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

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[.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.