The problem of one tube and two balls on a plane

[crazy lee]

Homework Statement

There is a long tube on a plane, inside which there are two small balls moving freely. The width of the tube is less than twice the diameter of a small ball and greater than the diameter of a small ball. All collisions are ideal collisions. The mass of the tube is infinite, and gravity and friction are not considered. Now, the questions are as follows:

1. After a sufficient number of collisions, can the positions and velocities of the two small balls return to the initial state simultaneously? Or can only the positions return to the initial state?

2. Does the velocity change of a single small ball have a certain periodicity? Does the distribution of the velocity of a single small ball in the two directions of the length and width of the tube have a certain pattern? And does this pattern have anything to do with the length and width of the tube?

3. Does the spatial distribution of the collision points have a certain pattern?

My English is not good. The questions were translated using an AI. Thank you all!

Relevant Equations

None

none

 

[.Scott]

I think there are two things missing here.
First, you need to take a crack at solving this yourself before expecting any help.
Second, you never explain why it is significant that the tube (of infinite mass) is on a plane.
I am guessing that this whole thing will go from the runway, fly through the air, and then return to a runway.

 

[kuruman]

What collisions? Why don’t the balls just sit at rest and not move? If they are moving, how do they start? If they start in opposite directions, they will never collide.

This question is not well-written.

 

[.Scott]

If they start in opposite directions, they will never collide.

I believe we are to presume that the tube is finite in length, closed at both ends, and provide the same "ideal collisions" that govern the balls. So once a ball is in motion, it will bounce back and forth along the length of the tube.
I am also guessing that the tube is oriented along the roll axis of the plane. And that the acceleration and deceleration of the plane is what drives the motion of the balls relative to the tube.

 

[kuruman]

believe we are to presume that the tube is finite in length, closed at both ends, and provide the same "ideal collisions" that govern the balls.

OK, I thought that the tube could be infinite.

I am also guessing that the tube is oriented along the roll axis of the plane.

You mean plane as in aero-plane? I think it is a plane of the xy-plane variety. We are to assume that the balls are given arbitrary initial velocities and then the two-ball system is allowed to evolve in time.

To @crazy lee : According to our rules, you must show some effort towards answering your questions before you can receive help.

 

[berkeman]

Homework Statement: There is a long tube on a plane

On an airplane, or the tube is just sitting on a flat plane?

 

[haruspex]

On an airplane, or the tube is just sitting on a flat plane?

Since the mass of the tube is infinite, it is surely a plane, not a 'plane. Besides, none of the questions make sense if the tube can move around arbitrarily. But since gravity is to be ignored, the plane is not relevant. Maybe it is just to indicate it is stationary.

The question surely should have mentioned the ends of the tube. Presumably it is closed with flat ends normal to the axis of the tube. (Which makes q1 easy.)

 

[.Scott]

There are so many problems you can make from the the OPs description! If it a geometric plane, does that suggest gravity? If it's a airplane, then clearly the tube needs to fit on the plane. The balls are apparently the same diameter - because there is no distinction between them when the OP uses as a measuring unit. But I would say that we get to decide whether they are the same mass - consider the situation when one is pi times heavier than the other.
Clearly the tube is of infinite mass to prevent collision with the balls from having any effect of the tube. But does the same deity that created this infinite mass also allow it to accelerate or roll? The collisions are ideal and therefore elastic. But we are still allowed to consider rotational energy - and since the balls are loose in the tube, we have even more degrees of freedom to play with.
There may be more recreation for us in describing the variety of problems that the OP may have been assigned than in solving any one of them.
Of course, the OP could return from his daily classes any time now - and then we'll need to focus on just his/her problem.

Actually, that tube of infinite mass is very interesting. If we presume that the mass density of the material is evenly distributed (hard to manage with infinite masses), then I don't believe tube mass will drive the balls towards the tube walls. But the end stops are a different matter. If a ball starts closer to one end stop than the other (which, of course, it will), it will accelerate towards that closer stop, presumably cross an event horizon (QM, notwithstanding), and then suffer an inelastic non-collision - or something.

