How far is darwin from Alice Springs ?

The distance from Darwin to Alice Springs is 1500kms as the others have said - however the speed limit from Darwin to Alice is 130kms per hour as it is a straight flat road with few towns where you need to slow down. Locals normally do the trip in one long day (depends on your preference) taking 12 -15hrs to complete including short breaks.
There is also few decent places to stay between the two - with only Mataranka hot springs and Katherine rating as reasonable in my books.

Distance Darwin NT to Alice Springs NT is 1497.87 kms.
Travel Time (not including stops for fuel etc) 17:26 hours

From RACQ Trip Planner.

I can't remember the distance. However, I drove from Darwin to The Alice about 10 years ago. It took me about 20-22 hours over three or four days to drive. Bear in mind this includes time for loo breaks etc and driving time to where I was staying each night. So you're looking about 18, maybe 19 hours, driving non stop. I think we were averaging out about 60 mph so you're looking about 1100 miles.

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