Wagon collision

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

Puzzle

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

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Solution

Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I = \{(x,y) : 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x=y=0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of \(I\) to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y = 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than \(2R\) and each wagon has radius exactly \(R\) both wagons must collide at this point.

References

Ordinary Differential Equations - V. I. Arnold's book on ODEs.