Solved: Velocity of Isolated System in Different Inertial Sys.

[liuxinhua]

TL;DR Summary

A problem of an isolated system's velocity in different inertial systems, in special relativity

A problem of an isolated system's velocity in different inertial systems, in special relativity

$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time.

$\ \ \ \ \$ The inertial reference frame $K'$ moves to the right at velocity $v$ relative to the inertial reference frame $K$ in $x$ direction,

$\ \ \ \ \$ In the inertial reference system $K'$, whether there is such a time, at this time, the speed of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.

 

[PeterDonis]

In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time.

So the "system" you are talking about has more than one component, and these components have different $x$ coordinates at the time you are talking about?

The inertial reference frame $K'$ moves to the right at velocity $v$ relative to the inertial reference frame $K$ in $x$ direction

Ok.

In the inertial reference system $K'$, whether there is such a time, at this time

If you don't know whether there is such a time, how can you say what happens at such a time?

the speed of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.

Are you asking if this is true, or claiming that it is true? If you are asking, the answer is no, it isn't true, at least not necessarily (it could be in some cases, but from the information you have given it will not always be true). If you are claiming it, you are mistaken.

In any case, what's the point? Do you have a question?

 

[liuxinhua]

I meet a problem.
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time.
$\ \ \ \ \$$\ \ \ \ \$the "system" we are talking about has more than one component, and these components have different $x$ coordinates at the time we are talking about.

$\ \ \ \ \$ The inertial reference frame $K'$ moves to the left at velocity $v$ relative to the inertial reference frame $K$ in $x$ direction. $\ \ \ \ \$In the first floor, left is written as $right$, which should be $left$.

$\ \ \ \ \$ In the inertial reference system $K'$, whether there is such a time, $\ \ \ \ \$ I find there exist a time,
$\ \ \ \ \$ $Result$ ：At some time, in the inertial reference system $K'$, the velocity of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.

Velocity is not a conserved quantity.
For an isolated system, momentum is a conserved quantity.

But I always feel that this $Result$ should not exist.

$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time.
I think, it is no different from the following:
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time. All components of the system are in the same $y-z$ plane.
$\ \ \ \ \$ This is relative to the inertial reference system $K$.
But I really can't give a reason.

 

[PeroK]

$\ \ \ \ \$ In the inertial reference system $K'$, whether there is such a time, $\ \ \ \ \$ I find there exist a time,
$\ \ \ \ \$ $Result$ ：At some time, in the inertial reference system $K'$, the velocity of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.

Velocity is not a conserved quantity.

If all components are at rest relative to each other (in $K$), then they must all be moving at $v$ in $K'$. If you get something different you must be doing something wrong.

Velocity is never an invariant quantity:

$v' = v + V$ (Newtonian)

$v' = \frac{v + V}{1 + vV/c^2}$ (SR)

Where $V$ is the relative velocity between frames of reference.

 

[PeroK]

@liuxinhua Is this what you meant?

If you have a system of equal mass particles, then in Newtonian physics conservation of momentum implies conservation of total velocity. Because the common mass cancels in conservation of momentum.

But, in SR, it is the total sum of $\gamma_n v_n$ that is conserved when you cancel the common mass.

 

[PeterDonis]

In the inertial reference system $K'$, whether there is such a time, I find there exist a time,

"Whether there is such a time" and "I find there exist a time" contradict each other. I assume you mean the latter.

At some time, in the inertial reference system $K'$, the velocity of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.

As I said before, this "Result" is not correct--it does not follow from your previous statements.

I always feel that this Result should not exist.

It doesn't. See above.

All components of the system are in the same $y-z$ plane.

Having all components of the system in the same $y-z$ plane is an additional condition that you can choose to put into your scenario, over and above the first one. But it does not follow from the first one.

 

[liuxinhua]

I'll state the whole problem tomorrow.
Let me rephrase the problem first.
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time, the "system" has more than one component, and these components have different $x$ coordinates at the time.
$\ \ \ \ \$ The inertial reference frame $K'$ moves to the left at velocity $v$ relative to the inertial reference frame $K$ in $x$ direction.
$\ \ \ \ \$Is this possible? $Result1$ (call it as Result1)：At some time, in the inertial reference system $K'$, the velocity of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.
I feel that this $Result1$ should not exist.
But I have meet $Result1$, so I ask you "Does the $Result1$ can exist?"

If all components are at rest relative to each other (in $K$), then they must all be moving at $v$ in $K'$. If you get something different you must be doing something wrong.

In the inertial reference frame $K$, the velocity of each component of an isolated system in $y$ direction is different at the certain time,
Can you sure: the velocity of each component of the isolated system is not less than $v$ in $x$ direction, they must all be moving at $v$ in $K'$ in $x$ direction. (SR)
Or sure it maybe wrong.

"Whether there is such a time" and "I find there exist a time" contradict each other. I assume you mean the latter.
As I said before, this "Result" is not correct--it does not follow from your previous statements.
It doesn't. See above.

I mean that this $Result1$ should not exist. But I have meet $Result1$, so I ask you "Does the $Result1$ can exist?"

Having all components of the system in the same $y-z$ plane is an additional condition that you can choose to put into your scenario, over and above the first one. But it does not follow from the first one.

This is not required in the scenario, all components of the system is not in the same $y-z$ plane.
I only mean :
I think, there is no difference between the following two sentences:
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time, the system’s components have different $x$ coordinates at the time.
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time, all components of the system are in the same $y-z$ plane at this time.

That's why I think $Result1$ shouldn’t exist. Of course, that's not a good reason.

Does anyone have reason to say that $Result1$ cannot exist.

 

[PeroK]

I'll state the whole problem tomorrow.
Let me rephrase the problem first.
$\ \ \ \ \$ In the inertial reference frame $K$, the velocity of each component of an isolated system in $x$ direction is zero at a certain time, the "system" has more than one component, and these components have different $x$ coordinates at the time.
$\ \ \ \ \$ The inertial reference frame $K'$ moves to the left at velocity $v$ relative to the inertial reference frame $K$ in $x$ direction.
$\ \ \ \ \$Is this possible? $Result1$ (call it as Result1)：At some time, in the inertial reference system $K'$, the velocity of each component of the isolated system is not less than $v$ in $x$ direction, and at least part of it is greater than $v$ in $x$ direction.
I feel that this $Result1$ should not exist.
But I have meet $Result1$, so I ask you "Does the $Result1$ can exist?"

This makes no sense. Why would part of the system have velocity greater than $v$ and part less than $v$?

What calculations led you to that conclusion?

 

[Ibix]

This makes no sense.

May be related to

I'll state the whole problem tomorrow.

 

[PeterDonis]

I'll state the whole problem tomorrow.

In other words, up to now you've just been wasting our time with an incomplete description of whatever it is you're asking about? Then this thread is closed.