A good problem about a hare with an interesting continuation.
The hare needs to hide in the forest from the dogs as quickly as possible. A hare with speed v starts running at the edge of the field along the lawn from point 0. Behind the field there is a saving forest (point 3), the distance to which is equal to Δ (width of the field). The cunning hare does not run straight to the forest, but diagonally to a round area (point 1), located in the middle of the field. This could be, for example, an agricultural aviation helipad with a smooth concrete surface, where a hare can run faster - at vr speed. The hare at point 1 abruptly changes the direction of its run, crosses a circular area along a chord, runs out of it with another change of direction (point 2) and finishes in the forest along the shortest path (point 3). Determine the running trajectory of the hare - the coordinates of points 1 and 2 at which the running time will be minimal. The origin point is at the center of the circle.
Attached is the solution to the problem.
Continuation.
How can the cunning hare run if the round area is not concrete, but a plowed field - v > vr?

Robit.sm (42kb) downloaded 1 time(s).