How do physicists think about Zeno's arrow?

[marplots]

I have been revisiting the pre-Socratics and have been pleasantly surprised at how 2,500-year-old ideas still have some resonance for me. In particular, Zeno's paradoxes of motion.

Here's a version that captures my query:
If you had an arrow sitting in front of you, motionless, and someone shot a second arrow past you, you could arrange it so that at some instant, the first and second arrows were in alignment and at that instant perfectly indistinguishable by observation. Yet the second arrow happily continues on toward it's target when time resumes.

So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?

Since there are many well-trained (even if not degreed) physics afficionados on this forum, I'd like to know, not necessarily what the resolution to the original "paradox" is, but how someone with a better understanding of physics (and more schooling) than me views the situation - how do you think about it?

 

[TubbaBlubba]

It has momentum. Although the quantity by its nature describes change over time, by means of the derivative, it can be described at any instant.

Obviously, performing an actual experiment to measure it requires time to pass, although you could arrange it so that the result could be read without needing the arrow to move. E.g. the kinetic energy means the arrow exerts additional graviational force, if you had a dynamometer above each arrow as you froze them, they would give (incredibly slightly) different readings.

 

[psionl0]

Zeno's paradoxes seem to revolve around trying to divide zero by zero.

 

[William Parcher]

A correct answer is identical to an absurd incorrect answer. This is because both statements are composed of letters, words and punctuation.

 

[marplots]

It has momentum. Although the quantity by its nature describes change over time, by means of the derivative, it can be described at any instant.

Obviously, performing an actual experiment to measure it requires time to pass, although you could arrange it so that the result could be read without needing the arrow to move. E.g. the kinetic energy means the arrow exerts additional graviational force, if you had a dynamometer above each arrow as you froze them, they would give (incredibly slightly) different readings.

That makes me wonder if momentum is a "thing" something has, or a description of what something does. Do you think of momentum like mass, or length, or some other intrinsic property of a material thing?

 

[marplots]

Zeno's paradoxes seem to revolve around trying to divide zero by zero.

Is it then that the material world does not decompose "properly" into mathematics? Is there a misalignment or failure somewhere?

 

[TubbaBlubba]

That makes me wonder if momentum is a "thing" something has, or a description of what something does. Do you think of momentum like mass, or length, or some other intrinsic property of a material thing?

I think of momentum as a fundamental property of an object in relation to a system - it is really completely analogous to energy in a lot of ways.

 

[GlennB]

I was also thinking of momentum, so a "momentum meter" would tell the difference, which is what TB said more clearly. But then the 'was moving' arrow now has no velocity and therefore no momentum.

So - when the moving arrow is frozen in time presumably all its properties are frozen in the state they were in right then? The flights would be slightly swept back from the wind it was experiencing at that moment?

 

[TubbaBlubba]

I was also thinking of momentum, so a "momentum meter" would tell the difference, which is what TB said more clearly. But then the 'was moving' arrow now has no velocity and therefore no momentum.

So - when the moving arrow is frozen in time presumably all its properties are frozen in the state they were in right then? The flights would be slightly swept back from the wind it was experiencing at that moment?

The "freezing in time" is non-physical, obviously, so it makes no sense to think too hard about it.

 

[Beerina]

The moving arrow would still be red or blue shifted depending on where you were. But that brings to mind what the question of "observation" means in a world with frozen time -- you could not observe via light, either.

You could probably walk around gathering up photons on your own little time-aware sensor, and detect the shift that way, I suppose.

 

[marplots]

I think of momentum as a fundamental property of an object in relation to a system - it is really completely analogous to energy in a lot of ways.

I have been thinking about it in terms of different reference frames. Frames that at that instant happen to be aligned. But then I am left without a way to distinguish which frame is which. Is this like what you mean by "system?"

 

[marplots]

The "freezing in time" is non-physical, obviously, so it makes no sense to think too hard about it.

That's always a danger - malformed questions. But do we have the same problem envisioning "frozen time" when something isn't moving relative to us? In other words, when the static arrow is observed, does it "have" momentum, even though we would say the momentum is zero relative to us?

 

[marplots]

I was also thinking of momentum, so a "momentum meter" would tell the difference, which is what TB said more clearly. But then the 'was moving' arrow now has no velocity and therefore no momentum.

So - when the moving arrow is frozen in time presumably all its properties are frozen in the state they were in right then? The flights would be slightly swept back from the wind it was experiencing at that moment?

Do you mean there would always be some internal tensions, some material difference captured by the molecules of the moving arrow? It seems to me to just shift the problem's scale, but not attack the root idea.

 

[lomiller]

I have been revisiting the pre-Socratics and have been pleasantly surprised at how 2,500-year-old ideas still have some resonance for me. In particular, Zeno's paradoxes of motion.

Here's a version that captures my query:
If you had an arrow sitting in front of you, motionless, and someone shot a second arrow past you, you could arrange it so that at some instant, the first and second arrows were in alignment and at that instant perfectly indistinguishable by observation. Yet the second arrow happily continues on toward it's target when time resumes.

So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?

Since there are many well-trained (even if not degreed) physics afficionados on this forum, I'd like to know, not necessarily what the resolution to the original "paradox" is, but how someone with a better understanding of physics (and more schooling) than me views the situation - how do you think about it?

If you measured perfectly the same thing would happen as when you accurately measure the position of an electron. That is they would stop acting like particles and begin to act like waves.

In practice there would simply be quantum fuzziness around the position and momentum of the arrows so you could never really say they were lined up perfectly.

 

[alexi_drago]

As different parts of the moving arrow are different distances from the observer there would be slight distortions along it's length, very very slight.

 

[lomiller]

It has momentum. Although the quantity by its nature describes change over time, by means of the derivative, it can be described at any instant.

You can describe momentum and any instant, but not momentum and position. The more precisely you measure one the more uncertainty there is in the other.

 

[marplots]

If you measured perfectly the same thing would happen as when you accurately measure the position of an electron. That is they would stop acting like particles and begin to act like waves.

In practice there would simply be quantum fuzziness around the position and momentum of the arrows so you could never really say they were lined up perfectly.

Does that also mean that I couldn't tell which was moving and which wasn't? Is it then correct to say that both arrows, at the time I "look" are identical?

I have to chew over the consequences of that. Very counterintuitive, because I think, instinctively, there ought to be some difference between them.

 

[TubbaBlubba]

You can describe momentum and any instant, but not momentum and position. The more precisely you measure one the more uncertainty there is in the other.

I really see no need to go beyond classical mechanics in this case.

 

[ehcks]

I really see no need to go beyond classical mechanics in this case.

If we can get electrons to quantum tunnel, we should be able to do the same with arrows. The ultimate armor piercing weapon.

 

[Mark6]

Zeno's paradoxes seem to revolve around trying to divide zero by zero.

Not sure about that, but some of them rely on the assumption that an infinite series of finite numbers cannot add up to a finite number. That is simply false, but 2500 years ago people did not know that.