Before reading the answer can I interest you in a clue?
First consider the diagram below:
R - is the radius
of the circle
X1 - is the radius of the inner dotted circle
X2 - is the radius of the outer dotted circle
Firstly I, like everyone else thought that
the most likely technique would be to stand in the middle and run away from
the dog. This would leave you running a distance R and the dog running πR this
would take you a time R/v and him πR/(4v) so clearly he would get there before
R/v > πR/(4v)
So clearly we need to be closer to the edge
than just in the middle, let us work out how far away from the centre we have
to be in order that we have the advantage over the dog. Let our distance from
the centre be xR where x is a number less than one eg. if R were 10m and we
were 1.5m from the centre then x=0.15 Assuming that we can get the dog on the
opposite side of the circle to us (I'll come back to that,) then the distance
we must run is (1 - x)R where as the dog must always run πR we can equate the
time associated with these distances, remembering the dog runs at 4v.
(1 - x)R/v = πR/(4v)
x = 0.2146
Any closer than this to the center (ie x being smaller,) and the dog will still
be able to catch us, we need to start with a value of x greater than 0.2146. Now how do we get the dog on the opposite side? Well if
we run in circles around the centre clearly if we are close enough to the center then the
dog will not be able to keep up. Consider if i am at a distance of x = 0.1 and
running in a circle, the dog will not be able to match my position on the
outside of the circle. Infact if I am running in a circle of x < 0.25 the dog
will not be able to keep up (remember the dog runs 4 times my speed.) So we
pretty much have it.
The strategy is to run in a circle of
0.2146R < radius < 0.25R until the dog is on the opposite side then make for
the fence like the clappers.
It seems possible that there are other solutions. Such as a spiral giving an angular component to your velocity. This would not work if your direction of travel was always away from the dog, (as in θdog + π,) which was my first thought because at some point all your velocity would be angular. We could assume the dog would always move so as it would always run to try to match θhuman. This seems solvable. You would need to consider that your radial velocity would be less than V because there is a tangental component. You could model with different angular velocities.
I like this one.