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

[ShaunM]

Hello

Ive have a problem with this simple question and hope someone (probably a lot of you) can help.

An object is traveling in a straight line between point A and point B. It always only goes half the remaining distance and so it will always be moving forward towards B but would never reach it. It would in effect be slowing down. If however its speed is increased with every half length traveled so the speed at which it is actually traveling is constant then it would eventually reach the speed of light, its overall speed however wouldn't be and it still would never reach point B.

Can someone help me see what I am missing here please.

 

[PeroK]

Hello

Ive have a problem with this simple question and hope someone (probably a lot of you) can help.

An object is traveling in a straight line between point A and point B. It always only goes half the remaining distance and so it will always be moving forward towards B but would never reach it. It would in effect be slowing down. If however its speed is increased with every half length traveled so the speed at which it is actually traveling is constant then it would eventually reach the speed of light, its overall speed however wouldn't be and it still would never reach point B.

Can someone help me see what I am missing here please.

This is Zeno's paradox. You and Zeno are missing the passage of time. While you are endlessly decomposing the motion, time has passed and the object has long since reached B.

 

[Ibix]

It always only goes half the remaining distance and so it will always be moving forward towards B but would never reach it. It would in effect be slowing down.

If however its speed is increased with every half length traveled so the speed at which it is actually traveling is constant

Decide whether it's speeding up, slowing down, or staying at constant speed. You can't hope to solve a problem when your self-contradiction means that you are describing two or three different problems as if they were one.

 

[ShaunM]

Decide whether it's speeding up, slowing down, or staying at constant speed. You can't hope to solve a problem when your self-contradiction means that you are describing two or three different problems as if they were one.

Thanks for the reply Ibix.

I can't however see how I can 'decide' what's going on (or can I?) and that's precisely the problem I have. What is the object actually doing? Speeding up, slowing down, is its speed constant and why doesn't it seem to reach point B?

 

[PeroK]

Thanks for the reply Ibix.

I can't however see how I can 'decide' what's going on (or can I?) and that's precisely the problem I have. What is the object actually doing? Speeding up, slowing down, is its speed constant and why doesn't it seem to reach point B?

If you stand up and walk steadily across the room, you get to the other side. Yes?

What's the issue?

 

[ShaunM]

This is Zeno's paradox. You and Zeno are missing the passage of time. While you are endlessly decomposing the motion, time has passed and the object has long since reached B.

Thanks PeroK

Thats really interesting and having just read a few things quickly it does seem to be exactly that.

The question is then actually whether or not our reality (Im not sure if that's the correct term, maybe 'space time'?) is discrete or not?

 

[ShaunM]

If you stand up and walk steadily across the room, you get to the other side. Yes?

What's the issue?

Im not sure and that's bothering me, hence my post:)

 

[PeroK]

Zeno's paradox won't help you with that. Going back to ancient wisdom is no use to modern physics.

The way to decide whether spacetime is discrete is a testable theory of quantum gravity.

You can't decide by pure thought inside a Greek ivory tower.

 

[PeroK]

Im not sure and that's bothering me, hence my post:)

Physics is an empirical science. If you're not sure motion is possible then you are not in any position to study physics.

The starting point for physics is that we observe motion and need a theory to explain it. Newton and Leibnitz invented calculus for this. Zeno didn't.

You're not a philosopher, I hope?

 

[ShaunM]

Physics is an empirical science. If you're not sure motion is possible then you are not in any position to study physics.

The starting point for physics is that we observe motion and need a theory to explain it. Newton and Leibnitz invented calculus for this. Zeno didn't.

You're not a philosopher, I hope?

I'm me and I'm here with a question. What I'm not sure about is whether you are giving me answer or simply telling me my question is stupid.

Thanks either way for your time.

 

[PeroK]

I'm me and I'm here with a question. What I'm not sure about is whether you are giving me answer or simply telling me my question is stupid.

Thanks either way for your time.

The answer is that motion is possible. I do consider Zeno's paradox to be particularly unparadoxical. I never understood the point of it.

