Cook scored a mammoth 766 runs during England's 3-1 series win in the last Ashes campaign in Australia in 2010/11. But in his first Ashes Test as captain, Cook was caught behind off James Pattinson for just 13.
Root, in his first Ashes Test and first as an opener after winning six caps as a middle-order batsman, made 30 before he was bowled by Peter Siddle's outswinging yorker.
But the pitch was a good one and at the interval Jonathan Trott was 37 not out, with 32 of his runs coming in boundaries, and fellow South Africa-born batsman Kevin Pietersen 10 not out.
Pattinson, bowling on the Trent Bridge ground where older brother Darren -- a one Test wonder for England against South Africa at Headingley in 2008 -- played for Nottinghamshire, started the match with an offside wide to Cook.
Next ball Pattinson, the 23-year-old son of an English father, speared one down the legside.