Ball bouncing on a planet (no atmosphere) follow up questions

[Jeronimus]

This thread will contain several follow up questions, but let me start with the most simple one.

Imagine a planet with no atmosphere and a ball in empty space. The ball is dropped on the planet from a certain height and starts bouncing.

We ignore friction and consider this to be the fully elastic case. Hence, if I am not mistaken, we are only concerned with energy and momentum conservation.
We do _NOT_ ignore that the planet has a high mass compared to the ball, but nevertheless it is not infinite but finite.

So my question is, will the ball keep bouncing forever and retain the same height, or will it eventually become "one", hence at rest with the planet, having transferred all it's momentum to the now planet+ball system as "one"?

 

[restart]

Since you have mentioned that the planet is indeed finite, I would think that the ball will eventually come to rest at the surface of the planet.
Upon each collision of the ball with the planet, a finite amount of the ball's momentum would be transferred to the planet, because of them being an action-reaction pair. Eventually, all of the ball's momentum would be imparted to the planet, and the ball would be at rest in the planet's frame of reference.

 

[Jeronimus]

Since you have mentioned that the planet is indeed finite, I would think that the ball will eventually come to rest at the surface of the planet.
Upon each collision of the ball with the planet, a finite amount of the ball's momentum would be transferred to the planet, because of them being an action-reaction pair. Eventually, all of the ball's momentum would be imparted to the planet, and the ball would be at rest in the planet's frame of reference.

This is what I thought would happen as well. If no one objects to this after some time, I will continue with my follow up questions.

 

[Jeronimus]

So my next question.

On the same planet with no atmosphere, the ball was bouncing on and came to a halt after some time, we now place a long, closed at the bottom, hollow cylinder/pipe, extending far into space. We are not concerned on if such a material to sustain the pipe without breaking exists.

We then place a box containing some pressurized gas, at several bar, at the bottom of the pipe and then open the box.

How will this gas behave over time? Will it expand at first, some of the gas reaching high inside the pipe? What will happen if we wait a long time? Can it be compared to the planet+ball scenario? Will the gas molecules pass energy/momentum onto the planet and the gas start dropping towards the base of the pipe, or what other kind of scenario is to be expected?

edit: I might add that the planet's gravity is high enough from allowing any of the gas to reach escape velocity.

 

[jbriggs444]

Upon each collision of the ball with the planet, a finite amount of the ball's momentum would be transferred to the planet, because of them being an action-reaction pair.
Eventually, all of the ball's momentum would be imparted to the planet, and the ball would be at rest in the planet's frame of reference.

This is wrong.
Yes, momentum is transferred at each collision. Momentum is also transferred between each collision due to gravity -- otherwise there would be only one bounce. Since it is stipulated that the collision is elastic, energy is not lost and the motion continues forever.

 

[jbriggs444]

How will this gas behave over time?

The original stipulation of "elastic collisions" can be applied in the case of the gas as "perfectly insulating walls". The gas never cools down and never settles to the bottom.

 

[Jeronimus]

This is wrong.
Yes, momentum is transferred at each collision. Momentum is also transferred between each collision due to gravity -- otherwise there would be only one bounce. Since it is stipulated that the collision is elastic, energy is not lost and the motion continues forever.

Energy is never lost. The question is, if a ball hits another bigger ball (in this case, the planet) in an elastic collision, then does the smaller ball transfer some of it's energy to the bigger ball because of energy and momentum conservation?
The answer to that, is yes I believe. It being an elastic collision does not imply that the smaller ball has to come out with the same energy as before after it hits the planet's surface.

What exactly happens when the ball moves up again after the collision, turning its energy from kinetic to potential within the gravity field, to drop back again onto the planet after, I did not fully wrap my head around, so I was hopping someone here would explain it in detail.
So maybe you are right and it does go on forever, but I cannot see it right now from your explanation.

edit: I gave this a bit more thought. So basically, as it appears to me, for you to be correct, when a small ball with a mass A hits a bigger ball with mass B, with B > A, then the _relative_ velocity, the ball has after reaching the surface shortly before colliding, and after the collision, would have to remain the same.
Only then, as it seems, would the ball be able to reach the same height as before, for the bouncing to go on forever.

Does the math tells us that?

edit2: I guess the math has to tell us that, because we can always consider the bigger ball with the greater mass at rest from some frame of reference and then the smaller ball will have to make up for the whole energy and momentum in this two balls system.
So deltaV has to be the same as before as it seems.

So yes, jbriggs444 right again :(

 

[Dale]

This is what I thought would happen as well. If no one objects to this after some time, I will continue with my follow up questions.

I don't think that is correct. If each collision is perfectly elastic then the ball will keep on bouncing at the same height forever. This is easiest to understand in the center of momentum frame.

 

[jbriggs444]

Energy is never lost. The question is, if a ball hits another bigger ball (in this case, the planet) in an elastic collision, then does the smaller ball transfer some of it's energy to the bigger ball because of energy and momentum conservation?
The answer to that, is yes I believe.

Energy is transferred, possibly. But in which direction? Energy is not an invariant quantity it depends on your choice of reference frame. Pick one in which the center of mass of the ball plus planet is at rest and no energy is transferred at all. [Edit: as @Dale hints]

 

[Vanadium 50]

This thread is going down the same rabbit hole that causes many of Jeronimus' threads to be locked: it starts with an incorrect premise, which is then adhered to despite correction.

For perfectly elastic collisions, the ball never comes to a halt. You can't have both requirements "perfectly elastic collisions" and "ball comes to a halt" at the same time. The easiest way to see this is in the center of momentum frame.

Edit: I see three people came to this explanation at the same time.

 

[Dale]

The question is, if a ball hits another bigger ball (in this case, the planet) in an elastic collision, then does the smaller ball transfer some of it's energy to the bigger ball because of energy and momentum conservation?

The answer is that it depends on the reference frame. Energy is conserved in all frames, but not transferred in all frames. In particular, in the center of momentum frame energy is not transferred during the collision.

At this point you have 3 experts recommending examining the problem in the center of momentum frame. Do you understand what is being suggested and how that simplifies the analysis?

 

[Jeronimus]

There is just one thing that puzzles me still.

If we can look at this from any reference frame and both energy and momentum are to be conserved. How is momentum conserved when we look at the case where the ball just turned all it's kinetic energy into potential and is at it's highest point relative to the planet. From the reference frame's perspective the planet is at rest in, the ball is now also at rest in for this brief moment. Both seem to have zero momentum combined.

Whereas, when we let the ball fall, all the potential turns into kinetic, the smaller ball now having a velocity v seen from the perspective of the reference frame the planet is at rest in. Hence it has some non-zero momentum while the planet has zero momentum in that reference frame. Hence non-zero when you combine both.

What did I miss this time?

 

[jbriggs444]

reference frame the planet is at rest in

That's not an inertial reference frame.

 

[Dale]

What did I miss this time?

The gravitational force of the ball on the planet.

Both seem to have zero momentum combined.

Use the center of momentum frame. The combined momentum is zero at all times.

Again, do you understand what we mean by "center of momentum frame" and do you see how it simplifies the analysis?

 

[SlowThinker]

What did I miss this time?

I am standing. In my reference frame, the Earth+I system has zero momentum.
I start to walk. In my reference frame, the Earth+I system has some huge momentum.
If you change reference frames, it is no wonder that the momentum changes.