And these inelastic balls, how much do they compress during a collision?

 

[kuruman]

Clearly the tube is of infinite mass to prevent collision with the balls from having any effect of the tube.

My take is that the tube has infinite mass so that one has to worry only about what the center of mass of the two balls is doing.

 

[phinds]

The question surely should have mentioned the ends of the tube. Presumably it is closed with flat ends normal to the axis of the tube. (Which makes q1 easy.)

If the tube is infinite in extent, what difference does it make whether or not there are closures on the ends? The balls will never get there.

 

[haruspex]

If the tube is infinite in extent,

It says long, not infinite.

 

[phinds]

It says long, not infinite.

I assumed that infinite mass would require infinite length, and I still think that's a good assumption since it's either that or infinite thickness, take your pick.

 

[haruspex]

I assumed that infinite mass would require infinite length, and I still think that's a good assumption since it's either that or infinite thickness, take your pick.

1. After a sufficient number of collisions, …

Multiple collisions if the tube is infinitely long and less than two diameters wide?
Maybe infinite density?
Whatever, it clearly is just a way of saying the tube is immobile.

 

[phinds]

Whatever, it clearly is just a way of saying the tube is immobile.

Then he should say that, not bring in an infinity.

 

[jbriggs444]

Let us consider a simpler and illustrative scenario.

Suppose that we have a rectangular area. Like an ideal air hockey table. We place a single ideal puck on this table. We launch the puck from one of the corners at a randomly chosen angle. We ask whether it will ever bounce back to its starting corner again.

For almost any launch angle we choose, the period for puck bounces between the short walls will not be a rational multiple of the period for puck bounces between the long walls. So there is no time after the start at which an exact integer number of round trips will have occurred for both directions simultaneously.

 

[kuruman]

If the pucks are identical, we know that their CM at the time of puck-to-puck collision is at the point of contact. This allows mapping the region where the CM can be at any time. It is shown in blue in the figure on the right with the pucks at one of the extremal positions. The ratio of the tube width to twice the tube diameter is 7/9. To describe the motion, one can imagine the CM bouncing elastically inside the blue hexagonal area and then write some equations.

 

[haruspex]

For almost any launch angle we choose, the period for puck bounces between the short walls will not be a rational multiple of the period for puck bounces between the long walls. So there is no time after the start at which an exact integer number of round trips will have occurred for both directions simultaneously.

Right, but q1 asks whether it is possible that the initial state is repeated, not whether it necessarily is.

 

[haruspex]

View attachment 358524If the pucks are identical, we know that their CM at the time of puck-to-puck collision is at the point of contact. This allows mapping the region where the CM can be at any time. It is shown in blue in the figure on the right with the pucks at one of the extremal positions. The ratio of the tube width to twice the tube diameter is 7/9. To describe the motion, one can imagine the CM bouncing elastically inside the blue hexagonal area and then write some equations.

Nice approach, but the left and right ends of the hexagonal area should be concave curves, not straight lines, no?

 

[kuruman]

Nice approach, but the left and right ends of the hexagonal area should be concave curves, not straight lines, no?

Yes, good point. Here is the modified picture. The CM is confined within the dotted closed loop and can be represented by a point following a trajectory subject specular reflection at the boundaries.

 

[haruspex]

Let's examine each question separately:

I take your post to represent proposed actual answers to the questions, rather than mere guidance.
If so, it is somewhat premature. A principle of the homework forums is to require an attempt from the OP first. However…

1. Return to Initial State:
In an ideal scenario (no friction, gravity, or energy losses, and perfectly elastic collisions), the system is deterministic and governed by conservation laws (momentum and energy). Under these ideal conditions, it's theoretically possible for both the positions and velocities of the two small balls to simultaneously return to their exact initial state after a certain number of collisions.

