Achilles and the tortoise problem

Question

You know the story - Achilles and the tortoise are having a race. The tortoise is given a head start of 100m. After some time, Achilles will arrive at where the tortoise was at, but the tortoise will have moved further. Question: find how long it will take for Achilles to catch up to the tortoise in seconds, and how far Achilles will travel when he reaches the tortoise in meters. Achilles runs at $10m/s$ and the tortoise $0.2m/s$. Use an infinite series.

Intuition

Obviously I would have to use infinite series to solve. As I see it, I should create two series: one for Achilles and one for the tortoise. For the turtle, it would roughly be $100 + \sum_{n=1}^{\infty}a_n$. The 100 is the 100 meter head start, and $a_n$ would be the series that represents the movement of the tortoise. For Achilles it would be similar.

Goal

The goal is to find $n$, which would depict the time in seconds, where Achilles will catch up to the tortoise. To do this, I would have to equate Achilles' series and the tortoise's series, evaluate each, and solve for n. To find the total distance traveled by Achilles, it would be substituting $n$ into the equation and evaluating the series.

Answers

So, any answers for $n$? I'm mainly confused as to how to formulate the infinite series, and how to solve for each series (which would involve seeing if it converges and such).

Answer 1

There is no reason that you have to use an infinite series. Achilles gains on the tortoise at a speed of $10-0.2 = 9.8 \frac {m}{s}$ and catches the tortoise in $\frac {100}{9.8} s$

If you really want to set this up as a series.

In 10 seconds the Achilles is at the tortoise's starting point. And the tortoise has moved 2 meters away. The new distance is 2 percent of the original distance. In $(10)(0.02)$ seconds Achilles will be an this new point, The tortoise will have move again creating a new gap $2\%$ of the previous gap.

$10\sum_\limits{i=0}^{\infty} 0.02^i$

Answer 2

You may regard this "answer" as an extended comment, since I have not proposed to answer the question in the way you asked for it to be answered. But I would prefer that this answer be viewed as a hint (though still not a complete answer).

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For the turtle, it would roughly be $100 + \sum_{n=1}^{\infty}a_n$. The 100 is the 100 meter head start, and $a_n$ would be the series that represents the movement of the tortoise.

There's no need to do anything "roughly". The formula $100 + \sum_{n=1}^{\infty}a_n$ is an exact formula, provided that you have a way to determine what $a_n$ will be for every $n.$

The goal is to find $n$, which would depict the time in seconds, where Achilles will catch up to the tortoise.

This is a conceptual error. A second is just an arbitrary unit of time. If you were told that Achilles ran at $600$ meters per minute and the tortoise moves at $12$ meters per minute, does that mean $n$ should depict the time in minutes when Achilles catches up with the tortoise? (This is the exact same problem as the original one, but with the speeds expressed in different units.)

The only thing $n$ has to represent in $100 + \sum_{n=1}^{\infty}a_n$ is the number of steps of some process. You can make the number of seconds in each step be whatever is suitable to the process you're trying to describe.

If you describe the race in the usual "paradoxical" way, the first step is for Achilles to reach the tortoises' starting point. This takes $10$ seconds. But now the tortoise is $2$ meters away, so the second step is for Achilles to run another $2$ meters. This takes $2/10$ of a second. In the second step the tortoise moves another $4/100$ meter, so the third step is for Achilles to run yet another $4/100$ meter, which takes another $4/1000$ second.

So we have:

For $n=1,$ $a_n = 2,$ total elapsed time so far is $10$ seconds.

For $n=2,$ $a_n = 0.04,$ total elapsed time so far is $10.2$ seconds.

For $n=3,$ $a_n = 0.0008,$ total elapsed time so far is $10.204$ seconds.

And so forth. There is no particular $n$ at which Achilles catches the tortoise; each $n$ just adds more decimal places to the repeating decimal that describes the exact amount of time required to catch the tortoise. The time at which Achilles catches the tortoise is the limit of the elapsed time as $n$ goes to infinity.

The solution without infinite series is much simpler, as this other answer has already shown.