The Dog and the Duck

Time is of the essence with this post so please excuse the strange mix of notations later on. One of my friends sent me this puzzle from and here is my solution to how the dog needs to catch the duck.
Lots of peoples first instinct is times faster but this is really just a lower bound, i.e. if the dog travels less than π times faster than the duck then the duck can just read radially to the antipoint of the dogs starting position.
There are a few key assumptions here that we can make from the beginning:

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