There has been a lot of talk of Special Relativity, so I'm just going to clear some things up so no one can BS it.

There is a coordinate system K with coordinates x, y, z and a time coordinate t.

There is a coordinate system K', moving at velocity v in the x direction relative to K, with coordinates x', y', z', and a time coordinate t'.

c is the speed of light.

These are the equations:

Where this is the Lorentz Factor:

So just multiply them together. I don't know why they put it like that. Just imagine the Lorentz factor, with the part in the original equation on the top instead of just 1.

The denominator in the Lorentz factor is extremely close to 1 when v is nowhere near c, so you just divide by very near to 1 and the rate of time and distance is not affected greatly, but as we reach the speed of light, as the Earth will after a lot of time, but it cannot reach or pass the speed of light. Time will slow down, which will decrease your relative speed, and distance will grow in the direction you're moving. This will avoid breaking the "cosmic speed limit" of the speed of light.

Relative to ...what? To something, Earth will be moving faster and faster, until time necessarily slows and direction grows, but the speed relative to, um, that same thing, will be decreased. If you go outside of the Earth to whatever the Earth is relative to, then the Earth will no longer be accelerating much, it will just be approaching asymptotically the speed of light, or it will get closer and closer to the speed of light, without reaching it. Which is awesome.

This is bad, but just look at the left half. The x-axis is speed, and the y-axis is the slowing of time or growing of distance. Take your pick.

Time will slow down until it is, essentialy, not moving. It will be lowered every second it constantly accelerates, and time will get slower and slower and never go on, simply be very nearly frozen for all eternity. This would have happened a long time ago, and the year 2011 would never have began. I have determined that it will take 30,559,883.588 seconds to accelerate to this speed, which is 353 days, 16 hours, 51 minutes, and 23.588 seconds.

Of course, time will have slowed down, so we won't be quite at that point yet, so, quite awesomely, this will probably happen after about a year.

But I've been alive for much more than a year, and the point is, time won't even go on, even slowly beyond all imagination, if it's accelerating at an entirely constant rate. To us, theoretically, the moment will pass without our noticing, but the problem is, the moment won't pass. Time will have, essentially, ended.

So, in other words, the end of time will come in one year.