An object traveling at $5m/s$ travels $5m$ in $1s$. The only way to deny that is to claim that $1s$ never passes. Which is more or less what Zeno did.

 

[ShaunM]

The answer is that motion is possible. I do consider Zeno's paradox to be particularly unparadoxical. I never understood the point of it.

An object traveling at $5m/s$ travels $5m$ in $1s$. The only way to deny that is to claim that $1s$ never passes. Which is more or less what Zeno did.

Thanks. So Zeno's paradox suggests time doesn't exist or stops in some cases?

 

[A.T.]

I'm me and I'm here with a question.

Have you done a search on Zeno's paradox?

 

[PeroK]

Thanks. So Zeno's paradox suggests time doesn't exist or stops in some cases?

No. Zeno's paradox is completely and utterly wrong. You can read up about it online.

Certainly in terms of classical physics, which is where you posted your question, it is wrong.

Time passes, things move, particles collide. That's classical physics.

 

[ShaunM]

Have you done a search on Zeno's paradox?

I had never heard of it until about 2 hours ago so any searches I have done have been very superficial. I had a problem trying to work out what was happening in the situation contained in my original post and thought this would be a place where someone could help:)

 

[PeroK]

I had never heard of it until about 2 hours ago so any searches I have done have been very superficial. I had a problem trying to work out what was happening in the situation contained in my original post and thought this would be a place where someone could help:)

What's not clear in post #11? Object moving at constant velocity of $5m/s$.

It travels $5m$ in $1s$, $10m$ in $2s$ etc.

Give me any distance and I'll tell you how long it takes to cover that distance.

It doesn't have to speed up or slow down or jump about.

What's unclear about that?

 

[A.T.]

I had never heard of it until about 2 hours ago so any searches I have done have been very superficial.

Now that you know what to look for, you should find plenty, even on this forum.

 

[PeroK]

I had a look at the Wikipedia page on this and I see I've been playing the role of Diogenes the cynic.

I like his style!

 

[ShaunM]

What's not clear in post #11? Object moving at constant velocity of $5m/s$.

It travels $5m$ in $1s$, $10m$ in $2s$ etc.

Give me any distance and I'll tell you how long it takes to cover that distance.

It doesn't have to speed up or slow down or jump about.

What's unclear about that?

I don't know if in my situation there is a set distance being travelled. I assumed point B would never be reached.

 

[PeroK]

I don't know if in my situation there is a set distance being travelled. I assumed point B would never be reached.

Then you are assuming the conclusion.

There are lots of ways not to reach B. What are you assuming about the motion?

 

[ShaunM]

Then you are assuming the conclusion.

There are lots of ways not to reach B. What are you assuming about the motion?

Im assuming that if an object only ever travels half the way between its current position and another point in space in which direction it is traveling that it doesn't matter how quickly it travels it would never reach said point in space. I am starting to think that I need to have a think about what 'travelling' means maybe. Is the halfway point the actual destination each time for instance.

 

[A.T.]

Is the halfway point the actual destination

Why call it halfway point then?

 

[phinds]

I'm me and I'm here with a question. What I'm not sure about is whether you are giving me answer or simply telling me my question is stupid.

No, the point is to help you learn to think. Rather than spoon-feed answers, we try to help people see how to work them out for themselves. That's what people in this thread have been doing.

EDIT: also, just FYI, it's "halving", not "halfing"

 

[jbriggs444]

Im assuming that if an object only ever travels half the way between its current position and another point in space in which direction it is traveling that it doesn't matter how quickly it travels it would never reach said point in space. I am starting to think that I need to have a think about what 'travelling' means maybe. Is the halfway point the actual destination each time for instance.

Zeno's paradox begins by assuming our commonly accepted notion of continuous distance. In particular, that given any two distinct points on a line, there is at least one point between them. It uses this to demonstrate that there must be at least one unending sequence of points within any non-empty line segment, each point being farther along than the last.

If the traversal of a point is thought of as an "action", one is invited to believe that we can only perform a finite number of "actions" in any finite extent of time. But I've never seen any logic extended to demonstrate that premise. It seems obviously false.

Alternately, one might be invited to believe that for any ordered set of actions, there must be a "last action" in the set. However, this is false.

