Discover it yourself:
Map-based motion at any speed
Level Three: This page continues a journey begun to discover 20th century methods for describing motion. It is designed like a multiple choice MAZE with challenges along the way that depend a good deal on the strategies that YOU adopt.
Although we've talked about the time on map-clocks between the two sneezes, we have not discussed the time between the two sneezes on the pilot's clock. A modern understanding of the problem can be easily summarized with Minkowski's extension of Pythagoras' Theorem for calculating the traveler or proper-time tau between events.
The fact that you are viewing this web page suggests that you have a feeling for Pythagoras' method for measuring the distance between two points, and that you are familiar with the meanings of velocity and speed (velocity's magnitude). Pythagoras' Theorem may be written as ds^2 = dx^2 plus dy^2, where ds is the distance between two points. In this spirit, Minkowski's metric equation for traveler-time elapsed dtau takes the form (c dtau)^2 = (c dt)^2 - ds^2, where c is the speed of light (around 3x10^8 m/sec^2) and ds the map-distance traveled. Here as usual, the letter d is used to denote a small change or difference in the parameter it preceeds.
This equation completely describes the connection between space and time in Einstein's theory of special relativity. It's simplest consequence: When considering motion (and simultaneity) with respect to a single reference or map-frame, traveler clocks move more slowly than clocks on the map. In the challenge below, we see if YOU can quantify this result.
Recall that our pilot sneezes as she flies over the red and then the orange clocks in the view below.
We have earlier determined that the map distance between those two sneezes is 141 meters, while the map-time elapsed between them is 7 seconds.
Question for YOU: What is the time elapsed between sneezes on a stop-watch carried along in the pilot's pocket?
Hint:One way to check your thinking is to rent a plane and perform the experiment with an extremely precise stopwatch in your pocket. A similar experiment has been done, so we know that this works. Another way is to play with Minkowski's metric equation until an answer emerges...
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This Discover-It-Yourself Series is Copyright 1987-1997 by P. Fraundorf,
Department of Physics & Astronomy, University of Missouri - St. Louis.
Send comments and/or questions to pfraundorf@umsl.edu.