In the fifth century B.C., the Greek philosopher Zeno proposed the following par

Question

In the fifth century B.C., the Greek philosopher Zeno proposed the following paradox of Achilles (along with other three paradoxes, the Dichotomy, Arrow, and the Stade, as retold by Aristotle). The Achilles paradox states that, given a head start, a tortoise will never be overtaken by the legendary runner Achilles because, Achilles must first reach the point P_1 at which the tortoise started, and by the time Achilles arrives at P_1, the tortoise will have advanced to a point P_2, which Achilles must reach before overtaking the tortoise. By the time Achilles reaches P_2, the tortoise will have moved ahead to a point P_3, and so on, ad infinitum. Thus Achilles will never be able to overtake the tortoise. Obviously this contradicts our common sense, but where is the flaw in this paradox? Aristotle called this and other paradoxes fallacies but was never be able to refute them. Bertrand Russell said that the paradoxes are "immeasurably subtle and profound". It took more than 2000 years (for mathematicians) to create a framework (infinite series) in which one could undertake the infinite process of calculus free from such logical difficulties. To quantify the Achilles paradox, suppose Achilles is running at the constant speed of 30 km/hr, and the tortoise at the constant speed of 0.1 km/hr with a 1 km head start. Follow the argument in Zeno's paradox and show, by using infinite series, that Achilles will indeed overtake the tortoise, in fact in a short amount of time.

Explanation / Answer

Time taken by achilies to travel 1km

1hr --> 30km so for 1km he takes 1/30hr = 0.033hr

In that 1/30hr the tortoise travels ==>0.1*1/30 =1/300 km

Time taken by achilies to travel 1/300 km = 1/(300*30)

Time taken by achilies to travel the distance travelled by tortoise = 1/30 + 1/300*30 + 1/ 300*300*30 + ....+1/300n*30= 10/299

Distance travelled by tortoise in that time = 10/299*0.1 =0.33km

Distance travelled by achilies in that time = 10/299*30 = 1.0033km

So here we can see that when time tends to infinity achilies travelled more distance than tortoise

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