Final answer:
Colonizing the Milky Way galaxy at a few percent of the speed of light would take a few hundreds of millions of years. This estimation considers the vast number of stars, the galaxy's size, and the challenges in energy requirements and technology.
Step-by-step explanation:
Colonizing the entire Milky Way galaxy at a few percent of the speed of light would be an immense challenge, not only because of the vast distances between stars but also due to the energy requirements and technological limitations. The closest star to Earth, Proxima Centauri, is 4.24 light-years away.
If we travel at 1% of the speed of light, it would already take us 424 years to reach just that one star. Considering the Milky Way galaxy contains about 100 billion stars and spans about 100,000 light years across, the journey to colonize all potentially habitable planets (a small fraction of total stars) would take many orders of magnitude longer.
Additionally, the passage of time for those traveling at high speeds and those remaining on Earth would differ greatly due to the effects of relativistic time dilation. Factors such as energy consumption, survival over extended periods, and evolution that could change our adaptability to different environments also come into play.