Multiple Choice Series A00 for Physcs111.E0A

Series-A00 is for submission by Midnight on the 2nd Sunday in September, if possible.

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1. Minkowski's space-time version of Pythagoras' theorem (also known as the flat space-time metric equation) takes the form:

where proper-time tau is the time elapsed on the clocks of a traveler whose motion is described by map-position x at map-time t, measured on yardsticks and clocks fixed with respect to an unaccelerated reference coordinate system or "map-frame". Here c = 3x10^8 [m/s] = 1[ly/y] is the speed of light in vacuo. Rearranging this equation shows that when simultaneity is so-defined by map-based observers, map clocks tick faster than traveler clocks by the "speed-of-map-time" ratio gamma:

Given this, if a car (or satellite) passes two bank clocks 1000 km apart, moving so quickly that the traveling clock experiences only half the time that elapses on the bank clocks during the traverse, how fast is the car (or satellite) traveling?

2. If we further define the proper velocity of a traveler as the map-distance traveled per unit time on the traveler clock, or w=dx/dtau, then unidirectional constant "felt" or proper-acceleration problems at any speed can be solved with the velocity conversions:

and acceleration alpha integrals...

Here the dimensionless quantity eta, involved with the proper-time integral, is sometimes called the hyperbolic velocity angle or rapidity. Imagine using these equations to plan a business taking customers on a space cruise, involving constant 1 gee = 9.8 [m/s^2] = 1.03 [ly/y^2] acceleration, that would bring them back 100 years in the future (e.g. where their investments may have grown A LOT). How long by their clocks would this cruise last (and how far would it take them away from earth)?

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