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.