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

In summary: 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.
  • #1
ShaunM
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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.
 
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  • #2
ShaunM said:
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.
 
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  • #3
ShaunM said:
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.
ShaunM said:
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.
 
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  • #4
Ibix said:
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?
 
  • #5
ShaunM said:
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?
 
  • #6
PeroK said:
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?
 
  • #7
PeroK said:
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:)
 
  • #8
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.
 
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  • #9
ShaunM said:
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?
 
  • #10
PeroK said:
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.
 
  • #11
ShaunM said:
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.
 
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  • #12
PeroK said:
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?
 
  • #13
ShaunM said:
I'm me and I'm here with a question.
Have you done a search on Zeno's paradox?
 
  • #14
ShaunM said:
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.
 
  • #15
A.T. said:
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:)
 
  • #16
ShaunM said:
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?
 
  • #17
ShaunM said:
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.
 
  • #18
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!
 
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  • #19
PeroK said:
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.
 
  • #20
ShaunM said:
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?
 
  • #21
PeroK said:
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.
 
  • #22
ShaunM said:
Is the halfway point the actual destination
Why call it halfway point then?
 
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  • #23
ShaunM said:
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"
 
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  • #24
ShaunM said:
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.
 
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  • #25
phinds said:
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.
 
  • #26
A.T. said:
Why call it halfway point then?

Indeed.
 
  • #27
ShaunM said:
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.
 
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  • #28
ShaunM said:
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.
 
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  • #29
Mister T said:
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.
 
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  • #30
A.T. said:
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.
 
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  • #31
Dale said:
@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.
 
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  • #32
ShaunM said:
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.
 
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  • #33
jbriggs444 said:
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.
 
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  • #34
A.T. said:
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.
 
  • #35
ShaunM said:
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.
 
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<h2>1. What is the concept behind "Reaching the Speed of Light: A Thought Experiment on Halving Distance Traveled"? </h2><p>The concept behind this thought experiment is to explore the implications of traveling at the speed of light, which is the fastest speed possible according to the laws of physics. The thought experiment involves halving the distance traveled at each step, which leads to interesting insights into the nature of time and space.</p><h2>2. Is it possible to reach the speed of light? </h2><p>According to the theory of relativity, it is not possible for an object with mass to reach the speed of light. As an object approaches the speed of light, its mass increases infinitely and it would require an infinite amount of energy to accelerate it further. However, this thought experiment allows us to explore the consequences of reaching the speed of light hypothetically.</p><h2>3. How does halving the distance traveled affect the time taken to reach the destination? </h2><p>As the distance is halved at each step, the time taken to reach the destination also halves. This is because at the speed of light, time dilation occurs and time slows down for the traveler. So, even though the distance is halved, the time taken to travel that distance is also halved.</p><h2>4. What is the significance of this thought experiment? </h2><p>This thought experiment allows us to understand the implications of the theory of relativity and the limitations of traveling at the speed of light. It also helps us to think about the nature of time and space and how they are interconnected.</p><h2>5. Are there any real-world applications of this thought experiment? </h2><p>While this thought experiment is purely hypothetical, it has practical applications in the fields of physics and space travel. It helps scientists to understand the limitations of the laws of physics and how they affect our understanding of the universe. It also has implications for the development of future technologies such as spacecraft that can travel at near-light speeds.</p>

1. What is the concept behind "Reaching the Speed of Light: A Thought Experiment on Halving Distance Traveled"?

The concept behind this thought experiment is to explore the implications of traveling at the speed of light, which is the fastest speed possible according to the laws of physics. The thought experiment involves halving the distance traveled at each step, which leads to interesting insights into the nature of time and space.

2. Is it possible to reach the speed of light?

According to the theory of relativity, it is not possible for an object with mass to reach the speed of light. As an object approaches the speed of light, its mass increases infinitely and it would require an infinite amount of energy to accelerate it further. However, this thought experiment allows us to explore the consequences of reaching the speed of light hypothetically.

3. How does halving the distance traveled affect the time taken to reach the destination?

As the distance is halved at each step, the time taken to reach the destination also halves. This is because at the speed of light, time dilation occurs and time slows down for the traveler. So, even though the distance is halved, the time taken to travel that distance is also halved.

4. What is the significance of this thought experiment?

This thought experiment allows us to understand the implications of the theory of relativity and the limitations of traveling at the speed of light. It also helps us to think about the nature of time and space and how they are interconnected.

5. Are there any real-world applications of this thought experiment?

While this thought experiment is purely hypothetical, it has practical applications in the fields of physics and space travel. It helps scientists to understand the limitations of the laws of physics and how they affect our understanding of the universe. It also has implications for the development of future technologies such as spacecraft that can travel at near-light speeds.

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