Some One-D Acceleration Rap

``Lightspeed is not a velocity but a units-conversion i.e. it's the number of meters in one second.''
Hermann Minkowski, perhaps, had he lived much beyond his 1908 talk.

``We have a wonderfully diverse and productive physics faculty,
all unfortunately in eigenstates of mutually non-commuting observables.''
Onana Namuh/4apr1998

``Just as absolute & unconditional power results in unbounded corruption,
so absolute & unconditional support results in unbounded contempt.
This happens 'cause it's natural for humans to choose orthogonal coordinates
embedded in the context of locally-curved spacetime.
Thus we bring out the best in those around us only in a context of conditional give and take.''
Proper Velocity/29mar2004

[What's New] [Empowering Kids] [Other's Inputs] [Cartoon Thoughts] [Map-Based SR] [Intro-Text Changes] [Twins w/o Transforms] [SR Text Questions] [Equation Dressing] [Microscope Facts]

What's New:
...two puzzles...
  • 25apr2005 - "Live remote access" to spacetime experiments just outside the Oort cloud.
  • 15nov2004 - One path from the metric equation to Lorentz transforms
  • 29mar2004 - A local-coordinate based equation-collection to check for errors.
  • 28jul2003 - Storyline for a ``spacetime smart'' intro-transition from kinematics-to-dynamics.
  • 06may2003 - Javascript calculator for one-frame views of unidirectional motion.
  • 29apr2003 - Some classroom-ready "web transparencies" on emerging science.
  • 04sep2002 - "Extreme Physics" motion simulator -- What patterns in nature do you detect?
  • 28oct2001 - A note on some subtle ways to modernize intro-dynamics.
  • 01jan2000 - Get all of kinematics at any speed, from the metric equation...
  • 09aug1999 - Three abstracts & a crackerbarrel for AAPT Summer 1999 in San Antonio, TX.
  • 07jul1999 - Get the metric equation from a simple air-track experiment?
  • 01feb1999 - Ideas, like other replicable codes, evolve. If not, here's about one peril they face!
  • 16oct1998 - Learn more about Minkowski in our web talk from the Jaynes Symposium.
  • 05may1998 - Our latest note for intro physicists on teaching Newton with anticipation.
  • 05may1998 - Puzzler answers are here and here.
  • 07oct1997 - Tired of describing motion as though the 20th century didn't happen? Try this!
  • 20sep1997 - A Java version of our browser-interactive anyspeed acceleration solver.
  • 12sep1997 - Three abstracts for the Winter 1998 AAPT Conference.
  • 22apr1997 - How Newtonian gravity flows from one non-flat metric and it's derivatives.
  • 02apr1997 - A light-meme on the relation between circular and hyperbolic angles.
  • 05mar1997 - Some laws of motion, like Newton's, but good at any speed.
  • 01mar1997 - Some 1D-acceleration web puzzlers solvable with these equations .
  • 28feb1997 - Invariance-colored 2-clock eqn tables: basic, 1d from rest, accel-3d, multi-map.
  • 22feb1997 - Abstract on two-clock relativity for the April APS/AAPT conference in DC.
  • 11feb1997 - Light-memes on 2-clock, 3-clock, metric-based, & magnet relativity.
  • 22jan1997 - MathCAD worksheets (example snapshot) on 1D and 3D map-based relativity.
  • 22dec1996 - A paper on "classy" ways to use Newton's laws at hi-speed .
  • 23jul1996 - (3+1)D equations for constant acceleration in the simplest possible form.
  • 01jul1996 - Universal constant acceleration plots in (3+1)D special relativity.
  • 08jun1996 - Pre-transform relativity primer for students and educators - With exercises!.
  • 26apr1996 - Table of Contents for these notes on 1D acceleration & special relativity.
  • 15apr1996 - Unsolved problems within reach of an inquisitive beginner?
  • 09apr1996 - An exploratory problem for my introductory physics class.
  • 30mar1996 - Refrigerator magnets are held up by relativistic length contraction! Say what?
  • 22mar1996 - Conceptual Physics Derivation of the universal acceleration plot for APSMAR96.
  • 15jun1995 - A browser-interactive 1D acceleration solver.
  • Notation Key A (used on most of this page) is:

    {map-time, coordinate-velocity}={t, v}, {traveler-time, proper-velocity}={tau, w}, and {chaseplane-time, "Galilean"-velocity}={T, V}
  • Cite/Link:
  • This release dated 12 April 2005 (Copyright by Phil Fraundorf 1988-2005)
  • How about getting small instead of going fast?
  • At UM-StLouis see also: cme, infophys, physics&astronomy, programs, stei-lab, & wuzzlers.
  • UM-StL map-based motion-at-any-speed table of contents.
  • Through July '96, the access log for this page shows 1887 visits.
  • Since 1 Aug 1996, you are visitor number [broken counter].
  • Mindquilts site page requests ~2000/day, hence more than 500,000/year.
  • Page requests to a stat-counter linked subset since 4/7/2005: .

  • 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

    Empowering students.

    Being able to solve hi-speed acceleration problems quantitatively and/or graphically, prior to introducing the Lorentz transform, might make anyspeed motion accessible to a much larger number of students. It could also create an appetite for understanding how all this could possibly be consistent from other inertial frames, so that more would be motivated dig into the abstractions and wonders of (3 plus 1)D space time! What do you think?

    Things others have said.

    On avoiding Newtonian misconceptions, Prof. Narendra Jaggi of Illinois Wesleyan University pointed out the loss if we don't share (with all students) Newton's useful, self-consistent, symmetric, and elegant 3D view of the world. For the majority of students who will not take further physics courses, it is indeed a heavy question. The real beauty in flat spacetime lies in a coordinate-free 4-vector view accessible only to students ready to abstract themselves from the visceral differences between space and time. Hence for intro-students the choices may be low-speed elegance plus any-speed engineering.

    On starting relativity with the metric equation rather than Lorentz transforms, Steve Gilham wrote:

    I approve heartily of this proposal. Having had during my time as a graduate student (working on fluid flow near black holes) unlearning the quasi-Lorentzian approach normally taken to SRT, and only years later finally having that "Aha!" moment when I realised what the coordinate free geometric approach was all about, I think that starting with the geometric approach would be a great improvement all round.

    Many [who find relativity counter-intuitive (my insert)] seem to base their approach and objections on the classical two-frame style of layman's relativity overviews, which cutting directly to the metric would avoid.

    Note: Although we cut to the metric here, we aren't shy about choosing a specific reference frame. Such shyness is something inherited from early relativity's rejection of a physically special frame, with respect to which the medium carrying lightwaves (or ether) was at rest. Rejecting this doesn't prevent us from making our own choice of a physically arbitrary reference frame, in terms of which to describe motion. J. S. Bell in Speakable & Unspeakable in QM (Cambridge, 1987) argues this, & drawings of the simplest embedding diagrams in general relativity (cf. Kip Thorne's Black Holes & Time Warps, W. W. Norton, 1994) require that this be done.

    On our 2-clock relativity paper for teachers, and new approaches to relativity in general, relativity-author William Shurcliff wrote:

    I wish you had pointed out this neat fact: because proper speed can be infinite, improper speed is necessarily limited -- cannot exceed 3x10^8 m/sec. If someone asks: "Why is the speed of light limited?", we answer: "It is not limited if defined in a manner that requires no synchrony, i.e. if defined as a ratio of distance (defined most simply) and time duration (defined most simply).

    It's a wonderful subject, fun to think about. You and I believe it could be explained better, understood more simply.

    But my firm and gloomy impression is that most of the professors who teach the subject regard it as old, finished -- no need for any further thought or new terms. "Its finished. Please don't bother me with any new ideas."

    About the Andromeda Problem here, e-mail correspondent Robert Williams wrote:

    What's up man? I am a senior in High School taking my second year of Physics. One of the privileges we have is to make a "scavanger hunt" for the regular physics students. It is composed of impossible problems with crazy calculations and conversions. I know this probably sounds unheard of to you, but, I think this is one of the coolest problems I have ever seen. I must admit I don't have a clue as how to do this and probably won't for a couple more years. But what I was wondering is if you could give me just the answer. NO work just a number. I need that to put this problem on the scavenger hunt. I am dying to see the student's minds meltdown. IF you won't do it I will understand but I would truly appreciate it. I will even give you credit for the problem. I need the answer by this coming Thursday so if you can do it please do it. This would be truly awsome. Thanks alot man and you have my word that I would never let the answer be known. Thanks. Hope to hear from you in a day or so.

