The problem of one tube and two balls on a plane

[crazy lee]

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.

I made supplementary explanations to the question in post 50. I would be extremely grateful if you could take a look at it.

 

[jbriggs444]

To describe the motion, one can imagine the CM bouncing elastically inside the blue hexagonal area and then write some equations.

The CM will not bounce elastically.

Total energy is conserved, of course. But that total energy can be seen to be divided into the bulk kinetic energy of the center of mass and the internal kinetic energy in the relative motion of the individual balls. The relative velocity of the balls is not constant. So the internal kinetic energy is not constant. So the center of mass energy is not constant.

 

[kuruman]

In post #1 you ask

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?

In post #50 you say

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.

What do you mean by "motion state"? Is it position and velocity or just velocity?

Also, if you consider what happens when two masses collide, you will understand that the velocity "at the moment of the collision" is not a well-defined quantity. The velocity before the collision is well-defined and so is the velocity after the collision. During the collision while the masses are in contact, the velocity of each mass is changing continuously from its "before" value to its "after" value. So what are we supposed to record, the velocity before or after the collision?

 

[haruspex]

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.

That is not possible because the sequence keeps forking according to whether a state value is more or less than a threshold. E.g. at one angle it will strike the side of the tube next, at a slightly different angle it will strike the end of the tube next.

Try to answer the first question, which I interpret to mean "is there an initial state (positions and nonzero velocities) such that it will eventually return to that state?".

Next, suppose that just after the balls have collided ball A has velocity vector $v_x\hat i+v_y\hat j$. What are the possible velocity vectors for A when next they are about to collide?

 

[kuruman]

The CM will not bounce elastically.

Total energy is conserved, of course. But that total energy can be seen to be divided into the bulk kinetic energy of the center of mass and the internal kinetic energy in the relative motion of the individual balls. The relative velocity of the balls is not constant. So the internal kinetic energy is not constant. So the center of mass energy is not constant.

I agree that the relative velocity of the balls is not constant. However, the CM cannot be said to "bounce" elastically (or otherwise) until the collision is complete. Let me explain.

Shown below is a sequence of frames illustrating the collision of the CM with the wall. The balls have equal speeds $v$ along the horizontal. Velocity arrows are omitted to reduce clutter. The color code is: Lime green → moves to the Left; Red → moves to the Right. The CTC (center-to-center) separation between balls is $d$ and the red dashed vertical line is parallel to the wall at the point of closest approach of the CM.

The onset of the collision of the two-mass system is when the left mass hits the wall at $t=0$ as shown in the first frame; it is over when both masses have reversed direction and the information that one part of the system has collided with the wall is transmitted to the other part. This is shown in the last frame. The collision duration is $\Delta t=\dfrac{d-2R}{v}.$

This is a simple 1D example with the common velocity in the CTC direction. For a tube of length $L$, the period of horizontal oscillations is $$T_{\text{hor}}=2\times\left(\Delta t +\frac{L-2d}{v}\right)=\frac{2}{v}(L-2R)$$and is independent of the separation $d$.

Similarly, it can be shown that if the CTC is horizontal, but the common velocity is vertical, the period of vertical oscillations for a tube of width $W$ is $$T_{\text{ver}}=\frac{2}{v}(W-2R)$$as the balls bounce up and down without colliding with each other.

I stop here. This is a live homework problem after all.

 

[crazy lee]

In post #1 you ask

In post #50 you say

What do you mean by "motion state"? Is it position and velocity or just velocity?

Also, if you consider what happens when two masses collide, you will understand that the velocity "at the moment of the collision" is not a well-defined quantity. The velocity before the collision is well-defined and so is the velocity after the collision. During the collision while the masses are in contact, the velocity of each mass is changing continuously from its "before" value to its "after" value. So what are we supposed to record, the velocity before or after the collision?

