When we return to these forumlae, they are assuming constant acceleration, both positive and negative, which is only "realistic" to a civilization with near-infinite energy capabilities.
It is much more realistic and practical to pick a target velocity high enough to make the biosphere and it's power supply reasonably plausible, and then coast to the "braking point", before slowing down.
Using the formula from this web site, for "low" relativistic values of V, we have.
V = velocity
M = mass of "stoppage" fuel in ideal anti-matter engine
m = mass of payload
M' = mass of accelerating fuel
N = Total Fuel
* note that all numbers are rounded UP at 4th decimal.
V = 0.1c, M = 0.1055m, M' = 0.1166m, N = 0.2221m
V = 0.2c, M = 0.2247m, M' = 0.2752m, N = 0.4999m
V = 0.3c, M = 0.3628m, M' = 0.4945m, N = 0.8573m
V = 0.4c, M = 0.5276m, M' = 0.8060m, N = 1.3336m
V = 0.5c, M = 0.7321m, M' = 1.2681m, N = 2.0002m
So, with ideal antimatter engines, you could accelerate and stop a ship ~9 times larger to 0.1c as compared to 0.5 c, for the same total energy.
We may surmised that the survivability, s, of the colony ship is some function of inverse time and all obstacles, O, also differentiated with respect to time, because the longer the mission the harder to survive.
s(t) = (f(O))/t
Now if we shorten the time, it's obvious the survivability goes up roughly AT LEAST linearly.
But in order to shorten the time costs exponentially more fuel. At some point, given a large enough ship and efficient enough internal technologies, we may be reasonably sure that it is actually easier to keep the ship and biosphere functioning for a longer period of time than it would be to further accelerate the ship.
Given the progression above, it may literally be easier to move a "Sealed Environments, type 1 civlization" on the surface of a moon with the right properties, than to make a "star ship". Let's take Enceladus, for example.
It has insane amounts of water, and probably a rocky core. It has respectable gravity, but not earth-like. This means that there are resources enough on the planet to prouduce fusion power for eons and eons, sufficient for the people living on it. This also means that the entire planet is covered in a fuel source for a direct fusion-based propulsion.
If you could get the entire moon moving at a relatively low speed, such as only a few hundred kilometers per second, as opposed to more relativistic speeds, the moon has enough energy (through controlled fusion) to support life for ages upon ages, for the same reason earth has: size. We know Enceladus has water, nitrogen, and carbon dioxide.
So if you moved the moon at 300km/s (0.001c), it would take ~20,000 years to reach the nearest star with a known planet. The amount of water on Enceladus would permit fusion for normal life purposes and normal civilization usage for billions of years actually.
Now to see if this is even possible.
Enceladus has a mass of 1.08022*10^20kg
Fusing 200kg of hydrogen equates to about 1kg of anti-matter.
According to that formula, to get Enceladus to 0.001c would take
M = 0.001m worth of anti-matter in an ideal engine
0.2m worth of hydrogen fusion in an ideal engine*
* not possible because by percentage, this is more than the mass of hydrogen on Enceladus.
Soooo....scale down another tenth of velocity.
v = 30km/s = 0.0001c, time = 200,000yrs
M ~ 0.0001m worth of antimatter
M ~ 0.020001m worth of hydrogen fusion.
M' ~0.020001m worth of hydrogen fusion
N ~ 0.040002m worth of hydrogen fusion
This MIGHT actually be doable, in theory, depending on the exact percentage of Enceladus' mass which comes from water, because by mass, water is 1/9 (or 11.11 percent) hydrogen, and 4 percent total lunar mass gives us approximately a factor of 2.75 to work with.
So Enceladus is large enough to support sentient life for geologic time periods through fusion of water, but small enough to be accelerated to a high enough speed to reach a distant star, at least in many, many ten thousands of years....using only materials on Enceladus (plus stuff imported from asteroid field ot make the actual habitats and engines.)
So at least this tells us one thing, even an artificial worldship with an ideal fusion engine (which may be easier and safer than antimatter) would be at least four percent "fusion rocket fuel" (hydrogen) by mass...and this is allowing the assuming that it is "big enough" to be self sustaining for the duration through an environment powered by controlled fusion life support.
If anti-matter could be processed from Enceladus and used in a purely artificial world ship, this would decrease the fuel mass to ship mass ratio by 200(ideal antimatter vs ideal fusion), or give 1/500th of the mass of the ship itself....this to obtain 30Km/s and require 200,000 years to reach the nearest star with a known planet....
An artificial world ship might be something as big as the Death Star I, and if it used anti-matter, it might be able to go as much as 10-100 times faster...
A cylinder with radius equal to the Death Star I and height equal to it's diameter would have mass significantly less than Asteriod Ceres, and "floor space" of as much as 32.5 million km^2 (6.36% surface area of earth). Even if a full 90% of the volume is dedicated to engines and fuel, that would still leavel about 3.25 million km^2 of floor space, or over half a percent of earth's surface area.
This should also be large enough to contain enough resources to support human beings across geologic time scales.