    On page 154 of Special Relativity by A. P. French (W. W. Norton, NY, 1968), one of the more respected older SR texts, Dr. French defines acceleration as the map-time derivative of coordinate-velocity, i.e. in our notation dv/dt, and then not surprisingly says:

    "The main lession to be learned from the above calculations is that acceleration is a quantity of limited and questionable value in special relativity. Not only is it not an invariant, but the expressions for it are in general cumbersome, and moreover its different components transform in different ways. Certainly the proud position that it holds in Newtonian dynamics has no counterpart here."

    One objective of these pages is to show that the proud position of acceleration in Newtonian dynamics does indeed have a counterpart in SR. That position is held not by coordinate-acceleration dv/dt, but by the frame-invariant proper-acceleration (ao=dw/dt=d(gamma v)/dt) as discussed in SpaceTime Physics by Taylor & Wheeler (W. H. Freeman, NY, 1963/92). Professor Taylor, when a way to solve relativistic constant acceleration problems with help from Galileo's* equations was first mentioned, wrote:

    "It is always interesting how much each older theory telegraphs the structure of the new theory -- but only after one understands the new theory!"

    Although we need a Galilean chase-plane to find out what Galileo's acceleration equations tell us about relativistic motion, Newton's equations provide considerable insight at high speeds if we only consider a proper-time clock along with the coordinates used to describe motion classically. We must also know how to define acceleration at high speeds. As shown elsewhere for the unidirectional case, at high speeds it is the second derivative of position with respect to time only in the Galilean kinematic, but it is the rate of energy gain, per unit distance-traveled per unit mass, in three co-existing kinematics! For unidirectional motion, proper-acceleration is also the coordinate-time derivative of proper-velocity, but in (3+1)D it is not the second derivative of position with respect to any time (cf. gr-qc/9607038). Although ``2nd time-derivative'' acceleration returns when we make the leap to 4-vectors, frame-invariance for acceleration (like that in Newtonian physics) stays with the proper-acceleration 3-vector alone.

    * The kinematic equations describing constant acceleration, sometimes referred to as "Newtonian" since they follow from Newtonian dynamics, were developed geometrically for describing the motion of falling objects by Galileo (cf. Dialogue Concerning the Two Chief World Systems, University of California Press, 1962).

    Interpreting the Cartoon: .

    There are a variety of levels.
  • One level (which underlies the original cartoon by John Lara as well) is of course that special relativity (SR) through Einstein enforces a speed limit on all of us! As we show on these pages, however, the speed limit applies only to a couch-potato's view of our traveler's motion, and not to the traveler's ability to get from point A to B on the map herself.
  • Another level tells us that in fact one might claim they were traveling "only two map-lightyears per traveler-year" to get out of an Einsteinian ticket! Yes, indeed, not only two but in fact any finite number of map-lightyears per traveler-year (or "roddenberries") is legal in special relativity terms!
  • Yet further, we might drop the inference that our traffic cop was being skeptical, and figure that he means to say "This IS a likely story!". After all, a (proper) speed of 2 [map-lightyears per traveler-year] is much more likely than a (coordinate) speed of 1 [map-lightyear per map-year] or c, since the latter implies a (proper) speed of infinity [map-lightyears per traveler-year]. This would have given our driver no time to hit the brakes before she arrived at (or worse, had passed up) her destination, not to mention the problems she would have seeing the police car in her rear view mirror to tell where to pull over. Thus ``Likely story!'' is also a literal evaluation of the driver's claim, and a conclusion ``Sheriff'' Einstein may have been quicker to draw than some of his deputies today (pun intended).

    Special Relativity for the "Frame-Dependent" among us. :)

    The title of Einstein's flat space-time theories, i.e. "relativity", implies that use of a special reference frame is somehow wrong. Of course, every time one writes out the coordinates of a vector, one has commited to an origin with respect to which those component values are referred. There is nothing wrong with this. The name stems instead from postulates, concerning the absence of some physically absolute reference frame (i.e. one chosen not by you, but by nature), which powered Einstein's insights to begin with.

    Unfortunately, I think, confusion about this may have resulted in a failure to fully explore the benefits of addressing relativity from the point of view of someone with a single (user-chosen) inertial reference frame. Single reference-frame thinking, of course, is the kind of thinking done by most earth-dwelling students in introductory physics courses. Because of this confusion, students are not told the relativistic things one can do with that kind of thinking. We have taken up that challenge here.

