Four dogs stand in four corners of a square . The side of the square is $1$ km . Now closing eyes, each dog runs at the same velocity to the dog residing to the right . By this, they cover half distance . After opening eyes , each dog runs at the same velocity to dog in the right and covers the same distance .
My questions is : 1)If all the dogs run like this , at last in what position will the dogs reside ? How much distance will they cover ?
My trying:
I have drawn figure . But what is the next procedure ?
By symmetry they will meet at the center of the square. The center of mass of the four dogs does not change and it starts at the center. The first run they each cover $\frac 12$ km. They are now $\frac 12\sqrt 2$ km apart. Presumably you mean that they cover half that distance on the second run, for a distance of $\frac 14 \sqrt 2$. Now find where they stop the second run and the distance between them. You are looking for a nice rule about how far they run in each step, then you will get a series to sum for the total distance.