Henry Ernest Dudeney/Modern Puzzles/211 - Another Wheel Paradox/Solution

Modern Puzzles by Henry Ernest Dudeney: $211$

Another Wheel Paradox

When a railway train is in motion, it is always the case that some parts of the train are travelling backwards.

Explain why.

Dudeney couches this paradox in a tediously long-winded conversation between a pair of cyclists leaning on a bridge.

Solution

Let us direct our attention to the wheels of the train.

A train wheel has a flange on it, so as to make sure it stays on the rail.

This flange descends below the level of the rail a short but measurable distance.

As the train rolls along the track, the part of the flange that is at any one time below the level of the rail is demonstrably moving backwards.

The graph of the motion of a point on this flange can be illustrated as:

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $211$. -- Another Wheel Paradox
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $295$. Another Wheel Paradox