As you wrote, rat can always reach and run along a small circle r=5v in opposite to cat. let say cat is at (50,0), and rat is at (-5v,0). Then we can imagine that there is no benefit for cat to run reversely, when rat run toward any point P (at the big circle) with x less than -5v. (just because 聪明的猫,应该会反向去追老鼠. So is a 聪明的鼠.)
Now the key is to find the minimum v. I just found some mistakes in my fomula. It should be
v=10*((Squre root((PI+A)^2+(cosA)^2-1)-cosA)/((PI+A)^2-1)
I didn't how you find v>1 and v<1 situation.