4.2.3. single-frame time-dilation

As long as only one map-frame is required, solving time-dilation problems may be much easier using proper-velocity in the traveler-kinematic, than using coordinate-velocity, since the former relates directly to the relation between map-distance covered in a given amount of traveler-time: Example Problem 4.2.3a: Muons with a lifetime of Δτ = 50 [micro-seconds] are created at Δx = 100 [km] altitude, in collisions between air molecules and high-energy cosmic rays. Since these muons routinely make it to sea level for detection before their on-board clocks cause them to decay, how fast (earth distance traveled per unit earth time) must they be moving?

Example Problem 4.2.3b: The timer carried by a high speed airtrack glider registers only one third of the elapsed time recorded by gate clocks, on the trip between a pair of synchronized gates fixed to the airtrack. How fast was the glider traveling?

Example Problem 4.2.3c: The highly-accurate timer carried by a fast-moving vehicle registers only 99% of the elapsed-time recorded by synchronized bank clocks that it passes. How fast was the vehicle traveling?

The caveat is that length-contraction always requires two frames of synchronized clocks, so careful treatment of length-contraction effects should likely await the introduction of Lorentz transforms. We do attempt to address some questions of extended radar-time simultaneity without them later on in this paper.