Ok then, how long would it take to fully charge up a launch system for one launch, where the only power source is a standard 240V plug socket?
Again, I can only really speak from my experience with buildings, but most plant (for example, air handling units, exhaust fans, chillers, many of these can have very large motors generating lots of power) are often only wired from the building's low-voltage supply - which is often at 240V or 400V (I think that's some sort of industry standard). These are seriously big pieces of equipment, pulling massive currents and really putting a lot of stress on the electrical infrastructure of the building. However, they could, essentially, be wired off the same panels as the plug sockets (you just have to make sure your breakers and fuses are up to it). You'll often run higher voltages for long distance distribution due to the reduced losses, but once you're down at the equipment level, it's the current you really want, so the voltage is stepped back down.
I would imagine that for most parks it's something similar. They'll have a high-voltage mains coming into the site, with a transformer stepping it down to medium-voltage for distribution around the park. Each ride likely then has it's own transformer (or smaller rides will share one) to take the voltage down even further to be usable. Magnetic launches will have large banks of capacitors to store up energy between the launches, meaning the grid seems a more steady load.
In fact, let's try some maths (with all the usual caveats of these sort of things - no friction, 100% energy transfer, I don't really know what I'm doing, etc).
Let's say we get an 8-tonne coaster train to 50 mph, at the end of the launch run the coaster would have [close to] 2 MJ of kinetic energy. Assuming a 240 V supply, with a 13 A fuse (similar to what you'd find for a laptop), that's delivering 3.1 kW of power. Dividing one by the other will, in theory, tell you how long it takes to deliver 2 MJ - in this case 645 seconds, nearly 11 minutes.
Capacitors aren't really that fast at charging, but this is where, without a bit more information on the size and properties of the capacitor, it would be hard to say for sure! Of course, they're likely running off much larger breakers than just 13 A, so they won't take anything like as long to charge as this! Plus, they could make parallel sets of capacity banks allowing one to remain charging while the other discharges, giving you double the time to charge a set.
I should add, that most of my knowledge of this stuff is A-level physics and some 2nd-3rd year courses at university, as well general cross-discipline learning from my colleagues at work.
EDIT: It's late and I can't sleep - so apologies to anyone who does actually know what they're doing for any errors in the above. Happy to stand corrected.
