In the fifth century b.c., the Greek philosopher Zeno posed a paradox involving a race between Achilles (the fastest runner at the time) and a tortoise. The tortoise was given a head start, but once the race began, Achilles quickly reached the point where the tortoise had started. By then the tortoise had moved on to a new point. Achilles quickly reached that second point, but the tortoise had now moved to another point. Zeno concluded that Achilles could never reach the tortoise because every time he reached the point where the tortoise had been, the tortoise had moved on to a new point. This conclusion was absurd, yet people had trouble finding an error in Zeno’s logic.
Suppose Achilles runs 10 m per second, the tortoise runs 1 m per second, and the tortoise has a 10-m head start. We wish to find how much time it takes until Achilles catches up with the tortoise.
(a) Solve this problem using a geometric series.
(b) Solve this problem using algebra.
(c) Explain the error in Zeno’s reasoning.