Note: If you echo this work in print or on web, a citation would be cool.
{Thanks. :) /pf}
Preparing for a Nomographic Solution
The first step in using a general purpose (one-for-all) velocity/time versus distance plot, like that developed on our derivations page, is to convert what you know into dimensionless units. To do this, it helps immensely if acceleration ao is one of the knowns. We therefore treat first the case when 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/ao, and the distance-traveled (if you know it) by c^2/ao. 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 18 decades of) dynamic range. For a smaller variable range near the origin, click on the graph below.
Interpreting the Result
Now you may wish to reconvert the absolute answers back into dimensioned units. To do this, multiply velocities by c, elapsed-times by c/ao, and the distance-traveled by c^2/ao. 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.