D SR only, on the non-coordinate time/velocity pair approach.