When I work out the time dilation and change in position during the acceleration phase, which is so complicated I usually ignore it for simplicity sake, I do see that the problem is a bit more subtle, but still is not "unatainable" as we keep thinking.
If we acclerate at 10m/s2 for what "should be" 29970000s <= 347 days.
We get some interesting results.
Time dilation due to acceleration formula:
http://en.wikipedia.org/wiki/Time_dilationg and a are the same, except you need to reverse the sign.
I used the following numbers:
g = 10m/s^2
v0=0 (since we are accelerating from rest relative to earth.)
t=29,970,000s
The after acceleration, we have:
Position = x = 8.598e15 meters (9.4608e15 meters is a light year)
So paradoxically, the ship travels almost an entire light year during the acceleration phase....
velocity = v = 212,025,888.75 m/s
Strangely, the velocity ends up lower than Newton says it "should be", but we actually travelled farther!
Proper time = T' = 585,518,761s = 6,777days = 18.57 years
Ok, proper time really, really hurts me
Wow, you really can't ignore the acceleration phase at all. Lol. That really hurts the travellers.
Still, this would suggest a 100ly journey could still be made in human lifetimes even with nothing but existing technologies.
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Ok, I'm in the middle of typing this, but I just found something that has got to be some sort of mistake. When I plug in numbers to this formula for accelerating at the absurdly slow rate of 1m/s2, I find that, according to the formulas as given by wikipedia, the ship reaching the exact same velocity in less time!
a = g = 1m/s^2
v0 = =
t = 299,700,000s
The according to the formula
x = 3.7216e16m (4 ly!?)
v = 212,025,888.75m/s (same as other way)
T' = 103,822,077.4s = 1201.64 days = 3.29 years.
This is a serious paradox in that the formula says you can actually travel almost ten times farther during the acceleration phase in 1/6th the time by accelerating more slowly, and actually expending far less energy in the process.
Ok, so I assume I've made a math error, but what that might be, I have no clue.
Can anyone be bothered to double check these results? I've re-checked them already with apparantly the same results.
This crazy formula literally suggests that the slower you accelerate the faster you will get to the destination, and literally get there faster than if you started out moving at the speed of light in the first place, which is ridiculous.