It can be possible given those conditions, but is it possible in this specific case? What if one end of the tube were to have its face set obliquely to the tube axis? can we be sure it could happen then?

Length Direction: Velocities will exhibit regular periodicity as each collision with the tube ends reverses velocity direction.

But the next collision between the balls may be at a different angle from previously, changing the longitudinal velocities. Why should that ever repeat?
Likewise latitudinal.

3. Spatial Distribution of Collision Points:
Collision points will exhibit clear patterns due to geometrical constraints. Along the length of the tube, collisions occur predominantly at the ends, regularly spaced due to periodic bouncing.

The geometric constraints certainly include the envelope @kuruman drew in post #19. Regular spacing within it is far from obvious. Do you have a proof?

Along the width, collisions will alternate sides in a regular sequence because of the restricted space.

I can construct a counter example.

 

[crazy lee]

I think there are two things missing here.
First, you need to take a crack at solving this yourself before expecting any help.
Second, you never explain why it is significant that the tube (of infinite mass) is on a plane.
I am guessing that this whole thing will go from the runway, fly through the air, and then return to a runway.

First of all, I have tried to solve this problem, but I haven't been able to. Secondly, the tube is placed on a plane in order to simplify the calculation.。Simplify the three-dimensional problem into a two-dimensional one. The tube is of infinite size to avoid the tube from overturning during the collision process between the tube and the small balls, and this is also to simplify the problem.。Thank you

 

[crazy lee]

What collisions? Why don’t the balls just sit at rest and not move? If they are moving, how do they start? If they start in opposite directions, they will never collide.

This question is not well-written.

The length of the tube is finite. The small balls are in motion, and their initial states are unknown constants. All collisions are ideal collisions. Each small ball has an initial position and an initial velocity.。Thank you

 

[crazy lee]

I think there are two things missing here.
First, you need to take a crack at solving this yourself before expecting any help.
Second, you never explain why it is significant that the tube (of infinite mass) is on a plane.
I am guessing that this whole thing will go from the runway, fly through the air, and then return to a runway.

I'm talking about a flat surface, not an airplane. This might be an error in the AI translation. Sorry

 

[crazy lee]

OK, I thought that the tube could be infinite.

You mean plane as in aero-plane? I think it is a plane of the xy-plane variety. We are to assume that the balls are given arbitrary initial velocities and then the two-ball system is allowed to evolve in time.

To @crazy lee : According to our rules, you must show some effort towards answering your questions before you can receive help.

Your opinion in this reply is completely correct. In fact, my English is very poor, and I don't know how to use this website either. Actually, where I'm from, no one discusses physics, so I have to come to this website to find answers to my questions.

 

[crazy lee]

On an airplane, or the tube is just sitting on a flat plane?

Flat surface. Thank you。

 

[crazy lee]

Since the mass of the tube is infinite, it is surely a plane, not a 'plane. Besides, none of the questions make sense if the tube can move around arbitrarily. But since gravity is to be ignored, the plane is not relevant. Maybe it is just to indicate it is stationary.

The question surely should have mentioned the ends of the tube. Presumably it is closed with flat ends normal to the axis of the tube. (Which makes q1 easy.)

Your judgment on this issue is completely correct. You have a very strong logical ability.The two ends of the tube are closed. I didn't mention it because I originally thought that this didn't need to be explained at all.

 

[crazy lee]

There are so many problems you can make from the the OPs description! If it a geometric plane, does that suggest gravity? If it's a airplane, then clearly the tube needs to fit on the plane. The balls are apparently the same diameter - because there is no distinction between them when the OP uses as a measuring unit. But I would say that we get to decide whether they are the same mass - consider the situation when one is pi times heavier than the other.
Clearly the tube is of infinite mass to prevent collision with the balls from having any effect of the tube. But does the same deity that created this infinite mass also allow it to accelerate or roll? The collisions are ideal and therefore elastic. But we are still allowed to consider rotational energy - and since the balls are loose in the tube, we have even more degrees of freedom to play with.
There may be more recreation for us in describing the variety of problems that the OP may have been assigned than in solving any one of them.
Of course, the OP could return from his daily classes any time now - and then we'll need to focus on just his/her problem.