    For example, the concept of a "traveler-kinematic" relies on a traveler whose distance measurements are referred to a reference inertial frame that he or she may be moving with respect to. In other words, all equations use "map distances", dx, measured with respect to some user-defined reference frame, such as the earth's surface or the rest frame of the galaxy center. If we thus shamelessly [:)] pick a user-defined inertial frame for measuring distances, many advantages result.

    For example:

  • We don't need to use or even understand Lorentz transforms for many problems.
  • Length contraction is not an issue, since we always ask observers stuck in our inertial reference for distance measurements.
  • When we talk about two events being simultaneous, we are unambiguous since we mean simultaneous in context of our pre-defined inertial frame.
  • Everything can be plotted with respect to a single set of axes on an x-ct chart.

    Putting this to use in Introductory Physics Textbooks.

    This single-frame strategy provides ways, both graphical and analytical, for showing introductory physics students how to solve relativistic constant acceleration problems, including twin-adventure problems, using the Galilean equations and a minimum of new concepts. Strategies that will do this are scattered about these pages. We provide an overview here:

    Newtonian Physics

  • 1.1 Lose: Rest as a Preferred State -> Gain: Perpetual Motion w/o Drag
  • 1.2 Accelerated Motion: Velocity, Time & Momentum vs. Position at Speeds << LightSpeed
  • 1.3 Accelerated Motion in 1D: Energy vs. Position at All Speeds.
  • 1.4 Action-at-a-Distance Forces: Gravity, Electrostatic & Magnetic at Low Speeds

    Introductory University, College, and Conceptual Physics Courses basically cover (and provide) some or all of the foregoing in their sections on dynamics (i.e. not including sections on waves, condensed matter, and thermodynamics). We show here that with a minimum of new concepts and abstraction, one can take the next step in these same courses and either graphically or with equations...

    Add path-dependent time wrt one user-selected inertial reference (or map) frame in 1D

  • 2.1 Lose: "v-perp" -> Gain: mass-energy & relativistic twin adventures
  • 2.2 hi-speed accelerated motion - velocity, time & momentum vs. position wrt chosen frame
  • 2.3 Single-frame x-ct, and universal x-tv plots

    In the long run, the foregoing material would then be needed only in review chapters of Modern Physics and dedicated Relativity Courses. These courses might then...

    Add Lorentz Transforms in 1D

  • 3.1 Lose: Frame-Independent Simultaneity and Rigid Bodies -> Gain: Length Contraction
  • 3.2 Multi-Frame x-ct Plots
  • 3.3 Mass-Energy & Momentum as One

    In Special Relativity and the early part of General Relativity Courses, one might then...

    Add Lorentz Transforms in 3D

  • 4.1 Lose: Newtonian v & t -> Gain: 4-vector Acceleration
  • 4.2 Minkowski 4-Vectors & the (3-1)D Metric: NAVSTAR Navigation
  • 4.4 Electricity & Magnetism as One: Maxwell's Equations, Magnetron-Based Radar

    The next step is a big one. You will probably want General Relativity if you want to make a living in the relativity sciences, or if you love abstract applied math. Many ideas associated with Pythagoras theorem (so beautifully extended by Special Relativity) are lost altogether, in every place where the special relativist might simply estimate using a gravitational field. Quantities of interest not only vary from point to point (you do get to keep a differential form of Pythagoras Theorem), but most practical parameters have between 16 and 256 components! This makes the quick working out of numerical problems an intuitive challenge at best. Of course, exciting things happen as well when you...

    Add General Relativity

  • 5.1 Lose: Extended Metric Altogether -> Gain: Gravity as Geometry
  • 5.2 Reimann Geometry & Tensors with 4x4x4x4 components (Oh Joy!)
  • 5.3 Cool Geodesic Transport & Field Equations
  • 5.4 Event Horizons, Black Holes, Traversible WormHoles...

    Solving the twin paradox problem without transforms OR acceleration!

    The twin paradox as commonly phrased requires that one understand both path-dependent elapsed-time, and space-time with that strange (3+1) Minkowski signature in the metric (with its Lorentz tranforms, length contraction, and frame-dependent simultaneity). The latter is quite counter-intuitive, and as a result the twin paradox, when considered in common parlance, leads to confusion for some.

