Physically Incorrect

This is a solution to this puzzle.

First, note that you cannot make a straight dash from the center since the dog will always outrun you. It will take you t=R/v seconds to get to the edge while the dog will make it before you in t=πR/(4v) seconds (recall that πR is simply half the circumference of the circle).

The insight I’ve used to solve the problem is that there is an inner circle of radius r, in which your angular velocity (ω=v/r) is greater than the dog’s angular velocity (ω=4v/R):

To find the maximal r for which this is true, you must solve (v/r) > (4v/R), which results in r<R/4. As long as you stay within that circle you can outrun the dog in the sense that you can complete a circle before he does; if you do so enough times you can eventually reverse the positions of yourself and the dog. At this point (assuming you’re very close to the edge, r=R/4), you can make a run for it and escape; the dog will not be able to close the distance in time:

The remaining distance from the edge of the inner circle is simply R - R/4 = 3R/4, so you can make it to the fence in t=3R/4v seconds. The dog will need  t=πR/4V seconds, which is slightly longer - enough for you to make your quick getaway!