Is something known about the following optimal trajectory problem?

Question

Given an arbitrary straight line L of which the only thing that is known is that it is located at a distance of 1 unit from the origin O
Find the optimum strategy to go from O to L in the mimimum amount of time. For that strategy, find the average time needed and the worst case time.

You can think of it as a ship in the fog 1 mile from the coast but the incompetent captain doesn't know the direction of the coast.

Answer 1

Travel one mile ahead, then turn 90 degrees to the right; from there, keep steering right in a circle with radius 1 mile.

You are bound to hit the coast in maximal 2 pi + 1, or 7.283 miles.