Conceptual Derivation of The Universal Acceleration Plot
Caution: Conversion to updated notation is in progress. 2005may/pf

This is a derivation first presented at a UM-StL Physics & Astronomy Journal Club session in March 1996, and then incorporated into our poster for the March 1996 American Physical Society Meeting. The posting here will begin with some scratch GIF files with figure labels missing, and be updated and improved for reference by students, educators, and textbook authors as time permits. For now, better plots may be found on the related pages listed below. A more graphical version of this derivation (in the form of GIF files) may be found here.

Relativistic Constant Acceleration with Galileo's Equations:

A Conceptual Physics Derivation

by P. Fraundorf/3mar96

Part I: The Universal Constant Acceleration Plot

Suppose you start with Galileo's equations for 1D constant acceleration, expressed with a coordinate origin at the time and place of zero velocity. We'll use capital letters for velocity and time because it turns out that at high speeds these equations only apply to motion examined with a special subset of all possible clocks. Thus we get V = aT, and x = ½aT2. If you multiply the 2nd equation by acceleration a, you get ax = ½(aT)2 = ½V2. If you further make this equation dimensionless by dividing by the speed of light c squared (something Galileo is unlikely to have done), you can rearrange to get the universal "x-parabolic" curve:
v/c = at/c = Sqrt[2 a x /c^2].
On a graph this equation, here used to solve a constant acceleration problem with a=9.8 [m/s^2], initial velocity Vo=-13.8 [m/sec], and elapsed time dT=4 [sec], looks like:

From this single curve in the variable range of interest, one can thus see the quantitative relationship between V/c, aT/c and ax/c2 for any unidirectional constant acceleration problem! Co-incidentally, if the acceleration is a = 0.969g (i.e. about one "earth-gravity"), the "dimensionless" x-axis reads out directly in lightyears, while times on the y-axis are directly in years. Regardless of acceleration, velocities on these plots are always in units of the speed of light.

Part II: Clock Behaviors during High Speed Acceleration

In order to describe motion at high speeds, Albert Einstein suggested that objects have, in addition to their kinetic energy, a rest energy equal to mc2. Thus in terms of Galilean-kinematic velocity their total energy is E = mc2 + ½mv2. If we divide through by mc2, this gives us the ratio between total and rest energy that Einstein called gamma: γ = E/mc2. Hence in Galilean terms γ = 1 + ½(v/c)2. Einstein further predicted that clocks associated with observers traveling, especially at high speeds with respect to one another, will experience time passing differently and not according to Galileo's equations. As result, Galilean-kinematic time T only approximates the behavior of physical clocks when relative speeds are low, or when the clocks are aboard a "chase-plane" whose motion is carefully selected to be intermediate to that of the traveler and the map-frame on whose yardsticks distance is being measured.

To quantify this, we capitalize the familiar Galilean variables time T and velocity V = dx/dT, but define coordinate-velocity v = dx/dt in terms of the map time t elapsed on synchronized clock's in the inertial frame in which distance x is measured, and proper-velocity w = dx/dτ in terms of the proper time τ (tau) elapsed on our accelerated traveler's clock. To keep the 3 velocities conceptually distinct, "coordinate" velocity in special relativity v is usually measured in units of [lightyears per map year] or c, while traveler (proper) velocity w can be measured in units of [lightyears per traveler year] or "rodden-berries". In these terms, Einstein's high speed clock predictions say that γ = 1/Sqrt[1-(v/c)2], and w = γv. From these, it follows also that γ = Sqrt[1+(w/c)2]. For our constant acceleration problem here, it is now easy to show that γ = 1 + ax/c2 = 1 + ½(aT/c)2 = Sqrt[1 + (at/c)2] = Cosh[a τ/c] as well.

These equations provide 3 more curves for our original universal acceleration plot (or 4 if we want to also plot γ = 1 + ax/c2 ). The 3 curves are:

As long as the velocities v and V are much less than the speed of light c, all of these curves superpose themselves on the "parabolic" curve plotted above, which in terms of gamma looks like: v/c = at/c = Sqrt[2(γ - 1)] . Hence the universal acceleration plot shows no change, and all velocities and times agree.

However, when we plot these curves for larger velocities, the velocity and time curves split up to reveal major differences in the experience of observers party to high speed adventures. In this way a single plot can be used to visualize any relativistic 1D constant acceleration problem, and particularly the relationship of all 6 variables describing time and velocity for inertial, accelerated, and "Galilean-kinematic" observers, as a function of traveler position x as measured in the map-frame of choice. Since all of this is done using distances and 3 kinematics referred to a single inertial frame, one does not need multiple inertial frames, Lorentz transforms, length contraction, or even calculus to understand and put these results to quantitative use.

Sections to follow here examine quantitative solution of relativistic unidirectional constant acceleration problems, using either: (i) the above graphs, (ii) Galileo's equations with added help from conversions to the kinematic variables of interest, or (iii) acceleration equations in inertial, traveler, and mixed kinematics as well. Analytic solutions for unidirectional acceleration problems are provided for most, but not all, possible combinations of input variables.

  • All "1-map 2,3-clock" equations of anyspeed acceleration are summarized here.
  • Copyright (1970-95) by Phil Fraundorf
  • Dept. of Physics & Astronomy, University of Missouri-StL, St. Louis MO 63121-4499
  • At UM-StLouis see also: accel1, cme, programs, 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/c3pderiv.html
  • Version release date: 22 Mar 1996.
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