I have, in a pinch, taken a few shots in dark rides with a slowish lens - can you get something 'usable'? Yes. Will it be as excellent as some examples you'll see here of folks using full-frame cameras, F1.4 lenses, and tons of RAW processing? No. I wouldn't let it discourage me from still taking some shots, but just accepting that they won't be as good as they would be with a faster lens.
My last camera, a Sony A550, had the 14MP sensor which is also in the D3100 - so actually not quite as good a sensor as the one in your D5100...with an F3.5 lens wide open, I could get usable but not great dark ride shots...here are some samples - these were shot strictly in JPG with no post processing, so they could possibly have been even slightly better results if they'd been shot in RAW and worked on a bit:
18-250mm lens at F3.5, ISO6400:
Tamron 10-24mm at F3.5 and ISO6400:
Sony NEX3, 18-55mm kit lens at F3.5, ISO6400 (same sensor, different camera):
Not my greatest dark ride results by any means - I can do much better with my 30mm F1.4 or 50mm F1.7 - but sometimes I didn't have those with me, and I still decided to give it a go just for fun. They'd be usable to show family, for slideshows, for small prints...so yes, you could get usable results even with your kit lens and shooting straight JPGs out of camera, and your D5100 is capable of even better results at ISO6400 with the newer 16MP sensor, which is quite good (I have the same sensor in my newer camera, Sony A580).