Actually, that tube of infinite mass is very interesting. If we presume that the mass density of the material is evenly distributed (hard to manage with infinite masses), then I don't believe tube mass will drive the balls towards the tube walls. But the end stops are a different matter. If a ball starts closer to one end stop than the other (which, of course, it will), it will accelerate towards that closer stop, presumably cross an event horizon (QM, notwithstanding), and then suffer an inelastic non-collision - or something.

And these inelastic balls, how much do they compress during a collision?

The two small balls are exactly the same. All the assumptions in this problem, and even the problem itself, are aimed at simplifying the issue. Even after a series of simplifications, I don't think it's easy to solve this problem. What I want to know is whether, after a series of collisions between the two small balls, there is a periodic change independent of the initial state in the changes of the position and velocity of a single ball.。The compression of the small balls is not considered either. Thank you for your participation.

 

[crazy lee]

My take is that the tube has infinite mass so that one has to worry only about what the center of mass of the two balls is doing.

Yes, that's exactly what I think.

 

[crazy lee]

I assumed that infinite mass would require infinite length, and I still think that's a good assumption since it's either that or infinite thickness, take your pick.

In fact, I made these assumptions just to simplify the problem. Actually, this is more like a math problem rather than a physics problem. The tube is of finite length. I want to know the position and velocity of the ball at any given time. In this way, I can determine whether there is a periodic variable included in its position and velocity. How does this periodic variable evolve? And what is it related to?

 

[crazy lee]

Multiple collisions if the tube is infinitely long and less than two diameters wide?
Maybe infinite density?
Whatever, it clearly is just a way of saying the tube is immobile.

You're so smart. My English is not good, especially when it comes to scientific English. So I can never accurately express what I mean at once.

 

[crazy lee]

Then he should say that, not bring in an infinity.

I said that the mass of the tube is infinite in order to prevent people from calculating the tube's rotation when considering the problem.I didn't expect that it would lead to everyone's misunderstanding. I'm sorry.

 

[crazy lee]

Let us consider a simpler and illustrative scenario.

Suppose that we have a rectangular area. Like an ideal air hockey table. We place a single ideal puck on this table. We launch the puck from one of the corners at a randomly chosen angle. We ask whether it will ever bounce back to its starting corner again.

For almost any launch angle we choose, the period for puck bounces between the short walls will not be a rational multiple of the period for puck bounces between the long walls. So there is no time after the start at which an exact integer number of round trips will have occurred for both directions simultaneously.

Your understanding of the problem is completely correct. When there is only one ball, I understand your answer. But when we need to consider the collision of two balls, is the above answer still valid?

 

[haruspex]

Your understanding of the problem is completely correct. When there is only one ball, I understand your answer. But when we need to consider the collision of two balls, is the above answer still valid?

In post #1, q1 is given as "can" the state return exactly, not "will" it.
Which is it? Showing that it can is easy.

 

[phinds]

First of all, I have tried to solve this problem

SAYING that you have tried to solve the problem is NOT trying to solve the problem. You have to show us what you have tried.

 

[crazy lee]

In post #1, q1 is given as "can" the state return exactly, not "will" it.
Which is it? Showing that it can is easy.

I'd like to ask whether the following problem is easy to solve. Given that the initial states of two small balls in a tube are unknown constants, find the specific position of the center of mass of the small balls in the tube when the two small balls collide, as well as the velocities of the small balls at that time and the specific time of the collision. Thank you. I want to ask about this specific problem first.

 

[crazy lee]

SAYING that you have tried to solve the problem is NOT trying to solve the problem. You have to show us what you have tried.

