[2] cperciva [http://news.ycombinator.com/item?id=1993399] used 3g. Based on that you would have to multiply the launch distance by (25/3) and get 2,018,750 meters, about 1,250 miles. Again please correct me if I got something wrong.
It's been a long time since I've thought about rail guns, but I think you might be off in your second equation. But seeing as my other comment today is about how I may not have fully recovered mentally from a debilitating disease, I have to doubt myself.
Anyway, aren't you forgetting that most of the time is spent at a lower speed, so that you want to integrate the acceleration rather than simply multiplying by the escape velocity?
While we're at it, I'm not sure what we really should be using for escape velocity. We should probably account for the deceleration after we leave the muzzle while still in the atmosphere. And if we presume a surface mounted rail gun, we'd probably want the escape velocity for a launch aimed just above the horizon, which involves more of this.[1]
Or if we're somehow presuming vertical, we'd want the escape from that height. But if vertical, we'd have to account for the extra gravitational force on the passengers, unless this is already accounted for in the human limits. In any case, I'm not sure that the speed of the Space Station (or the surface velocity I used above) really makes sense here.
[1] I vaguely recall that ignoring air resistance, the velocity is the same regardless of direction, but I don't have confidence in this, and my quick searching hasn't turned up anything definitive. Is this possibly right?
Forget about launching people or anything squishy or delicate. You'd still get a lot of use out of things that could survive 1000g. That would let you launch things into orbit with an accelerator only 4 miles long. Launching refueling stations or LEO->GEO tugs would be useful. How about rolled-up thin-film solar panels?
Your answer makes sense. Regarding [1] I do believe that your right, without air resistance the escape velocity is the same regardless of direction. The reason I chose the speed of ISS is because the escape velocity to enter into orbit it he speed needed to maintain that orbit. So in order to reach an orbit that parallels the ISS you would have to get up to the speed of the ISS and not necessarily any faster.
No, direction does matter. Perhaps we're confusing things by talking about escape velocity instead of orbital velocity. We want to reach a certain orbital velocity with respect to the Earth as a reference.
Think of it this way: if you were approaching Earth from space and wanted to enter orbit with the ISS, you need to reach a speed, e.g. 17,600. The speed is the same no matter where you are coming from, such as from the Earth's surface.
On the equator, if you take off east, you already have about 1,000 mph working with you. If you take off west, you need to get an extra 1,000 mph. Towards the poles, the (dis)advantage lessens.
I agree on the East vs West if we are aiming for orbital. I disagree on the speed being the same even at the surface, as (I think) you have to subtract the gravitational deceleration as you get from the surface up to the height of orbit. Coming from space you gain speed coming in, so this is not an issue.
Details aside, the overall conclusion remains that rail guns are not likely to be a useful means of propelling squishy cargo like humans into space.
Assuming:
Launch Speed: 17,180 mph (Speed of Space Station) [1]
Launch Acceleration: 25g (Based on my understanding of what humans can survive) EDIT [2]
Just using that as a rough estimate and Wolfram Alpha we get 242,250 meters, or about 150 miles.
http://www.wolframalpha.com/input/?i=17,239.2+mph+/25+gs
http://www.wolframalpha.com/input/?i=31.434+seconds+at+17,23...
Please feel free to check my math and yell at me.
[1] http://en.wikipedia.org/wiki/International_Space_Station
[2] cperciva [http://news.ycombinator.com/item?id=1993399] used 3g. Based on that you would have to multiply the launch distance by (25/3) and get 2,018,750 meters, about 1,250 miles. Again please correct me if I got something wrong.