The Achilles Paradox I know that it is a very old question but still I don't find any satisfactory solution for Achilles Paradox. Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.
 A: This is quite straightforward as long as you break it up into steps. 
Let the tortoise travel with speed $v_t$ and Archilles with speed $v$. We split up the steps Archilles takes as
\begin{equation}
d_A = d_0 + d _1 + ... 
\end{equation}
where $d_i $ is the distance travelled in a single step. The initial distance $d_0 $ is then just the initial separation between Archilles and the tortoise. We denote $\Delta t _i $ as the time of step $i $. The distance $d_1 $ is just how far the tortoise managed to step while Archilles was catching up to the tortoise's initial position. It is given by 
\begin{equation}
d_1=v_t \Delta t_1 =v_t \frac{d_0}{v}
\end{equation}
Similarly, the distance $d_2 $ is given by 
\begin{equation}
d_2=v_t \Delta t_2 = v_t \frac{d_1 }{v} = \left(\frac{v_t}{v} \right)^2 d_0
\end{equation}
Its easy to see that this trend will continue:
\begin{align}
d_A &= d_0 + \frac{v _t }{v} d_0 +\left(\frac{v_t}{v} \right)^2 d_0 + ... \\
&= d _0 \left( 1 + \frac{ v _t }{ v} + \left( \frac{ v _t }{ v } \right) ^2 + ...\right)
\end{align}
In the brackets we have a geometric series. If $v_t\le v$ then,
\begin{align}
d _ A & = d_0\frac{1}{ 1 - v _t / v } 
\end{align}
This is finite (and hence Archilles can catch up with the tortoise) as long as $v_t <v $. The apparent paradox is that Archilles needs to take an infinite number of steps. However, having to take an infinite number of steps doesn't mean it would take an infinite distance (or infinite time). An infinite series can either converge, as this one does, or diverge.

I think it's also interesting to solve this problem the simple way, avoiding splitting it up into steps. For Archilles and the tortoise representatively we have,
\begin{equation}
d_A=\frac{v}{t}\, , \quad d_t= \frac{v_t}{t}+d_0
\end{equation}
Setting $d_A = d_t $ and solving these equations gives,
\begin{equation}
d_A=d_0 \frac{1}{1-v_t/v}
\end{equation}
as required.
A: I don't know if I'll answer what you want, but is impossible not to talk about time; the point of the paradox is that both the Achilles will never reach the turtle, right? now, in the way the paradox is stated, it is said that at each "step" the turtle advances "x" space and Achilles will advance only a fraction of "x" space. The fallacy of the paradox lies in that the statement displaces the concept of time from real time to "step-time". 
So, Achilles in deed will reach the turtle only in infinity, but this doesn't happen in the real time, just in the mathematical steps, which are increasingly small. The place where they will meet can be solved as the result of a geometric series (see answer by JeffDror).
A: It's just a paradox, it's using weakness of human logic about infinite calculation in limited time. It convinces you think that you have to cover  half of the distance between Achilles and turtle every passing instant of time. If you think deeper, you will notice that most of the concepts such as time, motion, distance, instant, speed lose meaning, their definitions are relative to each other indeed.
A: Not Newtonian, but still relevant:
Zeno's Paradox is based on the premise that the distance between Achilles and the tortoise is infinitely divisible.  In theory, the Planck length is a lower limit on measurable distances.  If this is true, eventually Achilles must move more than half the distance to an arbitrary point between himself and the tortoise if he moves at all.
