Escape from goblin from circular lake

Question

You are in a boat in the exact center of a perfectly circular lake. There is a goblin on the shore of the lake. The goblin wants to do bad things to you. The goblin can't swim and doesn't have a boat. Provided you can make it to the shore — and the goblin isn't there, waiting to grab you— you can always outrun him on land and get away.

The problem is this: The goblin can run four times as fast as the maximum speed of your boat. He has perfect eyesight, never sleeps, and is extremely logical. He will do everything in his power to catch you. How would you escape the goblin?

Answer 1

The Optimum Strategies
Your optimum strategy has two parts: You start by rowing directly away from the goblin. As he pursues you, you turn so as to always keep the center of the lake between you and the goblin. You can maintain this strategy until you are 1/K times the radius of the lake (R) from its center and the path you row will be a semicircle. The second part of your flight is to continue in a straight line tangent to the circle in which you were rowing. (check the image)

The optimum strategy for the goblin (if you also adopt the optimum strategy) is to run as fast as he can in one direction around the lake. If you abandon your optimum strategy before you reach the the R/K circle, the goblin simply reverses direction as necessary to minimize your advantage. Regardless, the goblin can do no better than being as far away as possible when you cross the R/K circle. After you cross the R/K circle and are following the tangent to the shore, if the goblin reverses direction, you simply turn onto the chord centered on your position and head away from the goblin; you will reach the shore with an even greater margin.

The Answer
The threshhold value of K is 4.60333885 (to 8 decimal places). This is the solution to:

sqrt(1-1/k^2) = (pi + acos(1/k))/k.

The left side of the equation is the time you need to travel along the tangent line (for R = 1, and speed = 1) and the right side is the time for the goblin to arrive at the same point (speed = K).

Answer 2

The obvious plan is to make a beeline for the shore at the point farthest from where the goblin is right now. This gives you a substantial distance advantage. You have only to travel a radius (r) of the circular lake. The non swimming goblin has to run in a semicircular arc along the shore, amounting to half the lake's circumference. This comes to Jtr. The goblin thus has to cover π times the distance you do.

Pi is a little more than three. Were the goblin only three times as fast as your boat, you could narrowly beat him to the far shore. That's why the puzzle says the goblin is four times as fast. No matter where you choose to land, the goblin will be there to grab you.