By "dressing up" the variables in a 1-D constant acceleration problem, solutions for all variables in all such problems can be plotted onto a single graph! The horizontal axis of this graph represents the x-coordinate of events as viewed from within a single inertial frame. All kinds of stuff (actually 7 variables requiring 5 or fewer curves) can be plotted simultaneously along the vertical axis. The only requirement on the x-reference frame is that it be chosen so that any transverse (i.e. non-1D) velocities with respect to our acceleration direction be zero, or at least small in comparison to the speed of light.

In the long run, we hope to discuss and provide programs for scaling the graph to accomodate your particular problem. We will also discuss graphical strategies taylored according to which of the 3 of 11 possible variables are known, and therefore which of the 8 of 11 remain to be determined. For starters, our examples will begin with given values for acceleration, initial-velocity, and distance-traveled. Give some thought to how you might proceed in other cases, and let me know if you encounter either major problems, or cool strategies which work!

Notation Key B (used here) is: Galilean {time, velocity}={t, v}, coordinate {time, velocity}={b, w}, and proper {time, velocity}={T, u}.

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Preparing for a Nomographic Solution

The first step in using a general-purpose (1 for all) velocity/time versus distance plot of 1D constant acceleration, like that developed on our derivations page, is to convert what you know into dimensionless units. What you are really doing, of course, is plotting v/c and a_{o}t/c versus a_{o}x/c^2, using distance x and time t values measured from the zero velocity point in the constant acceleration trajectory. The speed of light c is thrown in to make all of this independent of the units chosen. It will also be easy to see from the plot when relativistic effects kick in (i.e. when and only when these dimensionless parameter-values get close to 1 or larger).

To figure where your acceleration problem fits on this plot, of course, it helps immensely if acceleration a_{o} is one of the knowns. We therefore first treat problems for which acceleration is known. In that case the first thing to do is to put the variables you have into absolute (or dimensionless) form, i.e. divide any velocities by the speed of light c, any elapsed-times you have by c/a_{o}, and the distance-traveled (if you know it) by c^2/a_{o}. Note: If acceleration is approximately 1g, then you can simply use times in years and distances in lightyears, since 1 g of acceleration is approximately 1 ly/yr^2. Write down the variables you know in this absolute form, and also list which of the remaining variables you want to find out.

Solving the Problem

Try plotting the knowns you have on the graph below, and using horizontal and vertical lines as appropriate to read off the unknown values of the variables of interest. There will be two vertical and two horizontal lines, unless one of the two endpoint velocities (initial or final) is zero, in which case there will be only one of each. If initial and final velocities are in opposite directions, plot them ignoring the sign but remember to figure time elapsed as time-elapsed going to zero speed from the initial value PLUS time-elapsed going to the final speed. If the variables you have fall outside the range shown on the graph, you may need to plot your own versions of the graph before taking these steps. The graph is simply the 2nd graph on our derivations page, plotted on log-log axes to clearly cover a wide (here four decades of) dynamic range. For an even larger variable range, click on the graph below.

For fun, I've thrown together a Windows program which can graph almost any constant acceleration problem you will encounter directly onto this universal plot, and then allow you to pan and zoom around on this plot to see where things are. You can download the program executable from here, and the Visual Basic runtime library (VBRUN200,DLL) it requires (if you don't already have it) from here.

Interpreting the Result

Once you've located the starting and ending points of your problem on the universal plot, you may wish to reconvert the universal coordinate answers back into dimensioned units. To do this, multiply velocities by c, elapsed-times by c/a_{o}, and the distance-traveled by c^2/a_{o}. If you are not sure what the various quantities on the graph represent, look over our examples and derivations.

Worked Example of a Problem with Known Acceleration

Suppose you needed to travel 160 lightyears in a spacecraft capable of extended 1g acceleration. The first 80 lightyears would be done accelerating, while the second 80 lightyears would be done decelerating. How much older would you be on arrival, and how much older would the place you sought to visit? Basically, all you do is draw a line up from 80 on the x-axis, and see what times it intercepts on the exponential traveler time (tau) curve (a bit over 5 years of your time) and on the hyperbolic inertial time curve (about 80 years of time for those at your destination). Multiply these times by 2 to get the total duration of both legs of the trip (around 11 years traveler time, but 160 years of inertial observer time). The hyperbolic curve also reads out your maximum speed in traveler units (near 80 roddenberries). The asymptotic curve, of course, shows that your inertial velocity remains just under the speed of light. These solutions can be documented by simply drawing on the graph, as in the figure below. Also, you might want to check here for a graphical solution using a linear version of this same plot..

Unknown Accelerations and the Rubber Graph

This is a topic for future work.

Send ideas/ comments/ questions/ complaints, and register for updates to these pages with an email subject-field containing "accel1", to philf@newton.umsl.edu.