When is the optimal time for humanity to send colonists out to another star?" Definitely not when it takes 19'000 years and 600 generations .After 600 generations of living in space there would be no way the people on...
The solution to the wait calculation is to keep track of everyone you have launched already, so that a faster ship travelling in the same direction can rendevous with the older, slower ship, transfer everyone aboard, and continue at the faster speed.
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