...consider what happens to the swimmer reversing his direction for an equal time period.
Whatever gain he derives from supplementation by the river current while going downstream is wiped out by an equal loss
going upstream. This bothers me.
The time taken to swim a given distance upstream is greater than (not equal to) the time taken to swim the same distance downstream.
That's the point: the distance is the same, but the time is different. The time would be the same only if there were no river current (no 'ether wind').
Example:
Distance to swim, d, = 15 metres
Swimmer speed, v, = 1 metre per second
Stream flow rate, r, = 0.5 metres per second
Upstream time = d/(v - r), Downstream time = d/(v + r)
Upstream: 15/(1 - 0.5) = 30. Downstream: 15/(1 + 0.5) = 10
So, after 40 seconds, the swimmer returns to his starting point.
That's 10 seconds longer than it would have taken with a stream flow rate = 0.
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If the swimming time were the same in each direction, e.g. 30 seconds:
Distance to swim = t*(v - r) + t*(v + r) = 30*(1 - 0.5) + 30*(1 + 0.5) = 45
That takes the swimmer 15 metres downstream from their starting point!