SAYING that you have tried to solve the problem is NOT trying to solve the problem. You have to show us what you have tried.

Firstly, this is not a homework problem. Secondly, I've been out of school for more than 20 years, and I've pretty much forgotten all the English and physics I learned back then. Finally, for me, just accessing this website is already a challenge. I would really appreciate it if you could understand me.The most crucial thing is that I haven't found any effective and simple train of thought.

 

[crazy lee]

I'd like to ask whether the following problem is easy to solve. Given that the initial states of two small balls in a tube are unknown constants, find the specific position of the center of mass of the small balls in the tube when the two small balls collide, as well as the velocities of the small balls at that time and the specific time of the collision. Thank you. I want to ask about this specific problem first.I communicate with everyone through AI translation, so there may be some translation errors. Thank you all for your help.

 

[crazy lee]

In fact, it's still a bit difficult to discuss physical problems with others in a language without having a good command of that language. Besides, I'm not familiar with some of the rules on this website. I have seriously thought about this problem, but I haven't come up with any simple and effective ideas. Even if I take pictures of my notes and upload them, it won't be of any help to you, and the notes are all written in Chinese. So I hope some friends can be understanding. Thank you all.

 

[.Scott]

Since the objective is to make things simple, we should add these restrictions:
1) The balls track along the floor of the tube - so they follow a strictly 1-dimensional path.
2) When the balls collide with each other, they are not only elastic, but they reverse direction without deforming at all.
3) At least one of the balls has a non-zero starting velocity.

It has already been mentioned in this thread that without end stops, there is no hope of ever returning to the original state. So ...

4) There are end stops and when the balls collide with and end stop, the collision in not only elastic, but the ball reverses direction without deforming.

Initial conditions:
L: The length of the track (inside of the tube) as measured from one collision contact point to the other.
D: The diameter of each ball.
F = L-D: The range of travel for the balls. Positions will be measured from -D/2 (collision contact point 1) to F+D/2 (collision contact point 2).
P1: The initial position of ball 1 (from 0 to F-D).
P2: The initial position of ball 2 (from D to F).
V1: The initial velocity of ball 1.
V2: The initial velocity of ball 2.

Here is a specific case:
Can the balls ever return to their initial conditions:
P1: Any (from 0 to F-D).
P2: Any (from P1+D to F).
V1: 1.
V2: pi

 

[kuruman]

I'd like to ask whether the following problem is easy to solve. Given that the initial states of two small balls in a tube are unknown constants, find the specific position of the center of mass of the small balls in the tube when the two small balls collide, as well as the velocities of the small balls at that time and the specific time of the collision. Thank you. I want to ask about this specific problem first.

The answer to your question is obvious. If the initial states of the small balls in the tube are not known it is impossible, not just difficult, to find the specific position of the center of mass when the balls collide or at any time. Suppose I tell you that at noon I start moving from some unknown position moving with unknown but constant velocity. Would you be able to find my position and velocity some specific time later if I didn't tell you where I started, how fast I am moving and in what direction?

I understand that English is not your language but you need to formulate your question in a way that makes sense. The final velocities and positions depend on the initial velocities and positions. This means that final answers at a given specific time, say 2 minutes, can be found as numbers if the initial positions and velocities are given as numbers or at least as symbols corresponding to numbers. You cannot say that the initial state is unknown and expect to find the final state at some specific time.

 

[.Scott]

The answer to your question is obvious. If the initial states of the small balls in the tube are not known it is impossible, not just difficult, to find the specific position of the center of mass when the balls collide or at any time

There is a difference between the initial conditions being unknown vs. simply not provided. Even without knowing the initial conditions (or all of the initial conditions), one can describe under what conditions the balls will return to their initial state and under what conditions they will not.

As an example, in post #40, I do not specify the initial positions of the balls, but the question (will the balls return to their initial states) can still be answered.