    However, to predict what happens to a twin who leaves on a voyage and then returns, only path-dependent elapsed-time need be understood. The rule is this: traveler time in context of a single inertial frame goes at a rate that is a factor of 1/gamma slower than that of a non-traveler, where gamma is defined as usual (see below). Even acceleration of the traveling twin can be ignored in solving the problem!

    Let's do it quantitatively: Twins 1 and 2 are together at event A (say their 10th high school reunion) and back together again at event B (say their 20th reunion), but twin 1 spends most of the intevening time going at coordinate-speed v traveling who knows where (it doesn't matter as long as he makes it back for the 20th reunion), while twin 2 stays in town. At event B, twin 2 will be 10 years older, while twin 1 will be 10/gamma years older, where gamma = 1/Sqr[1-(v/c)^2] as usual.

    We've ignored the accelerations and decelerations, and only require that our two twins be present at events A and B having taken different paths to get there. We've also solved the problem in that particular inertial frame for which events A and B occur at the same place. If you like, accelerations and decelerations can also be considered in such problems graphically and quantitatively without transforming to other frames, but that is not required to figure differential aging for the simplest twin-paradox adventures.

    Consistency with other inertial frames (i.e. frames for which A and B are separated in space) is further questioned in the original twin paradox problem. As you probably know, with help from the Minkowski metric or Lorentz transforms you can solve the problem in any other inertial frame, and you get the same result for aging of the twins. For example, if twin 1 spends most of the first half of their time going at speed v to the left, with the remainder being spent going at speed v back to the right, you might pick a frame moving at speed v to the left. The x-ct diagram for this solution will look quite different, but the answers will be the same.

    Anyone first seeing it might be confused by the counter-intuitive nature, when expressed in everyday parlance, of the strange linkage between space and time required to solve the problem in this second case. But this is not needed to get kids started, at least as far as relativistic-twin adventures are concerned, since you can get the answer simply by considering the path-dependence of time between events in the context of a single inertial frame.

    Some questions to ask about Special Relativity Books.

    Some of the questions I am asking of SR texts presently available are:
  • do they show students how to solve constant acceleration problems (regardless of how they define acceleration)?
  • do they show how constant proper acceleration works?
  • do they do this from the within a single-inertial frame, in terms of two instantaneously inertial frames in relative motion, in terms of Minkowski 4-vectors, or in some other way?
  • do any point out the "kinematic-pair" relationship between spatial four-velocity and proper time, as something that indeed might belong to a traveler, i.e. someone who measures distances in context of a frame with respect to which she is moving.
  • do any exploit dimensionless constant-acceleration x-tv plots to illustrate the relationship between the various velocities and times in special relativity?
  • do any make a conceptual distinction between relativistic inertial (coordinate) velocity/time (with it's hyperbolic x-ct curve under constant acceleration), and Newtonian velocity/time (with it's parabolic x-ct curve), in the relativistic regime.

    If anyone has opinions in this regard about the various books out there now, or can suggest some other questions to ask as well or instead, let me know! If you know someone working on an SR or introductory physics text, share your impressions with them as well.

    The equations, dressed up.

    Caveat: No time yet to update the variable notation in these tables. Hence these equations use the older Notation key B:

    {map-time, coordinate-velocity}={b,w}, {traveler-time, proper-velocity}={T,u}, and {chaseplane-time, "Galilean"-velocity}={t, v}

    Acceleration from Rest, 3 Kinematics:

    Four-Vector Form:

    Note in the first set of equations that, when velocities are dressed up in units of c, times in units of c/ao, and distances in units of c^2/ao, the six equations involving x plot onto only 4 curves, since v/c = aot/c, and u/c = aob/c. For relationships between kinematics, note that the right-hand side of all 6 x-equations is gamma-1, so that you have gamma expressed separately in terms of all six kinematic variables: t, v=dx/dt; tau, u=dx/d(tau); and b, w=dx/db.

    The simplest complete set of interkinematic conversion relations may be...

    Some remarkable TEM facts.

    This section now has a page of it's own, here.

    Note: Send comments and questions to This page might contain original material. Hence if you echo, in print or on the web, a citation would be cool. {Thanks. :) /pf}