A rook beats a lone pawn only when its king arrives in time. Count the moves for both sides and bring your king first. Rook checks that only push the enemy king next to its pawn waste the moves you need.
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A rook should beat a pawn, and often does not. Here your rook on h1 faces a black pawn on e4 supported by its king on f3, while your own king is far away on a5. White to move wins, but only by counting first and moving the right piece. Ten moves to win the pawn.
What you will learn
01How to count the race between your king and the pawn.
02Why the rook on the first rank is already doing its job.
03Why rook checks that look active throw the win away.
The technique
The rook on h1 already guards e1, the queening square. The trouble is the black king: next to its pawn it protects it, and the rook cannot win a protected pawn on its own. So the win depends on your king arriving in time. Kb4, then Kc3 and on to d3 or d2: from there the king covers the squares beside the pawn and guards the rook when it captures.
The natural mistake is a check. Rf1+ or Rh3+ drives the black king forward, next to its pawn and closer to e1, and both checks throw away the win. The rook does its work later: once your king is close, a check along the second rank or the h-file pushes the black king off the pawn, and the rook takes it with your king beside it.
Kb5 and Ka4 also win, and so do Ra1 and Rb1, but Kb4 is the fastest. Most other rook moves only draw, among them Re1 and Rd1, which look natural. So do Ka6 and Kb6, which walk the king the wrong way.
How to practise it
Play White. Count first: how many moves the pawn needs to reach e1, and how many your king needs to reach d2. Then move the king and leave the rook where it is until the king arrives. Taking the pawn inside ten moves ends the drill.
No. If the pawn is far advanced with its king beside it and the attacking king is far away, the defender often draws. The count of king moves decides it.
Why not check the king away from its pawn?
Because from f3 the black king answers a check by stepping next to its pawn or towards e1, which is where it wanted to go anyway. Each check costs you a move and helps the king you are checking.