Analyzing Infinitesimal Motion

[Chenkel]

Hello everyone!

I was wondering about this physics problem.

First example:

If a rocket is traveling in a straight line continuously in uniform motions from position 0 to position 1000 in 10 seconds then it will move through an infinite number of points.

Since it is always changing position due to continuity, then for each point it must spend a very small amount of time in each position.

If there are an infinite number of points it moves through, then the time it spends at each point is a = (10 seconds) / (infinite number of positions) which I conclude is an infinitesimal.

Second example:

If a rocket is traveling in a straight line continuously in uniform motions from position 0 to position 1000 in 5 seconds then it will move through an infinite number of points.

Since it is always changing position due to continuity, then for each point it must spend a very small amount of time in each position.

If there are an infinite number of points it moves through, then the time it spends at each point is b = (5 seconds) / (infinite number of positions) which I conclude is an infinitesimal; and furthermore it must have spent less time at each point in the second example than in the first example if the number of points are the same.

Precisely the rocket must have spent 50 percent less time at each point in the second example compared to the first example, which means a/b = 2.

Is this correct logic?

Let me know what you think, thank you!

 

[phinds]

a/b when a and b are arrived at the way you did is like dividing zero by zero. When you allow ANY division by zero, you can come to any ridiculous conclusion that you like. You can, for example, readily prove that 1=2

 

[hmmm27]

If there are an infinite number of points it moves through, then the time it spends at each point is a = (10 seconds) / (infinite number of positions) which I conclude is an infinitesimal.

As the size of the infinitesimal approaches zero, the number of infinitesimals in a line segment approaches infinity.

But a point is not an infinitesimal : an infinitesimal has length ; a point does not.

The sum of the lengths of any number of points is still zero, because a point has zero length.

 

[Chenkel]

a/b when a and b are arrived at the way you did is like dividing zero by zero. When you allow ANY division by zero, you can come to any ridiculous conclusion that you like. You can, for example, readily prove that 1=2

Are you saying we can't cancel "like" infinities?

If so I might be able to see that point of view, because infinity might be considered an abstract idea, not a number per se.

But the time spent at each point for the rocket in the second example is less, this could mean it's either moving through more points or spending less time at each one with the same number of points. What is physically going on in the universe here?

Let me know, thank you!

 

[hmmm27]

It's not a matter of "infinities". If you cut a piece of string, the cut is not a piece of string.

 

[Chenkel]

It's not a matter of "infinities". If you cut a piece of string, the cut is not a piece of string.

Could you explain this idea a little more? I'm having trouble following.

 

[hmmm27]

I think I explained in my previous previous post which you may have missed seeing due to posting overlap.

 

[Chenkel]

I think I explained in my previous previous post which you may have missed seeing due to posting overlap.

I just found the post and I'm studying it now, thank you.

 

[Chenkel]

As the size of the infinitesimal approaches zero, the number of infinitesimals in a line segment approaches infinity.

I agree with this.

But a point is not an infinitesimal : an infinitesimal has length ; a point does not.

I also agree with this.

The sum of the lengths of any number of points is still zero, because a point has zero length.

But I don't think I'm trying to sum lengths of points, I agree that points don't have lengths.

I think my calculations for a and b are correct, but the part where I made a mistake is dividing infinitesimals.

So all this comes down to the question, "what is going on at the very tiny level?" Is the object moving through points quicker in the second example, in which case we can conclude b is less than a?

If b is less than a, can we have a logical way to find out how much b is less than a?

Let me know what you think, thank you!

 

[Ibix]

If b is less than a, can we have a logical way to find out how much b is less than a?

Both your a and b are zero, not infiniesimal. Trying to calculate their ratio doesn't make sense.

 

[Chenkel]

Both your a and b are zero, not infiniesimal. Trying to calculate their ratio doesn't make sense.

But if the object spends zero seconds at each position, then it will take zero seconds to get from position 0 to position 1000 which doesn't make sense.

 

[hmmm27]

So all this comes down to the question, "what is going on at the very tiny level?"

That's infinitesimals. A point is not "very tiny".

