Coming a bit late to this party but I'll do my best...
Johnny Canuck wrote:At exactly what (faster-than-light) speed would you have to travel in order to escape a black hole?
There's no speed fast enough, it's just not possible in the geometry of the universe. A very closely equivalent question, which is much easier to visualize, is something like "Where can I go that is further north than the north pole?". Clearly there's no such place, because no matter what you do, you're still bound by the fact that the Earth's surface is a sphere and the north pole is as far north as you can get. Likewise as you travel around the 3-dimensional space of the universe, you're bound by other geometrical rules
which (among other things) make it logically impossible to go faster than the speed of light. It's hard to imagine because, at the sort of speeds we're used to dealing with, it seems like you can just go as fast as you like, but at high speeds that's not the case. (All of the above assumes general relativity, which is our current best guess of how the universe works at large scales.)
Ian Volante wrote:Depends on the mass of the black hole. The formula is (2GM/R)^1/2.
That's the classical formula, it doesn't apply in the general relativistic case. The speeds and masses involved in black holes are so large that the classical formula doesn't apply. (In fact, a black hole doesn't even make sense in the classical model, and you need general relativity to describe them.)
Ian Volante wrote:The heavier the black hole is, the larger the event horizon will be, and the more energy it'll take you to escape though.
I think this is a misunderstanding of escape velocity. The escape velocity is the initial speed you need to set off at to escape; you don't need to spend more energy as you go. If you do expend more energy then your initial speed can be much lower, which is how e.g. a satellite can escape without being flung off the Earth at 11 km/s. (I think Gevin made the same point already but I didn't fully understand his post.)
Dave Preece wrote:Surely there isn't ever any specific speed, for any size of black hole, as whatever speed you come up with, the speed can infinitely be slowed fractionally, bit by bit forever; I'd say it's an impossible question to answer (as specific speed that is)? I'm probably wrong though?
It's not really a problem. Say the escape velocity of the Earth was exactly 11 km/s. If you set off at any speed less than 11 km/h, even 10.9999999, you would be able to get a certain distance but no further; the gravity of the Earth would force you to stop or turn back. If you set off at 11 km/h or faster, you'll be able to go as far as you like. The escape velocity is the lowest speed at which you'll be able to travel as far from the surface as you like. So it's quite precisely defined. (PS: beware