B Reaching the Speed of Light: A Thought Experiment on Halving Distance Traveled

[ShaunM]

I'm talking about this:Do you see that each line sums up to 1? Do you see that repeating the subdivision infinitely many times won't change that?

I obviously see that each line sums up to 1. I don't see where you are getting the second 'half' from each time you reduce. This is the missing half we are talking about and there it is each time. It seems to me this is being magicked out of thin air but I understand I don't know enough to comprehend why this isn't so.

 

[Ibix]

I obviously see that each line sums up to 1. I don't see where you are getting the second 'half' from each time you reduce. This is the missing half we are talking about and there it is each time. It seems to me this is being magicked out of thin air but I understand I don't know enough to comprehend why this isn't so.

See the first two paragraphs of my last post.

 

[ShaunM]

He's generating each...

Thanks for this reply.

I do know about atoms and I understand fully what you are saying about the ruler, no matter how small its cut up, is still the length it always was. What I don't think it will ever be however is no ruler. No matter how many times its cup up you would never reach 0.

"sorry, you're wrong"

I really like this answer and I mean that sincerely:) I assume I am wrong I am just trying to understand why.

 

[A.T.]

I understand fully what you are saying about the ruler, no matter how small its cut up, is still the length it always was.

Then the time to travel along will also be always be the same.

What I don't think it will ever be however is no ruler. No matter how many times its cup up you would never reach 0.

The point was to reach the end in finite time, not in zero time.

 

[jbriggs444]

In practice, with a real ruler, it will of course end when you get to the atomic scale.

And before. As a carpenter you learn not to ignore the width of the kerf.

 

[ShaunM]

Then the time to travel along will also be always be the same.The point was to reach the end in finite time, not in zero time.

I think the way I am seeing it is that trying to reach 1 on the way 'up' is the same as trying to reach '0' on the way down. Is this not so?

 

[jbriggs444]

I think the way I am seeing it is that trying to reach 1 on the way 'up' is the same as trying to reach '0' on the way down. Is this not so?

Either way you have to pass by an uncountable infinity of points to get there. Passing a countable subset is a trivial matter.

 

[russ_watters]

Im actually quite comfortable with the thought that 1 is never reached as it seems to object isn't actually trying to get there anyway.

See my graphs in post #41 (the first two in particular): there is zero difference in the motion when described using Zeno's logic, it just presents slightly differently as an uneven spacing of points on the graph. The object gets to the goal just as surely as you have no trouble walking across a room.

What would cause the object never to reach the goal is if each step took equal time. Then it would be obvious that the total time would be infinite.

 

[Herman Trivilino]

What would cause the object never to reach the goal is if each step took equal time. Then it would be obvious that the total time would be infinite.

Right, because the number of steps is infinite.

 

[sophiecentaur]

Right, because the number of steps is infinite.

There is no actual need to bring Infinity (which is yet another of those concepts that make people uneasy) into it explicitly if you just do what the introduction to differential calculus does when it describes velocity as the limit of δx/δt as δt approaches zero. The concept of Limits is not too intuitive but it is a good way into a lot of these sort of problems.
That Zeno has done us no favours over the years,

 

[PeroK]

Quite right. I am having problems however understanding how 1 is reached. I appreciate you all trying to explain.

This is largely irrelevant to Zeno's paradox. Either it never reaches 1, in which case not even 1s can pass; or, in some sense it eventually reaches 1 in which case the universe lasts at most 1s.

Mathematically the finite sums never reach 1. But, we would like to identify 1 somehow as the limit of that sequence of sums. This can be done using the rigorous mathematics called real analysis, developed in the 19th century.

These infinite sums are extremely useful in physics but don't necessarily map to physical processes.

The real issue with Zeno's paradox, IMO, is that there is no reason to decompose time into smaller and smaller increments. It serves no purpose. Also, the conclusion is inherent in the approach. By only considering smaller and smaller increments, you can never consider any phenomena that take place outside that first second.

 

[jk_er_gamma]

That Zeno has done us no favours

What about giving calculus a day to save?

 

[sophiecentaur]

, IMO, is that there is no reason to decompose time into smaller and smaller increments. It serves no purpose.

The 'paradox' only goes half way with it so it falls over. Once Calculus, with its LIMITS, came along, it dealt with that problem and many others.
Perhaps Zeno and his paradox were just an early example of a Conspiracy Theory. Find an awkward question and then blame someone / something for it. Rationality is / was never very popular.
Edit: The function is actually not differentiable at the instant of impact, if you use a simple mathematical model. There will be a 'damped harmonic' type of motion as the ball and the wall collide. The actual time that the ball stops can be well defined that way - and the velocity approaches zero in a nice well behaved fashion. But Zeno can still muck about with the time of impact.

 

[sysprog]

The formulations that I've seen of Zeno's Paradox all begin with something like: before you can traverse the stated distance, you have to traverse half of it, and then you have to traverse half of the remainder, etc.. My reaction to that is that if that's so, you have to traverse the first half of the first half before you can traverse the second half of the first half, and before you can do that, you have to traverse the first half of the first half of the first half, etc., wherefore you could never even get started. This of course contradicts the postulate that you have a velocity, i.e., that you are in fact moving, with a speed. and in a direction, and so abuses the facts that all motions over finite distances may be described as inclusive of traversal of an infinite number of infinitesimal distances, and that all durations may be described as comprising an infinite number of infinitesimal moments or instants.

 

[scottdave]

I apologize that I haven't read all three pages of responses yet @ShaunM . In what context was this question posed to you? Is it possible that it was posed as a thought experiment? Another thing to think about - did they tell you that points A and B are fixed? What happens if B is also moving? Just thought I'd throw something out there to consider.