Stadium Paradox

Paradox

Consider three rows of bodies:

$\begin{array} {ccccc} \text {(A)} & 0 & 0 & 0 & 0 \\ \text {(B)} & 0 & 0 & 0 & 0 \\ \text {(C)} & 0 & 0 & 0 & 0 \\ \end{array}$

Let row $\text {(A)}$ be at rest, while row $\text {(B)}$ and row $\text {(C)}$ are travelling at the same speed in opposite directions.

$\begin{array} {ccccccc} \text {(A)} & & 0 & 0 & 0 & 0 & \\ \text {(B)} & 0 & 0 & 0 & 0 & & \\ \text {(C)} & & & 0 & 0 & 0 & 0 \\ \end{array}$

By the time they are all in the same part of the course, $\text {(B)}$ will have passed twice as many of the bodies in $\text {(C)}$ as $\text {(A)}$ has.


Therefore the time it takes to pass $\text {(A)}$ is twice as long as it takes to pass $\text {(C)}$.

But the time which $\text {(B)}$ and $\text {(C)}$ take to reach the position of $\text {(A)}$ is the same.

Therefore double the time is equal to half the time.


Resolution


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Historical Note

The is one of Zeno's Paradoxes, as famously raised by Zeno of Elea.


Sources

  • 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Zeno's paradoxes
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): stadium paradox
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Zeno of Elea (5th century bc)
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Zeno of Elea (5th century bc)
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