In Question 4, the state of motion refers to the velocity of the small balls before the collision, specifically at the instant just before the collision. If the component of the velocity along the y-axis is positive, it is marked as 1, and if it is negative, it is marked as 0. Then, the velocities of the two small balls at the instant just before each collision can be represented by a set of two-bit binary numbers. I would like to ask if there is any pattern in the change of this set of two-bit binary numbers. Thank you.

 

[crazy lee]

I agree that the relative velocity of the balls is not constant. However, the CM cannot be said to "bounce" elastically (or otherwise) until the collision is complete. Let me explain.

Shown below is a sequence of frames illustrating the collision of the CM with the wall. The balls have equal speeds $v$ along the horizontal. Velocity arrows are omitted to reduce clutter. The color code is: Lime green → moves to the Left; Red → moves to the Right. The CTC (center-to-center) separation between balls is $d$ and the red dashed vertical line is parallel to the wall at the point of closest approach of the CM.

The onset of the collision of the two-mass system is when the left mass hits the wall at $t=0$ as shown in the first frame; it is over when both masses have reversed direction and the information that one part of the system has collided with the wall is transmitted to the other part. This is shown in the last frame. The collision duration is $\Delta t=\dfrac{d-2R}{v}.$
View attachment 358733

This is a simple 1D example with the common velocity in the CTC direction. For a tube of length $L$, the period of horizontal oscillations is $$T_{\text{hor}}=2\times\left(\Delta t +\frac{L-2d}{v}\right)=\frac{2}{v}(L-2R)$$and is independent of the separation $d$.

Similarly, it can be shown that if the CTC is horizontal, but the common velocity is vertical, the period of vertical oscillations for a tube of width $W$ is $$T_{\text{ver}}=\frac{2}{v}(W-2R)$$as the balls bounce up and down without colliding with each other.

I stop here. This is a live homework problem after all.

Thank you for typing so many words and drawing the diagrams. After studying for a long time, I finally understood it. You gave a special example and explained the oscillation periods of the center of mass of the two balls in the horizontal and vertical directions. Thank you.

 

[crazy lee]

That is not possible because the sequence keeps forking according to whether a state value is more or less than a threshold. E.g. at one angle it will strike the side of the tube next, at a slightly different angle it will strike the end of the tube next.

Try to answer the first question, which I interpret to mean "is there an initial state (positions and nonzero velocities) such that it will eventually return to that state?".

Next, suppose that just after the balls have collided ball A has velocity vector $v_x\hat i+v_y\hat j$. What are the possible velocity vectors for A when next they are about to collide?

I also know it's quite difficult. So is it possible to calculate the first three terms or the first term of this sequence? Besides, I'd like to ask you, how can I find the post I made two years ago? Which icons or texts should I click on? I also want to ask, can a post be modified after it is published? That is, attach the modified content to the end of the original post instead of directly modifying the original post. Thank you.

 

[crazy lee]

In fact, this problem of two balls in a tube is the first one in a series of problems. It is a problem that has been repeatedly simplified from other problems. This problem of two balls in a tube has two simple variations. For the first variation, with all other conditions remaining the same, another identical small ball is added into the tube. The question is whether the speed of the small ball in the middle is slightly slower than that of the other two small balls, or vice versa. For the second variation, the width of the tube is widened to 2.1 times the diameter of the small ball. The question is what is the probability that when the two small balls meet in the length direction of the tube, they pass through each other through the gap and reach the other end of the tube. It seems that it will be even more difficult to figure out these two problems.

 

[berkeman]

In fact, this problem of two balls in a tube is the first one in a series of problems.

Thread is closed temporarily for Moderation...

 

[berkeman]

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.

After a long and productive PM discussion with the folks who have been trying to help you, this thread will remain closed just like your previous one. It seems that you have not been able to better define your question(s) since your original thread:

https://www.physicsforums.com/threa...ution-of-rigid-balls-in-a-vast-space.1050265/

You have an ill-defined problem statement and you keep moving the goalposts as others try hard to help you. That is not a good way to participate in a technical discussion forum. Please do not start yet another thread on this ill-defined problem.