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Unlike coordinate-velocity dx/dt, proper-velocity is proportional to momentum at any speed, has no upper limit, and only uses one clock (the traveler clock) and thus does not rely on an extended array of synchronized map-clocks. One might also however wonder about its relative utility. At high speeds, or in the dream-world of George Gamow's Mr. Tompkins (Gamow 1940) where lightspeed was not much more than 2.5 miles per hour, would highway speed-limits be more usefully expressed in proper or coordinate velocity units? Example Problem 4.2.2a: On a car trip with 300 map-miles remaining, is it proper-velocity or coordinate-velocity which determines most directly how much longer you have to remain in the vehicle? Is it proper-velocity or coordinate-velocity which determines most directly what time you can be expected to arrive on clocks at your destination?
Example Problem 4.2.2b: What upper limit on vehicle momentum, vehicle kinetic energy, driver reaction-time for obstacle avoidance, and pedestrian reaction-time for oncoming-vehicle avoidance, is provided by a proper-speed limit of w/c in the range of positive real numbers? How would these relationships change for a coordinate-speed limit of v/c instead?
Example Problem 4.2.2c: Read the first chapter of George Gamow's book about Mr. Tompkins, and see if you can improve on our estimate (above) of the speed of light in his dream.
Example Problem 4.2.2d: Your 1000 kg space-roadster accelerates from rest to 1 [ly/ty] i.e. ~0.707 [lightseconds/map-second]. The kinetic energy acquired on this “jump to lightspeed” is equivalent to how many gallons of gasoline, each of which contains 100 million Joules of energy available to do work?
The answer to this last question may also be estimated with help from the lightspeed curtain (K versus p) plot in section 4.1.1.
Related references:
G. Gamow (1940,1965) Mr. Tompkins in Paperback (Cambridge University Press, Cambridge, UK).
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