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**A3**

__Achilles and the turtle__

20-7-2012

a—-“The swift-footed Achilles is in position A_{0} and in rectilinear distance of 100m from position X_{0} where a turtle is. Achilles, who is going to try to reach the turtle in a race, is running at the speed of 10m/sec, while the turtle is running at the speed of 5m/sec. The two competitors start simultaneously towards the same direction A_{0}X_{0}.”

b—-Zeno believes that “Achilles cannot reach the turtle if they both run in the following way: When Achilles reaches the position X_{0}, the turtle then will have reached the position X_{1}, having covered the distance X_{0}X_{1}= 50m, which will be half of the distance A_{0}X_{0}= 100m which Achilles ran, since the turtle will always be running at half speed of that of Achilles.”

c—-When Achilles reaches the position X_{1}, the turtle will have reached the position X_{2}, having covered the distance X_{1}X_{2}=25m.

It is clear that Achilles will never be able to reach and overtake the turtle because the latter will always preceed, even on minimum.

__Note:__

The walker, the arrow or Achilles and the turtle are arbitrarily chosen moving objects. In their position, any material mobile object can be placed – for example an Electron or a galaxy. Zeno chose to study the principle of movement with the use of these ordinary objects.

It is understood that neither Achilles nor the turtle are able to start (see note of paradox of the arrow.)