The farmer and the pig

Found while scanning through one of my old books in search of some notes on constrained nonlinear optimization techniques.

(a) At t=0, a pig, initially at the origin, runs along the x axis with constant speed v. At t=0, a farmer, initially 20 yd north of the origin, also runs with constant speed v. If the farmer’s instantaneous velocity is always directed toward the instantaneous position of the pig, show that the farmer never gets closer than 10 yd from the pig.

(b) Now suppose that the pig starts over from x=0, y=0, and t=0 and starts running at speed v. The farmer still starts 20 yd north of the pig but can now run at a speed of 3v/2. The farmer is assisted by his daughter who starts 15 yd south of the pig at t=0 and can run at a speed of 4v/3. If both the farmer and the farmer’s daughter always run toward the instantaneous position of the pig, who catches the pig first?

(c) At t=0 a pig initially at (1, 0) starts to run around the unit circle with constant speed v. At t=0, a farmer initially at the origin runs with constant speed v and instantaneous velocity directed toward the instantaneous position of the pig. Does the farmer catch the pig?

This is exercise 1.30 from Advanced Mathematical Methods for Scientists and Engineers (1978).