Question

Zeno’s famous paradox, as recounted by Aristotle, says that “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point where the pursued started, so that the slower must always hold a lead.” Although the paradox is philosophically interesting, we now know how to solve this problem – and as expected Achilles will eventually catch up to the tortoise. However, the slower of Zeno’s two protagonists can still win a race if given a sufficient head start.

QUESTION: Achilles and the tortoise run a Marathon (42.2 km), which Achilles will run at an average speed of 5.6 m/s and the tortoise at 0.28 m/s (the fastest land speed recorded for a tortoise). How much of a head start (in meters) does the tortoise need to win the race if both contestants start at the same time?

Answer 1

Average speed of Achilles = 5.6 m/sec

Distance traveled in marathon = 42.2 km = 42200 m

So time taken by Achilles to finish the race = distance/speed = d/V = 42200/5.6

t = 7535.7 sec

Now in this time distance covered by tortoise will be:

d1 = V1*t

V1 = speed of tortoise = 0.28 m/sec

d1 = 0.28*7535.7 = 2110 m

So head-start given for tortoise to win = D = d - d1 = 42200 - 2110 = 40090 m

So tortoise should be given a head start of 40090 m (40.09 km) to win the race

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