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24.6 Lorentz Transformation: Examples

Module 24: Lorentz Transformation: Examples - Problem 1

One of the most potent but confusing aspects of special relativity is how each constant velocity observer can claim to be at rest and so rightly concludes that their own clock is ticking off time faster than all others. The question is: How is this possible? How can each of two observers each conclude that their own clock is ticking off time faster than the other observer’s clock, and yet not run into a contradiction? Let’s investigate this in a specific example.

Imagine a sprinter is running a 3000 foot race along a straight track. The sprinter is wearing a watch that starts ticking from zero when the starting gun goes off. The track is equipped with a clock at the start line and another at the finish line, to record the elapsed time for the sprinter to cover the 3000 foot distance. Ignore unnecessary complexities like (i) the travel time from the gun to the sprinter’s ear and (ii) the fact that the sprinter is accelerating from rest. Assume that the sprinter has a constant speed $v=\frac{3}{5}c$ throughout the problem. Use our standard approximation that $c=1 \,\textrm{ft}/\textrm{ns}$.

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