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Achilles and the tortoise


TheDivineFool
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For those who don't know what this paradox is about...

 

Achilles and a tortoise are in a race. Achilles gives the tortoise a head start. Then Achilles begins to run toward the finish. Before Achilles can overtake the tortoise, he must first reach where the tortoise was. When he reaches the tortoise's last position, the tortoise has moved a little further.

Now Achilles must reach the tortoise's new position. In the time taken to reach this new position, the tortoise has moved a little further...and so on.

Achilles can never catch up with the tortoise. BUT in reality he does. Thus, the paradox.

 

My knowledge of math is limited so I hope what I'm going to say isn't wrong. Anyway here goes...

 

Assume the following for the mathematical analysis of the problem. I've used concrete numbers to make the math easier.

1. The tortoise starts at 10 meters ahead of Achilles

2. Achilles' speed is 5 m/s and the tortoise's speed is 1 m/s.

 

Math analysis 1

D1 = distance traveled by Achilles in time t seconds = 5t

D2 = distance traveled by the tortoise in time t seconds = 10 + 1t = 10 + t

Achilles will catch up and then overtake the tortoise when

D1 >= D2

5t >= 10 + t

4t >= 10

t >= 2.5 seconds

That means that Achilles will catch up with the tortoise after exactly 2.5 seconds and then overtake the tortoise to win the race.

No paradox here. It agrees with reality.

 

Math analysis 2

Achilles must cover the 10 m distance between him and the tortoise.

It will take him 10/5 = 2 seconds to reach where the tortoise was.

In this 2 seconds, the tortoise moves 2 m.

Now, Achilles takes 2/5 = 0.4 seconds to reach where the tortoise was

In this 0.4 seconds, the tortoise moves 0.4 m.

Now Achilles takes 0.4/5 = 0.08 seconds to reach where the tortoise was

In this 0.08 seconds, the tortoise moves 0.08m ahead

And so on and so forth....

 

We notice 2 things:

1. The gap between Achilles and the tortoise is decreasing

2. The time taken to reach the previous position of the tortoise is also decreasing

 

However, the gap will never reduce to ZERO, the required condition for Achilles to catch up with the tortoise.

Therefore, Achilles never catches up with the tortoise.

This math analysis results in a paradox.

 

As far as I can see, I don't see any problem in both mathematical analyses (1 and 2).

1 is in harmony with the real world while 2 is not. But both are correct.

 

How do we resolve the paradox?

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The approach described looks at ever-decreasing times and distances, but even though there are an infinite number of terms, these series each converge to a finite number. It also misdirects you into thinking about motion that doesn't model reality. The result for the series for the time intervals is the time it takes to overtake the tortoise, but the motion doesn't stop at that time. The ratio of those infinite series is the speed, and Achilles moves at a faster speed than the tortoise.

 

IOW, Achilles doesn't move to where the tortoise is, he moves a fixed distance in a fixed time.

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Yea - he doesn't actually STOP at the tortoises last position, he keeps on going. If he had to actually stop each time he came to where the tortoise HAD been, then he would never overtake - just get closer and closer. But, if like reality, he has no need to actually stop each time, then he will sail past the tortoise as your maths suggest. :)

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The approach described looks at ever-decreasing times and distances, but even though there are an infinite number of terms, these series each converge to a finite number. It also misdirects you into thinking about motion that doesn't model reality. The result for the series for the time intervals is the time it takes to overtake the tortoise, but the motion doesn't stop at that time. The ratio of those infinite series is the speed, and Achilles moves at a faster speed than the tortoise.

 

IOW, Achilles doesn't move to where the tortoise is, he moves a fixed distance in a fixed time.

The approach described looks at ever-decreasing times and distances, but even though there are an infinite number of terms, these series each converge to a finite number. It also misdirects you into thinking about motion that doesn't model reality. The result for the series for the time intervals is the time it takes to overtake the tortoise, but the motion doesn't stop at that time. The ratio of those infinite series is the speed, and Achilles moves at a faster speed than the tortoise.

 

IOW, Achilles doesn't move to where the tortoise is, he moves a fixed distance in a fixed time.

I'm aware, in a very elementary sense, that the infinite series converges. Is that the correct explanation?

 

Also, you mention that this 'model' is not an accurate representation of real world motion. I wasn't quite satisfied with the infinite series sum solution. So, I was looking for an explanation our model of motion. As you suggest, could there be a problem in this? Where do you think the problem lies?

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It's not an infinite series because he doesn't keep stopping each time he reaches the last position of the tortoise... other wise it would be. His motion would be continuous and thus he catches the beast in the real world situation.

 

That's how I see it anyway - please correct me if I am wrong, but I don't think I am. :)

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