Question is ultimately asking why would we do this? A few parameters first...
- Our current knowledge of physics is correct and there is no such thing as faster than light travel, nor is there any 'wormhole' or other connecting device between the two systems. Communication is no faster than light
- Human lifespan is ~100 years on average by this time, immortality is not obtainable
- We are capable of travelling around 5% the speed of light for this journey (or 1 light year every 20 years of travel)
- 14-15 light year journey, or slightly under 300 year journey (time dilation ignored at this point), meaning 4 generations of humans will live before arriving.
This puts several limitations on this new colony. They are fully isolated from Earth as communication is at least 15 years in transit, and is at most capable of sending new technological ideas/inventions as blueprints with a 15 year lag time (in this story line, quantum entanglement eventually comes along as a method of faster than light communication, but when the ship leaves this idea is nothing but theory). Trade between the colonies is impossible due to the 300 year in-transit delay. There is really little gain for Earth, other than knowing humans now exist outside of the solar system.
It's the 22nd century and humanity is more and more reaching into the stars, but is completely unable to resolve the issue of interstellar distance. On Earth, mobility of populations has greatly increased and the world is far more homogeneous in general. A global government has formed with countries still existing, but more as states of this global whole. People of all races and from any nation (with a couple exceptions) are now relatively free to travel and settle where ever they would like. Freedom and individuality is still a core belief, and though the pro-humanity concept is there, most people do whats right for them before 'the interests of humanity'.
This move isn't the most popular and is dubbed 'viral colonization tactics' by those voices disagreeing with the move to send humans like this. It gets referred to as sending our spores to infect the rest of the galaxy. Assume support is around 50/50, such that delays, risks, and other factors may mean a politician is risking their career by doing this, however there is enough support to get the idea off the ground.
We have had contact with another alien race and it's generally understood that we are not alone. However the contact has been very minimal and we understand little of 'galactic politics' or the species involved in it
Signs of an alien race were found spotted throughout the system as technological artifacts the we recovered and studied. Most of the artifacts were discovered in orbit of Jupiter as a crashed ship, however other jettisoned parts were discovered around in orbit of Neptune. Actual contact with a single alien representative was made after an explorer from that species came into contact with human explorers looking for more artifacts. Contact was brief, however entirely recorded. Alien was hesitant as he could not speak for his race as he was 'just an explorer'. There is mention of a second alien race as well, though not described. Alien insisted repeatedly that they are peaceful and intend no harm.
The Moon has been partially colonized and holds the space dock in which the ship could be created.
What motivations would a worldwide government have to put together this mission? Is the knowledge that humanity now exists elsewhere in the universe and some distant time where alternate forms of interstellar travel exists that may allow us contact with them enough to make this investment worth while? On a government scale:
Money and time invested by Earth to make the space craft need to be justified
Staking political careers on the idea that this is worth it
And the question #2
- What would motivate an individual to sign up for this mission, well knowing that they will never see the new world they are to colonize and are forcing their children, grandchildren, and great-grandchildren to the same wordless fate so their great-great-grandchildren have the possibility of colonizing a new planet isolated on their own?
Side note - this is integral to the story line as a future holds these colonies linking up under new technologies. But is the idea that we might one day be able to enough to start it now?
More than one colony ship will make the journey, all to separate stars...hence the 'viral tactics' reference where we blindly spore and hop to 'infect' as many planets as possible. The more reasons the better.