# Achilles and the tortoise

At what time and at what point will Achilles reach the turtle, if he does not stop in the infinite-trap of Zeno to make calculations.

You could change their speeds and initial distance and get the corresponding results.

All 3rd fields must have values greater than 0.

Achilles speed must be greater than the turtle.

km

## The time it will take to reach it

hours minutes seconds

## Info

In a road race, even if Achilles (point A) runs k times faster than the turtle. The turtle (point B) that is slower is given a lead d1.

When Achilles reaches point B the turtle will reach point C by traveling a distance d2=d1/k. When Achilles reaches point C the turtle will reach point D by traveling a distance d3=d2/k=d1/k2. By the same procedure the turtle will be preceded by d4=d3/k=d1/k3, d5=d4/k=d1/k4, ...
dν=(dν-1)/k=d1/kν-1, that is, Achilles will always be behind the turtle.

### Mathematical solution

The total distance that Achilles will travel until he reaches the turtle is S=d1+d2+d3+d4+d5+... = d1+d1/k+d1/k2+d1/k3+d1/k4+...
which is the sum of infinite terms of decreasing geometric progress with first term d1 and ratio λ=1/k. So Achilles will reach the turtle traveling space S=d1/(1-λ)=d1/(1-1/k)