A line segment may contain an infinite amount of points, but an infinite amount of points does not make a line segment.

Since it is always changing position due to continuity, then for each point it must spend a very small amount of time in each position.

A point in time is not a duration of time ; again "very small" is not non-existent.

 

[Ibix]

But if the object spends zero seconds at each position, then it will take zero seconds to get from position 0 to position 1000 which doesn't make sense.

But if it spends any non-zero time at each of the infinite points then the total time is infinite. This is not a productive line of reasoning.

It takes an infinitesimal time to move an infinitesimal distance, and the time it takes to travel zero distance is the same at any speed: zero. To put it another way, you can ask what is the slope of a line at a point but it makes no sense to ask what is the slope of a point.

 

[Chenkel]

A line segment may contain an infinite amount of points, but an infinite amount of points does not make a line segment.

Perhaps an infinite amount of points can make a line segment depending on the distribution of points.

A point in time is not a duration of time ; again "very small" is not non-existent

I agree with that.

I'm trying to see the big picture by understanding the very small, but I'm not sure where to look.

So I will attempt to recap:

Example 1:

Rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 10 seconds.

There are an infinite number of movements, and each movement is an infinitesimal amount of distance da in an Infinitesimal amount of time dt

Example 2:

A rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 5 seconds.

There are an infinite number of movements, and each movement is an infinitesimal amount of distance db in an infinitesimal amount of time dt, it just so happens that db is more than da for the same amount of dt.

Is it safe to assume that db is twice as large as da for a given dt?

 

[hmmm27]

Zeno's Paradoxes
Wikipedia
https://www.physicsforums.com/search/7395084/?q=zeno+paradox&o=relevance
.

 

[Chenkel]

Perhaps an infinite amount of points can make a line segment depending on the distribution of points.I agree with that.

I'm trying to see the big picture by understanding the very small, but I'm not sure where to look.

So I will attempt to recap:

Example 1:

Rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 10 seconds.

There are an infinite number of movements, and each movement is an infinitesimal amount of distance da in an Infinitesimal amount of time dt

Example 2:

A rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 5 seconds.

There are an infinite number of movements, and each movement is an infinitesimal amount of distance db in an infinitesimal amount of time dt, it just so happens that db is more than da for the same amount of dt.

Is it safe to assume that db is twice as large as da for a given dt?

I just posted my recap into an AI assistant and got a cool response if anyone is interested :

"Yes, it is safe to assume that the infinitesimal distance traveled (db) in the second example is twice as large as the infinitesimal distance traveled (da) in the first example, given the same infinitesimal amount of time (dt). This is because the rocket in the second example is traveling at a faster speed and therefore covers more distance in the same amount of time.

The infinitesimal distance traveled (db) in the second example can be calculated as follows:

db = speed * dt

where speed is the speed of the rocket and dt is the infinitesimal amount of time.

If we let the speed of the rocket in the first example be v1 and the speed of the rocket in the second example be v2, we can say that v2 is twice as fast as v1, or v2 = 2 * v1.

Substituting this into the formula for db above, we get:

db = (2 * v1) * dt

Since v1 is the speed of the rocket in the first example, we can substitute it for da/dt, where da is the infinitesimal distance traveled in the first example and dt is the infinitesimal amount of time:

db = (2 * (da/dt)) * dt

This simplifies to:

db = 2 * da

Thus, we can see that db is indeed twice as large as da for a given dt."

 

[Drakkith]

I'd like to think the problem is trying to analyze continuous motion by breaking it down into discrete parts. Of course it doesn't make intuitive sense when we have to break out infinity to explain something. Infinity often indicates we're doing something wrong or analyzing something the wrong way (or at least in a sub-optimal way). Don't analyze continuous motion as an infinite series of discrete movements through points. Because that doesn't make sense. Motion isn't discrete. It is continuous.

Perhaps the reason continuous motion is often counterintuitive is because we mostly use math and numbers in our everyday life to describe discrete things. I mean, the natural numbers, the first ones you learn and the ones you use the most, are discrete. You don't have 2.5 kids, you have either 2 or 3. You don't have a 12.333... pack. You have a 12 pack. So perhaps we just aren't used to dealing with continuousness until we're already set in our discrete ways. Or perhaps our brains just aren't built to understand it well.

