Little Jo and her pig

This happened in a square-shaped corral ABCD (corners labeled anti-clockwise). There was a hole in the fence at the point C. A pig, which was initially at the point B spotted the hole and started running along the fence from B to C with a constant speed 1 (you choose your favorite units). Rancher Jo*, who was initially at the point A, noticed this immediately, and started running with the constant speed 1.5, such that at every moment her direction is straight towards the pig**. Will she catch the pig before it escapes through the hole?

Do you see any similarity between this and the `Airplane and the wind' problem? Maybe after you solve both?

*According to the movie `The Ballad of Little Jo' this rancher really existed (sometime during the Gold Rush in California), but the accident with a pig seems to be an invention of differential equations teachers.

**If Jo had time to think a little, she would rather run straight to the hole (using the fact that the ratio of speeds is 1.5, which is greater than the square root of 2). I don't want to imply that Jo was not smart enough. Her reaction was instinctive reaction. When your pig tries to escape, you have no time to think about the ratio of the speeds, even less to solve differential equations...

This problem can be seen frequently in the older differential equations textbooks, though the stories vary: a fox trying to catch a rabbit, a soldier firing an infrared self-guided anti aircraft missile and so on. I prefer Little Jo, because I like the movie. The problem was offered in 1994 on a qualifying exam for engineering graduate students.

Hints.

Let us choose a rectangular coordinate system, so that A=(-1,0), B=(0,0) and C=(0,1). Let y=f(x) be the trajectory of Jo. Let t be the time variable, so that both Jo and the pig start running at t=0. What was said about Jo's direction means the following. If at some moment Jo is at a point (X,Y) and the pig is at a point (0, t), then the tangent line to the graph of f at the point (X,Y) passes through (0,t).

It is recommended to find the trajectory y=f(x) first. To do this it is convenient to calculate the moment t, when Jo is at a point (X,Y) in terms of f and x. The length of the piece of Jo's trajectory from A=(-1,0) to (X,Y) is given by the formula for the arclength from calculus: is it the integral with respect to x, from -1 to X of the square root of {1+y'(x)y'(x)}. The time t is this length divided by Jo's speed. Now the pig's position (0,t) can be expressed in terms of X and f. This leads to a second order differential equation. After you solve it, and find f(x) explicitly, it will be easy to answer the question.

Solution