 

[ShaunM]

No, the point is to help you learn to think. Rather than spoon-feed answers, we try to help people see how to work them out for themselves. That's what people in this thread have been doing.

EDIT: also, just FYI, it's "halving", not "halfing"

Thanks and yes, I see my thinking was flawed, as was my spelling. I thought that word looked odd.

 

[ShaunM]

Why call it halfway point then?

Indeed.

 

[Herman Trivilino]

Im assuming that if an object only ever travels half the way between its current position and another point in space in which direction it is traveling that it doesn't matter how quickly it travels it would never reach said point in space.

As you approach the destination you will eventually get close enough that, as the Heisenberg Uncertainty Principle describes, you will not be able to state both the speed and the position with enough precision to be able to continue halving the distance.

Zeno had several paradoxes, all intended to refute the atomist philosophy. Some are resolved using calculus, but the resolution of this particular one requires quantum mechanics.

 

[A.T.]

Thanks and yes, I see my thinking was flawed

It might help you to learn about infinite series. Mainly that the sum of infinitely many non-zero positive numbers can still be finite. For example:

1/2 + 1/4 + 1/8 + ... = 1

If you move 1m at 1m/s you have these steps:

1/2m + 1/4m + 1/8m + ... = 1m

Which take these times:

1/2s + 1/4s + 1/8s + ... = 1s

So the total time is still 1s.

 

[phinds]

As you approach the destination you will eventually get close enough that, as the Heisenberg Uncertainty Principle describes, you will not be able to state both the speed and the position with enough precision to be able to continue halving the distance.

Zeno had several paradoxes, all intended to refute the atomist philosophy. Some are resolved using calculus, but the resolution of this particular one requires quantum mechanics.

You are seriously overthinking the problem. It does NOT require QM.

 

[Dale]

It might help you to learn about infinite series. Mainly that the sum of infinitely many non-zero positive numbers can still be finite. For example:

1/2 + 1/4 + 1/8 + ... = 1

If you move 1m at 1m/s you have these steps:

1/2m + 1/4m + 1/8m + ... = 1m

Which take these times:

1/2s + 1/4s + 1/8s + ... = 1s

So the total time is still 1s.

@ShaunM I would strongly recommend going over this post multiple times until you understand it completely. This post completely resolves Zeno’s paradox, and it does so on Zeno’s own terms without avoiding the issue raised by Zeno.

 

[jbriggs444]

@ShaunM I would strongly recommend going over this post multiple times until you understand it completely. This post completely resolves Zeno’s paradox, and it does so on Zeno’s own terms without avoiding the issue raised by Zeno.

If you look for it, there is a quibble that can be raised. Although we talk about the "sum" of an infinite series as if it were a sum and treat it [correctly, as long as it is absolutely convergent] as a sum, the "sum" of an infinite series is technically a limit rather than a sum. It is the limit of the sequence of partial sums.

The naive picture of computing a sum by adding to a running total and reporting the result after the last term has been added in does not work for infinite series because there is no last term. So instead, one reports the value that is approached, if any.

 

[russ_watters]

I can't however see how I can 'decide' what's going on (or can I?) and that's precisely the problem I have. What is the object actually doing? Speeding up, slowing down, is its speed constant and why doesn't it seem to reach point B?

It's *your* scenario. You get to decide how to make it. But hopefully if you try applying math/numbers to it you will recognize it is self-contradictory. Specifically if your time steps are equal in length your speed is dropping, not increasing. 1m/s, 1/2m/s, 1/4m/s, etc.

 

[Dale]

If you look for it, there is a quibble that can be raised. Although we talk about the "sum" of an infinite series as if it were a sum and treat it [correctly, as long as it is absolutely convergent] as a sum, the "sum" of an infinite series is technically a limit rather than a sum. It is the limit of the sequence of partial sums.

The naive picture of computing a sum by adding to a running total and reporting the result after the last term has been added in does not work for infinite series because there is no last term. So instead, one reports the value that is approached, if any.