Or perhaps this is all a long-winded way of saying, "Shut up and do the calculations."
I dunno.

 

[phinds]

Are you saying we can't cancel "like" infinities?

Canceling "like infinities" is not what you are doing. What you are doing is dividing by zero, which will give you nonsensical results. Denying that it is so does and/or not liking that it is so does change the math. You can't divide by zero and get meaningful results.

 

[Dale]

If there are an infinite number of points it moves through, then the time it spends at each point is a = (10 seconds) / (infinite number of positions) which I conclude is an infinitesimal.

Ugh, I wish people had stopped you right here. This operation is not a valid one using real numbers. That means that before you can even ask this question you need to decide what number system you are using.

My preference is the hyperreals, but there are others like surreal numbers, or extended real numbers, etc. If you don’t have a preference then go with hyperreals, but you need to make a choice before your question can be meaningful.

 

[f95toli]

It might be worth pointing (again) that this is not a physics problem; the same "issue" would crop of in many, many other situations unrelated to physics. This is about understanding the math.

Hence, trying to "solve" this by somehow understanding how motion works is not useful. You need to learn basic calculus.
The solution to this problem (also known -as has already been pointed out above- Zeno's paradox) then follows.

 

[jbriggs444]

Since it is always changing position due to continuity, then for each point it must spend a very small amount of time in each position.

The set of times at which the rocket is exactly at a particular point is a singleton. For most reasonable measures, the amount of time for a singleton set of times is exactly zero. [For all measures, the set of singleton sets with non-zero measure will be countable].

If there are an infinite number of points it moves through, then the time it spends at each point is b = (5 seconds) / (infinite number of positions) which I conclude is an infinitesimal

Yes. Zero is an infinitesimal.

Note that although a countable sum of zeroes is zero (we have a reasonable definition for countable sums in terms of series, limits and convergence), there is no definition for an uncountable sum of zeroes.

Or, if you prefer, the definition for an uncountable sum of zeroes is called "measure theory" and does not always fit with one's intuition. In particular, an uncountable union of sets of measure zero can yield a set with non-zero measure.

 

[Chenkel]

After reading all of your replies and looking at the output to the analysis of my recap by AI, I do believe I have a better way of looking at the problem by measuring an infinite number of infinitesimal movements with each movement having an infinitesimal amount of time to make the movement, to bring it through an infinite number of states as it moves continuously along the line.

Btw on a side note, I am amazed by modern AI, is anyone else?

This analysis from both the physicists on this forum and the AI Assistant however makes me wonder about the "state" of an object, if an object is always changing because it's moving continuously, then is the object in a particular "state" for any period of time during the movement?

If the object is in each state for zero seconds through an infinite number of movements, then there is no "time period" that is countable, or measurable by a clock that one can say the object was in a particular state, so at any "present" moment is the object in a discernable state?

Can we even "experience" a moment that is zero seconds long? That might be what is required to observe a state, so perhaps we can, but I'd like to hear people's feedback on how this happens in nature.

During seconds 1 to 2 in the first recap example, the object moves through an infinite number of states, of those states we can empirically see it moves through the state position 100 to the state position = 200, it also goes through states 101, 102, and so on; and also all the fractional "real numbers" in between, but what is going on with "state", any ideas?

For example, as the object moves from position 100 to 200, you can never tell your friend "the object is definitely in the state of having position 150" you can only realistically say "the object is approaching state of position 150" and then you can say "the object is leaving state of position 150" but you and your friend can logically determine that you must have observed an object for zero seconds in state of the object being in exactly position 150, or else there would be a discontinuity in reality, which might mean that there is no time required to make a scientific observation.

How does observation of a zero time event work?

 

[Mark44]

Rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 10 seconds.
There are an infinite number of movements, and each movement is an infinitesimal amount of distance da in an Infinitesimal amount of time dt

Zeno's Paradoxes

This

 

[Dale]

Zero is an infinitesimal.

