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On the other hand - just got off the phone with my younger son - who rode an annual 66 mile bike race last weekend at Corpus.
Dr. Denker's comments (below) remind me of something I mention to runners that I coach/advise: everybody in the race is working equally hard, but the fastest runners hurt for a lot less time than everyone else.
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The singularity in equation [2] can be taken as a
warning. If you are trying to finish a race in the
least amount of time, going slowly even for a rather
short distance is devastating. This is actually
true, and quite noticeable if you do the experiment.
To say the same thing in slightly more positive terms,
if there is a fast half of the course and a slow half
of the course (perhaps due to hills) you can gain a
lot more by speeding up the slow half by 1 mph than
you can by speeding up the fast half by 1 mph. Also
true and quite noticeable.
====================
This occasionally comes in handy as an answer to the
athlete who thinks physics equations could not possibly
explain anything he is interested it.