I wouldn't call that a quibble. That is part of Zeno's problem setup. He computes this limit (incorrectly) and claims that it is not finite. You cannot answer the problem on Zeno's terms without computing the same limit (correctly) and showing that it is finite.

 

[ShaunM]

It might help you to learn about infinite series. Mainly that the sum of infinitely many non-zero positive numbers can still be finite. For example:

1/2 + 1/4 + 1/8 + ... = 1

If you move 1m at 1m/s you have these steps:

1/2m + 1/4m + 1/8m + ... = 1m

Which take these times:

1/2s + 1/4s + 1/8s + ... = 1s

So the total time is still 1s.

Many thanks. That is cooincidentally exactly what is contained in this video I have just found:



I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

Im enjoying this very much however and I am glad to have stumbled across such an old idea and you lot.

 

[PeroK]

Many thanks. That is cooincidentally exactly what is contained in this video I have just found:



I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

Im enjoying this very much however and I am glad to have stumbled across such an old idea and you lot.

The mathematics of infinite sums can be put on a rigorous basis, but that is not necessary.

Zeno adds $1/2 + 1/4 +1/8 \dots$. Now,

1) This sum might eventually reach any number. In fact the series $1/2 + 1/3 +1/4 \dots$ does just that.

In which case, time passes and things move.

2) this sum might never reach $1$. In which case it is a poor model for the passage of time, which experimentally would appear to tick along.

3) this sum might have no meaning, in which case again it is a poor model.

Fundamentally, case 2 is the critical one. What Zeno is really saying is: here is a model for the passage of time. Time moves in increasingly small steps, in some sense, and can never reach $1s$.

The refutation is simply to let $1s$ pass by whatever measure of time you adopt.

It's a simple mathematical fact that the finite sums above never reaches the value $1$. But, that makes it a poor model for the passage of time. And, it certainly doesn't compel time to behave according to Zeno's model.

 

[A.T.]

I still don't understand why that makes 1m

You start with 1m. You subdivide it into more and more, smaller and smaller pieces. But you never add to or abstract from it, so it remains 1m:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

The sum doesn't change in those steps, so no matter how often you repeat it, it is still 1.
This way to write it also avoids the issue mentioned by @jbriggs444.

 

[russ_watters]

I think this point is critical:

It's a simple mathematical fact that the finite sums above never reaches the value $1$. But, that makes it a poor model for the passage of time. And, it certainly doesn't compel time to behave according to Zeno's model.

Zeno did the math wrong, but even if you do the math right, it is still a poor way to describe motion. It's the model that's wrong, not reality.

Specifically:
If you graph time over displacement of a constant walking speed with a zero in the middle instead of at the end, you get a simple sloped line.

if you graph the number of samples over distance you get a hyperbole with a discontinuity where the model breaks.

 

[jbriggs444]

I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

The sequence is bounded. [There is a single fixed value such that every sequence element is less than or equal to that value]

The sequence has no last term. [For every element in the sequence, there is another element with a larger value]

The two are not contradictory.

 

[Herman Trivilino]

This post completely resolves Zeno’s paradox, and it does so on Zeno’s own terms without avoiding the issue raised by Zeno

Not this particular one. You still have to add an infinite number of terms to get the final finite sum. Thus requiring an infinite number of steps to get there.

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

 

[russ_watters]

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

The problem gets worse if you don't intend to stop walking when you cross the goal line. The function is discontinuous at 0.

 

[russ_watters]

Zeno's model vs the normal model, in graphical form:

I think the problem gets clearer if you arbitrarily move the "0" displacement.

 

[PeroK]

Not this particular one. You still have to add an infinite number of terms to get the final finite sum. Thus requiring an infinite number of steps to get there.

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

I don't remember saying that!

It was actually @Dale who said that.

You need to take a course in real analysis.

The HUP is not relevant to Zeno's argument. QM shows that the naive analysis of motion, ultimately for small enough particles, doesn't apply. It's actually the expected values of measurements that behave classically.

 

[ShaunM]

You start with 1m. You subdivide it into more and more, smaller and smaller pieces. But you never add to or abstract from it, so it remains 1m:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

The sum doesn't change in those steps, so no matter how often you repeat it, it is still 1.
This way to write it also avoids the issue mentioned by @jbriggs444.

