The Theory of Relativity 125
T
total
= 2dv′/[(v′ + u)(v′ − u)] = 2dv′/(v′
2
− u
2
)
To make our answer look similar to the one we obtained when the river was not
moving, we divide both numerator and denominator by v′
2
to get our nal answer:
T
up/down
= (2d/v′)/(1 − u
2
/v′
2
) = (2d/v′)/(1 − (u/v′)
2
) (12.3)
The numerator is identical to what we had for the time when the river was not mov-
ing, but now we have a denominator that is less than 1. That means that with the
river moving, the up/downstream boat takes a longer time to get back than when the
river is still. We can see why this is so by considering the special case when the river
is owing with the same speed as the boat moves in still water (i.e., u = v′). Then,
when the boat turns around ...