What follows is a short puzzler about a high speed air-track experiment: Clocks in the track frame are synchronized, and then located at two contact points along the track, so that the travel-time of signals does not complicate life. These results then follow...
A glider carrying its own clock passes two contacts separated by dx=one foot, while traveling at various speeds. When track clocks say dt=2 nanoseconds (ns) during the traverse, the glider clock says that only dT=1.732 ns has elapsed! Here, the letter "d" precedes each variable name to denote "change in". When the track clocks say dt=1.5 nanoseconds, the glider clock says that dT=1.06 ns has elapsed. When track clocks say dt=1.15 ns, the glider clock says that dT=0.575 ns has elapsed. Lastly, when track clocks say dt=1.06 ns, the glider clock says that dT=0.353 ns has elapsed.
As you can see, when glider speeds approach one foot per nanosecond, the "speed of glider time with respect to map time" (dT/dt) begins to decrease! But what rules of decrease are suggested by this data? Does this explain why everyone might want to carry her or his own watch? And what other consequences might those rules have...
[Help! How might one begin to address such questions?]
Need more incentive? The results above, for example, provide everything you need to discover Minkowski's space-time version of Pythagoras' theorem (i.e. the "metric equation"). This describes our current view of the relationship between map-distance, map-time, and traveler-time in the universe we inhabit! Is that cool, or what?
A more general high-speed air-track experiment, accessible via computer if not yet in the lab, will soon (we hope) be made available on the web and on campus instructional technology servers as well. A screen capture from the experiment is shown below. Some data and a bit of analysis therefrom is provided here. This experiment also allows one to determine how the definition of both kinetic energy and momentum must differ, if motion at any-speed is to be considered...
Suggestions for modeling tools that may be helpful to incorporate into such programs are invited, as well as interest in the program that results. Write pfraundorfSPAMX@umsl.edu. Although the program above is in Visual Basic, a Java applet that will run through a browser is in the works as well. For more on this map-based approach to anyspeed motion, whose origins lie with H. Minkowski and J. S. Bell, see our note on "teaching Newton with anticipation...", as well as our web-page on "map-based motion at any speed".