Airplane and the wind

An airplane starts from the point A and flies to B. The speed of the airplane with respect to the air is v (constant). There is also a wind of speed 1 in the direction perpendicular to AB. Suppose that the pilot steers so that the point B is always straight ahead. Will the airplane ever reach B? The answer depends on v, of course.

Do you see any similarity between this and the `Little Jo and her pig' problem? Maybe after you solve both?


Let us choose a rectangular coordinate system, so that A=(0,0), B=(1,0). Let the trajectory of the airplane be described by a function y=f(x). Then the direction of velocity with respect to air is given by the vector (1-x,-y), and the velocity with respect to air is the collinear vector of length v. To obtain the velocity with respect to land one adds the wind velocity vector, which is (0,1). All this leads to a differential equation which is not difficult to solve. After you solve it, and find f(x) explicitly, it will be easy to answer the question.