Unsolved Mysteries in 1D Constant Acceleration

Did you think 1D constant acceleration was a trivial and long-finished subject? Well, this is almost true, since even relativistically the equations are simple as long as we stick with a flat metric (i.e. with special relativity).

Nonetheless, some combinations of given input variables (such as distance-traveled, initial velocity, and elapsed proper-time) don't yield closed-form solutions obvious to us. If you think you might find explicit solutions for some of these puzzling combinations, check out here the input combinations we don't have solutions for yet. A list under construction of example problems which could use these solutions can be found here.

If you come up with the first closed-form solutions seen (by us) for any of the (presently 10 of) 31 input combinations still in parentheses in our "remain to be found" tables here, we'll post your solution (if it is manageable), and credit you as well for sending it in.

Note: six of the 10 presently un-solved combinations involve Galilean-kinematic time-elapsed, as well as either coordinate or proper time-elapsed. These are likely to be of little practical interest, as they presume input data about time-elapsed on the clocks of an observer "following" our accelerated traveler in a "Galilean chase-plane", whose offset-speed affects chase-plane clocks so that Galileo's original constant-acceleration equations describe the traveler at any speed.

We are also interested in literature solutions to the same problems, and will cite same if they are brought to our attention.


  • Notation Key B (used on this page) is: Galilean {time, velocity}={t, v}, coordinate {time, velocity}={b, w}, and proper {time, velocity}={T, u}.
  • Copyright (1970-96) by Phil Fraundorf
  • At UM-StLouis see also: a1toc, cme, fzx&astr, progs, stei-lab, & wuzzlers.
  • More on this: derivations, slow-example, fast-example&twins, x-tv Plots, x-ct Plots, 4-vectors, rap.
  • For source, cite URL at http://newton.umsl.edu/~philf/unsolved.html
  • Initial release: 16 Apr 1996. Current version: 3 Mar 1997.

    Which 1D Constant-Acceleration Problems have Analytic Solutions?

    As illustrated in these web pages, constant acceleration problems in (1+1)D special relativity involve the acceleration, the distance traveled, time-elapsed in three self-consistent kinematics, and initial & final velocity variables for each of those three kinematics as well. Only three of these 11 variables are independent. Hence if all were fair game, there might be as many as 165 ways (combinations of 3 from 11) to setup a problem!

    Of course, not all triplets work. For example, any velocity variable can be trivially converted from one kinematic to the next. Some acceleration problems can thus be solved within the kinematic of choice. Using conversions post facto, all remaining variables can be determined once ao and any initial/final velocity pair are in hand. If we ignore differences in the kinematic that any velocity is specified in terms of, one finds only 25 distinct choices possible for the three given (or independent) variables which might define a 1D constant acceleration problem.

    For three of these 25 choices, no elapsed-time is given. Hence their inputs are either independent of kinematic (i.e. ao & dx) or can be trivially converted to any kinematic (e.g. velocities wo, wf, vo, vf, uo or uf). One can solve relativistic acceleration problems like these using acceleration equations from any of the three kinematics, including the well-known (and notationally simplest) equations discovered ages ago by Galileo!

    Equation sets for such input triplets are assigned numbers in Table 1. For example, equation Set 0 is the set of equations which solve for dt and dx given ao, vo & vf. From any introductory physics book, or from the dressed up equations or derivations pages here, one can show that equation Set 0 is simply:

    Look familiar? I hope so. The other 8 equation sets numbered in Table 1 can also be determined from material in these web pages. Can you determine Set 1? Set 10? Set 20? Set 2?

    For each of the remaining choices for our 3 independent-variables, elapsed-time in at least one kinematic is given. Hence the problem may have to be worked in that given kinematic, rather than in a kinematic of choice. The 12 independent-variable triplets which include one elapsed-time are assigned numbers in Table 2 below. Note that the triplet 26 (dT, dx & a velocity given) is in parentheses. That's because we haven't come up with an explicit (closed-form) solution, even though graphical and numerical methods for solving this problem are in hand. Does one exist? Perhaps the question has not yet been asked in the right place!

    Thus Galileo's age-old acceleration equations (with conversions as appropriate) can be used to exactly solve 13 of the 21 "intrakinematic" problem types listed in Table 1 and Table 2, even if the velocities are relativistic! Of the 10 "interkinematic" triplets not listed there, 9 involve two given elapsed-times. The last involves all three. These are listed in Table 3 and Table 4 below. An explicit solution exists for input triplet 28 below (db, dt & a velocity given). Otherwise this territory may be relatively uncharted! Anyone out there ready to explore?

    Keep me posted about progress you make in finding closed-form solutions to acceleration problems with input triplets associated with parentheses in the Tables above. In return, I will try to keep readers of this page posted on progress that I learn of in this regard.


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