Using which number system? It is not infinitesimal in the real numbers because there are no infinitesimal real numbers.

I recommend the hyperreals where indeed 0 is infinitesimal, but I think it is worth being clear here about what numbers you are using in that statement.

 

[Dale]

by measuring an infinite number of infinitesimal movements with each movement having an infinitesimal amount of time to make the movement, to bring it through an infinite number of states as it moves continuously along the line.
...
If the object is in each state for zero seconds through an infinite number of movements,
...
the object moves through an infinite number of states
...
you must have observed an object for zero seconds in state of the object being in exactly position 150, or else there would be a discontinuity in reality

You still need to choose some number system other than the real numbers for these questions to be well formed. Which number system do you want to use?

Please do not ignore this request again.

 

[Chenkel]

You still need to choose some number system other than the real numbers for these questions to be well formed. Which number system do you want to use?

Please do not ignore this request again.

I'm an amateur, could you please explain what I need to do as if I am ignorant? (Which I am btw, just not willingly.)

 

[Dale]

I'm an amateur, could you please explain what I need to do as if I am ignorant? (Which I am btw, just not willingly.)

Then I think that you need to hold off on your questions. If you only know the real numbers then there is no infinite number so a statement of the form "a = (10 seconds) / (infinite number of positions)" is undefined. It literally has no meaning.

Asking a meaningless question is not useful. You need to either change your question into a form that has meaning in the real numbers (i.e. by rephrasing it as a limit) or you need to learn a new number system where your question has meaning.

I personally recommend learning a new number system, specifically the hyperreals. Only then your question can be asked and answered meaningfully.

Here is a place to get started on hyperreals:
https://en.wikipedia.org/wiki/Hyperreal_number

Here is an overview of other number systems that you could use if you decide you don't like hyperreals:
https://en.wikipedia.org/wiki/Infinitesimal#Number_systems_that_include_infinitesimals

 

[Chenkel]

Then I think that you need to hold off on your questions. If you only know the real numbers then there is no infinite number so a statement of the form "a = (10 seconds) / (infinite number of positions)" is undefined. It literally has no meaning.

I agree that idea was not well formulated, that's why I created a "recap" with a more correct way to conceptualize the motion of the rocket with additional examples, perhaps you were unable to see it because a forum bug?

The example recap, although certainly lacking in analytical rigor because my lack of knowledge of hyperreals, is something that I'm absolutely willing to look at for flaws, but this last post I made is about a work in progress (which is my understanding), I'm not omniscient and I never claimed to be, therefore I just wanted to show my new way of looking at the problem in the recap to see if at the very least, it may have proved in the to be less flawed than the examples given in the initial post.

On a side note I will check out hyperreals, I've heard about them for a long time, but they never made too much sense, perhaps now that I am a little more mature in my understanding of math I might be able to answer certain questions I have about them (hyperreals) by researching them.

Additionally tell me exactly what I am to not use in my examples until I understand it, I heard the word hyperreals, is there anything else?

Please summarize what I have to do so this thread doesn't wind up in the black hole.

 

[Mark44]

Example 1:
Rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 10 seconds.

Example 2:
A rocket moves continuously in uniform motion on a straight line from 0 to 1000 in 5 seconds.

You don't need to be a rocket scientist to figure out that the rocket in the first example is moving at a rate of ##\frac{1000}{10} = 100## units per second, and that the one in the second example is moving at a rate of ##\frac{1000}{5} = 200## units per second.

As mentioned a couple of times already in this thread, your questions are little more than paraphrases of Zeno's paradox, in which Achilles fires an arrow through the air, and it paradoxically seems unable to move. The thesis there is more-or-less that at any instant (i.e., a point in time) the arrow isn't moving, and since it isn't moving, it can't traverse any finite distance. About 2000 years later, after the advent of calculus, we have definitions for the notions of average velocity and instantaneous velocity.
Average velocity is defined as the quotient ##\frac{s(t + \Delta t) - s(t)}{\Delta t}##. IOW, as the distance traveled during some time interval ##\Delta t##, divided by the length of that time interval. Instantaneous velocity is defined as the limit of this difference quotient as the time interval decreases to zero. Notice, though, that one does not simple substitute 0 for ##\Delta t##, because that would result in the indeterminate form ##[\frac 0 0]##.

