Narrow square paths ABCD and abcd have sides of $$200\text{ m}$$ and $$100\text{ m}$$. Two persons P and Q are initially located at a and A respectively.<br>With same initial positions as earlier and the speeds of $$1\text{ m/s}$$ and $$2\text{ m/s}$$, if Q were to move anticlockwise along the path, find the largest distance (in m) between them.
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The greatest possible distance between points of the two paths is from an inner corner to the opposite outer corner, $$150\sqrt{2} \approx 212.13\text{ m}$$. This value is actually reached. At $$t = 700\text{ s}$$, P (clockwise, 1 m/s) has walked 700 m, i.e. 300 m into its lap, and is at corner d, while Q (anticlockwise, 2 m/s) has walked 1400 m, i.e. 600 m into its lap, and is at corner B. These are opposite corners, so the largest distance is $$150\sqrt{2} \approx 212.13\text{ m}$$.