Most folks require 7-9 hours of sleep per 24 hour cycle. If one was sleeping regularly and if one got a full night's sleep (8 hours) them one COULD perhaps drive for 16 hours before pulling over to sleep. You would also have to subtract stopping for food, gas, restroom breaks, stretching, etc.). Additionally, driving during the day makes more sense. So: although you could go for 16 hours, 12 is a more reasonable outside figure in one day (although still a lot of time on the road).
Skipping a night of sleep and then driving would NOT be recommended. If one did this, one could conceivably drive all night and the next day - gaining an additional amount of driving time without sleep - although this would be reckless, dangerous and potentially life-threatening.
A more common practice for those pressed for driving time might be to drive with a partner and switch off driving duties with sleeping time. Some people are also fine taking shorter naps of 1-4 hours and then driving again.
Better to be safe than sorry if one is the least bit sleepy - and pull over at a rest stop for a nap.
Here's a link to check out:
http://www.aaafoundation.org/resources/index.cfm?button=drowsyfaq
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