A ship capable of carrying live adult humans, yes.

Something carrying nanotechnology that will use raw materials at the destination to construct a colony including new humans, not so much.

Sure, the latter isn't going to be within our capability soon. But on a time horizon of thousands of years or more, it will.

https://en.wikipedia.org/wiki/Nuclear_pulse_propulsion

Single digit percentages of the speed of light are not that hard to achieve, but would probably be sufficient to launch some von Neumann probes (or generation ships) and start colonies.

-- Lady Deirdre Skye, "Conversations With Planet"

Can we propel a ship that can sustain 100 years of human (or AI) life and protect it from cosmic rays and provide it with enough equipment to build a self-sustaining colony on arrival?

Like, is the mass required for a self-sustaining interstellar generation ship, with all the required shielding needed to handle high-velocity interstellar travel (blasting through the interstellar medium an appreciable fraction of c means absorbing a crapload of high-energy particles, and the occasional micrometeorite that can't be detected until it's too late that hits the ship with the energy).

For example, hitting a little 10 gram rock at 0.1C would be a kinetic energy transfer in the gigajoules - like a small (1 kiloton) nuke.

So we've established our ship needs to have armor that's thick as hell. In order to make that economical, it must be huge. Also, in order for it to do useful work there, it needs to contain factories worth of machinery.

The sheer scale of this ship would be staggering. And then that raises the question: is there enough energy available to propel such a thing?

And this seems to happen everywhere in everything. For instance as early late as the early 20th century there were many scientists who felt that physics was basically solved. There were a few inconsistencies to work out and some more measurements to be performed more solidly, but that all of the most fundamental laws of the universe had been discovered and that we knew everything key. Of course quantum mechanics and relativity would be discovered within a couple of decades showing us that we, to this day, know extremely little.

I think the reason for this is that as one reaches advanced mediocrity, it's easy to look back and see how little people who knew less than you truly knew. And so, without knowing what you do not know, it is quite difficult to imagine how you could not genuinely be near the top of the knowledge scale. Yet something that's shown true time and again in our universe is how little we truly know.

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Beyond this, there's also a technical response to your comment there. Even given our current level of knowledge interstellar travel is not only possible but practical, though communication is not. A common misconception conflates the relativistic effects of fast moving effects between observer and observed. And in particular as objects approach the speed of light relative to an observer, their mass approaches infinity thus sending the amount of energy required to further accelerate them also approaching infinity.

However, this is not true from the perspective of what's being observed. Indeed they'd perceive quite a different outcome. The energy for a vessel in a vacuum to accelerate 1000 km/s does not suddenly change after they've performed such acceleration 300 times. All sorts of strange things would start happening from their perspective including time for everything else seeming to speed up, and physical distances starting to literally contract (such that they themselves would also never perceive themselves as traveling faster than the speed of light), but they could happily keep accelerating to no physical limitation. And the net effect of this is that you can travel millions, billions, and really a practically unlimited number of light years within a single human lifetime.

Here [1] is a handy calculator for how long it would take to travel certain distances given a certain fixed acceleration. For instance 45 years at 1g acceleration would be sufficient for 15 billion light years of travel. The catch of all of this, and the reason communication is impossible, is that for everything 'observing' our traveling ship indeed 15 billion years would pass. So if our ship turned around and made its way back to Earth (I'm ignoring universal expansion for this discussion), though only 90 years would have passed for the ship - 30 billion years of time would have passed for the ship.

This effect even shows up in particle accelerators, in a really hand way. The collisions create unstable particles that though they would decade after let's say 1 millisecond, often can be observed for many times that. It's the same effect in play. Relative to us, time for the particle slows down due to its acceleration. Perhaps this all makes an even better point. Our universe is extremely weird and defies any notion of intuition. And the more we learn, the weirder it becomes. To think that we have even scratched the surface of its secrets shows that though the universe may be vast, it is incomparably small compared to the bounds of our arrogance!

[1] - http://convertalot.com/relativistic_star_ship_calculator.htm...

The amount of shielding this vehicle would require is massive - solid feet of shielding to protect the occupants. So the economies of scale make it obvious: the vehicle would have to be enormous. More of a hollow asteroid than a spaceship. Is there enough energy available to launch that to another system?