 

[russ_watters]

Or perhaps this is all a long-winded way of saying, "Shut up and do the calculations."

Yes, and in this case the calculations are the normal souvat equations of motion. The OP's calculations messing with infinities and infinitesimals are pointless. It's trying to use a screwdriver to turn a hex nut. Just the wrong tool for the job.

 

[Mark44]

A line segment may contain an infinite amount of points, but an infinite amount of points does not make a line segment.

Perhaps an infinite amount of points can make a line segment depending on the distribution of points.

It gets even worse. The segment [0, 1] contains both an infinite number of points with rational coefficients (such as 1/2 and 2/3) and points with irrational coefficients (such as ##\frac{\sqrt 2} 2## and ##\frac \pi 4##). In a field of mathematics called measure theory, the measure of the rationals in [0, 1] is 0, while the measure of the irrationals in the same interval is 1.

 

[Chenkel]

Yes, and in this case the calculations are the normal souvat equations of motion. The OP's calculations messing with infinities and infinitesimals are pointless. It's trying to use a screwdriver to turn a hex nut. Just the wrong tool for the job.

I appreciate any feedback, but I need to be honest here, I'm a fool just like everyone in the universe, but some fools are wiser than others.

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I really don't want to create animosity because it's unproductive to an intellectual conversation, but I am not the one who started with this language and additionally I see no value in silencing anyone, so just to ask for clarification, who should shut up exactly and why?

 

[russ_watters]

I appreciate any feedback, but I need to be honest here, I'm a fool just like everyone in the universe, but some fools are wiser than others.

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I really don't want to create animosity because it's unproductive to an intellectual conversation, but I am not the one who started with this language and additionally I see no value in silencing anyone, so just to ask for clarification, who should shut up exactly and why?

The context is that it's more of a slang saying - a catchphrase - than a order. It just means you should follow the math where it leads and not get bogged down in non-mathematical logic or philosophical questions about if/how the math relates to reality. Just do what works. Yes, the wording is aggressive, but it really isn't meant that way.
https://physicstoday.scitation.org/doi/10.1063/1.1768652?journalCode=pto#references-1

That being said, you seem pretty determined to stay on the course you are on, despite being repeatedly told it is the wrong one. So maybe aggressive correction is warranted?

I doubt even Zeno himself was as troubled by this issue 2000 years ago. I bet he recognized he was using the wrong math.

 

[Dale]

I created a "recap" with a more correct way to conceptualize the motion of the rocket with additional examples, perhaps you were unable to see it because a forum bug?

You mean post 22? I saw and responded to you about it. It suffers from the same problem as the OP. It is no more or less correct than the OP. You simply need to either use different numbers than the reals (if so which?) or express your questions in terms of limits.

Additionally tell me exactly what I am to not use in my examples until I understand it, I heard the word hyperreals, is there anything else?

You cannot treat infinity as a real number. Specifically, you cannot divide by an infinite number because there is no such thing as an infinite real number. Instead you can divide by a real number, ##n##, and take the limit of that division as ##n## increases without bound. This is written as $$\lim_{n\rightarrow \infty} \frac{1}{n}$$ but it is important to recognize that in this notation ##\infty## is not a real number, but shorthand notation for "##n## increases without bound".

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I am not telling you that. I am telling you to slow down (a lot) and think seriously about the question you are asking. It is not meaningful in the way that you asked it, so I am encouraging you to think about how you should ask it instead to produce a meaningful question. Either you need to change the form of the question itself of you need to choose an alternative representation where that form of question is meaningful.

You are trying to jump to answers before you have a meaningful question. Slow down.

 

[jbriggs444]

Using which number system? It is not infinitesimal in the real numbers because there are no infinitesimal real numbers.

In the reals. Zero is an infinitesimal in the real numbes. The only infinitesimal in the standard reals.

Edit: It has been forever since I was taught that. But I see that the definition on Wikipedia disagrees and requires that "infinitesimals" be distinct from zero.