 

[haruspex]

Since the objective is to make things simple, we should add these restrictions:
1) The balls track along the floor of the tube - so they follow a strictly 1-dimensional path.

With the given relationship between tube width and ball width, it seems clear we need to allow two dimensions at least, and the intent may even be to allow three.

4) There are end stops and when the balls collide with and end stop, the collision in not only elastic, but the ball reverses direction without deforming.

… and the end stops are flat discs normal to the tube axis.

Two simple cases to start with would be an initial contact at the centre of the tube, equal and opposite longitudinal velocities, equal or equal and opposite lateral velocities.

 

[haruspex]

I'd like to ask whether the following problem is easy to solve. Given that the initial states of two small balls in a tube are unknown constants,

Do you mean known but arbitrary, i.e. not special values? If so…

find the specific position of the center of mass of the small balls in the tube when the two small balls collide, as well as the velocities of the small balls at that time and the specific time of the collision.

Since the system is deterministic, yes, but the equations would be messy.

 

[.Scott]

Since the system is deterministic, yes, but the equations would be messy.

No, with the limits I put on it, there is nothing messy.
Although, I will also assume no friction and no spin. Just two balls bouncing back and forth along a linear track.

 

[haruspex]

No, with the limits I put on it, there is nothing messy.

Sure, but as I posted, I feel sure it is supposed to be 2D and maybe 3.

 

[crazy lee]

With the given relationship between tube width and ball width, it seems clear we need to allow two dimensions at least, and the intent may even be to allow three.

… and the end stops are flat discs normal to the tube axis.

Two simple cases to start with would be an initial contact at the centre of the tube, equal and opposite longitudinal velocities, equal or equal and opposite lateral velocities.

Your judgment in this post is completely correct. This is exactly what I meant to express.

 

[crazy lee]

Do you mean known but arbitrary, i.e. not special values? If so…

Yes, I mean that the initial state of the small balls is known, but it's not represented by definite numbers.

 

[crazy lee]

No, with the limits I put on it, there is nothing messy.
Although, I will also assume no friction and no spin. Just two balls bouncing back and forth along a linear track.

Thank you very much for your help. This is indeed a two-dimensional problem. A friend of mine previously compared my problem to ice hockey pucks moving on an air hockey table, and that metaphor is really quite appropriate.

 

[crazy lee]

I would like to add three more questions to help everyone understand the original intention of my question. The direction of the length of the tube is set as the x-axis, and the direction of the width is set as the y-axis. The two small balls A and B in the tube are exactly the same, and the collisions are ideal collisions. Friction and spin are not taken into account. The initial state is known, but not represented by specific numbers. It is expressed by letters representing constants. The total energy of the two small balls is e. Then, after a sufficient number of collisions, the questions are as follows:

4. When the two small balls collide, record the motion state at the moment of the collision. If the component of the velocity of the small ball on the y-axis is positive, mark it as 1. Otherwise, mark it as 0. Then, for each collision, a set of two-digit binary numbers can be obtained. Does the change of this set of binary numbers follow a certain pattern? Is there any periodicity?

5. Is the average velocity of the components of the velocity of the small balls on the x-axis and y-axis exactly equal to the ratio of the length to the width of the tube?

6. If the sum of the energies of the two small balls and the ratio of the length to the width of the tube are fixed, does it mean that regardless of the initial states of the two small balls, the motion states of the two small balls will eventually show similar periodic oscillations? What rules should these oscillations follow?

In fact, I want to obtain two sequences. The first sequence records the position of the center of mass of ball a each time it collides, and the second sequence records the velocity of ball a after the collision. However, I don't know how to calculate them. Or is there any other way to get the answers to the above six questions without calculation? Thank you all very much.

 

[crazy lee]

Sure, but as I posted, I feel sure it is supposed to be 2D and maybe 3.

Your judgment is completely correct. Besides, I made supplementary explanations to the question in post 50. Thank you.