Thanks. To my (admittedly untrained) eye that's hard to understand. Working backwards from 1m to 0 in this way seems to to me to lead to the exact same problem. Theres always one discrete part of 1 missing.

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.

 

[A.T.]

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

Theres always one discrete part of 1 missing.

How much is missing in which line?

Im actually quite comfortable with the thought that 1 is never reached

You shouldn't be, because it's nonsense.

 

[PeroK]

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.

Given this is a physics forum and not a new age philosophy forum, I'm going to quote Richard Feynman:

If your theory doesn't agree with experiment, then it doesn't matter what your name is or how clever you are, it's wrong.

 

[ShaunM]

Given this is a physics forum and not a new age philosophy forum, I'm going to quote Richard Feynman:

If your theory doesn't agree with experiment, then it doesn't matter what your name is or how clever you are, it's wrong.

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

 

[ShaunM]

How much is missing in which line?

You shouldn't be, because it's nonsense.

The lines don't seem to me to be what we are talking about, sorry if I am not getting something.

Ive never studied math or physics outside of school but I do think about these things and some things I stumble across bother and interest me. To quote Manuel however, 'I know nothing'.

I know its nonsense simply from my own experience but 1/2 + 1/4 + 1/8... doesn't seem to me to ever reach 1 either.

 

[ShaunM]

It seems my understanding of math is not sufficient to allow me to understand how 1 is ever reached but thanks anyway to all of you, I really appreciate your time. Its enough for me to know that people with more knowledge can assure me that 1 is reached. The fact that this conclusion is also visible in my real experiences is a bonus:)

 

[A.T.]

The lines don't seem to me to be what we are talking about, sorry if I am not getting something.

I'm talking about this:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

Do you see that each line sums up to 1? Do you see that repeating the subdivision infinitely many times won't change that?

 

[Ibix]

Thanks. To my (admittedly untrained) eye that's hard to understand.

He's generating each line by copying the line above but with the last term replaced by two halves of it. The sum is one, and it never stops being one no matter how many times you split the last term.

Think of actually doing it - start with a 1m ruler and cut it in half. The total length is still 1m. Cut one half in half (two quarter-meters). The total length is still 1m. Cut one of the quarter meters into two eighth meters. The total length is still 1m. That's all @A.T. is doing, just stated algebraically. The point is that no step in this process ever changes the length of the ruler - the total length is always 1m. That's a (rather boring) pattern that won't change no matter how many times you keep subdividing the last piece.

In practice, with a real ruler, it will of course end when you get to the atomic scale. But you only know this because we've done experiments and we know that matter is made of atoms - extra knowledge you bring to the problem. If you don't know about atoms, you can propose just keeping dividing the ruler into smaller and smaller segments forever, and the total length stays 1m.

So the only problem with dividing a ruler into infinitely many pieces is that we know a ruler is actually made of a finite number of pieces. We don't know that about space itself, though, so there might be absolutely no problem about thinking about subdividing it infinitely many times.

The other problem you have, I think, is that you are thinking that you have divided the task of moving from A to B into infinitely many subtasks, so it can't be completed. But you've forgotten to think about how much time each subtask takes. Assuming that the object is moving at constant speed (which is why I asked a couple of pages ago), the amount of time each step takes is half the previous one. Just like with the ruler, simply dividing the amount of time up infinitely many times isn't a problem in principle. It may turn out to be one in practice, but we have no evidence of that, and your argument doesn't provide one.

The fundamental problem is that you are trying to impose discrete thinking onto an example that can be well explained by continuous variables. If time and space are continuous (which is a reasonable assumption as far as we are aware) then you can continue subdividing them infinitely, and the total distance and time remain unchanged. If your objection to that is "there must be a first step in moving" then the only answer we can give you is "sorry, you're wrong" and ask you exactly when you think the "keep on subdividing" will suddenly stop adding to one, or when a further subdivision is impossible - and you cannot answer that. It feels strange, I agree, but it's perfectly self-consistent.