Farmer Chasing a Goose

Farmers know that to catch a stray goose one does not run after the goose in an open field. A better strategy is to corner the goose. However, suppose the farmer and the goose are in an open field and they both run with the same speed, V, to provide us with some semblance of fair play. Furthermore, restrict the farmer to chasing the goose along the instantaneous line of sight to the goose. When will the farmer catch the goose?

Answer

If the farmer is restricted to chasing the goose along the instantaneous line of sight to the goose, the farmer will never be able to catch the goose unless there is a head-on encounter. The best strategy for the goose is to run in a straight line, for then the farmer’s velocity is eventually in the same direction as the goose’s direction, and the relative distance remains constant. Note that even when the goose changes direction often, the farmer cannot close the gap completely because the closer they are, the more toward being parallel are the velocities!

In the real open-field experience without the restriction, one strategy that might work if the goose is inexperienced is for the farmer to anticipate the position of the goose and to get there at the same time the goose arrives. However, most geese “read” the game plan and change course in midflight.