Achilles, the tortoise and the sum that ends

Question

Achilles runs at 10 m/s. A tortoise crawls at 1 m/s and starts 100 m ahead. Zeno argues: by the time Achilles reaches the tortoise's starting point, the tortoise has moved 10 m ahead; by the time Achilles covers those 10 m, the tortoise is 1 m ahead; and so on forever, so Achilles never catches it. 1. Where does Achilles actually overtake the tortoise, and when? 2. What is wrong with Zeno's argument?

Accepted Answer

Achilles overtakes after 11 and 1/9 seconds, having run 111 and 1/9 metres. Direct calculation. Achilles closes the gap at 10 minus 1 = 9 m/s. A 100 m gap closes in 100/9 = 11.11 seconds. In that time Achilles runs 10 × 100/9 = 1000/9 = 111.11 m, and the tortoise has crawled 11.11 m to the same point. Zeno's stages add up. Zeno's stages have Achilles running 100 m, then 10 m, then 1 m, then 0.1 m, and so on: a geometric series with ratio 1/10. Its sum is 100 / (1 minus 1/10) = 100 / 0.9 = 111.11 m, the same point. The stages also take 10 s, 1 s, 0.1 s, ..., summing to 11.11 s. What is wrong w…

Achilles, the tortoise and the sum that ends

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