# Two trolleys on the rail

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### Luigi Fortunati

Nov 7, 2022, 3:15:04 AM11/7/22
to
In my animation
https://www.geogebra.org/m/ytws9kbr
there are two trolleys that start from stationary on the rail and
accelerate to speed v=0.866c (range=2).

In one case, the two carriages remain at the same distance (and the belt
does not break), in the other the trolley B moves twice as far away from
the trolley A (and the belt breaks): which of the two conditions is
correct?

### Richard Livingston

Nov 8, 2022, 11:37:37 AM11/8/22
to
The animation appears to be as seen in a frame that is accelerating with
carriage A, and I think you intend that both start at the same time, in the
initial rest frame. In that case the top animation with the carriages pulling
apart is mostly correct.

I say mostly correct because the animation does not take into account how
the observers in carriage A will see only a delayed image of B, and thus
when starting up will initially see B stationary for a moment before the
light from B gets to A. In fact A will always see a delayed image of B,
yet the string will stretch and break. Likewise, initially the string will
not be pulled by B, as seen at A, until the acceleration of B can
propagate down the string at the speed of sound.

Also you do not show the Lorentz contraction of the railroad ties as the
carriages speed increases. That is happening also.

Rich L.

### Sylvia Else

Nov 9, 2022, 1:33:05 AM11/9/22
to
The issue here lies in the expression "same acceleration" because it
turns out that the accelerations cannot both remain the same as
experienced by the trolleys, and the same as determined by an observer
who remains at rest relative to the initial state.

So you have to choose which it will be, and your choice will affect the
outcome.

Sylvia.

### Jonathan Thornburg [remove -color to reply]

Nov 10, 2022, 10:36:37 AM11/10/22
to
Luigi Fortunati <fortuna...@gmail.com> wrote:
> In my animation
> https://www.geogebra.org/m/ytws9kbr
> there are two trolleys that start from stationary on the rail and
> accelerate to speed v=0.866c (range=2).

There's a very nice discussion of this in the physics FAQ:
https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html

There's also a nice discussion of this paradox (including a link to
the above physics FAQ article) in

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### Luigi Fortunati

Nov 10, 2022, 11:44:26 AM11/10/22
to
Consider both cases individually.

In the first case, which of the two conditions is correct?

And in the second case?

Luigi.

### Luigi Fortunati

Nov 11, 2022, 11:33:57 AM11/11/22
to
Jonathan Thornburg [remove -color to reply] giovedě 10/11/2022 alle ore
16:36:34 ha scritto:
> Luigi Fortunati <fortuna...@gmail.com> wrote:
>> In my animation
>> https://www.geogebra.org/m/ytws9kbr
>> there are two trolleys that start from stationary on the rail and
>> accelerate to speed v=0.866c (range=2).
>
> There's a very nice discussion of this in the physics FAQ:
>
> https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html
>
> There's also a nice discussion of this paradox (including a link to
> the above physics FAQ article) in

Yes, my animation reproduces Bell's paradox, set on Earth instead of in
space.

The two trolleys have the same acceleration in the reference of the
tracks where they maintain the same distance.

But is the initial distance of only one rail and, therefore, neither of
my two animations are correct?

### Sylvia Else

Nov 12, 2022, 12:57:33 AM11/12/22
to
On 11/11/2022 3:44 am, Luigi Fortunati wrote:
> Sylvia Else marted=EC 08/11/2022 alle ore 15:33:00 ha scritto:
> Consider both cases individually.
>
> In the first case, which of the two conditions is correct?
>
> And in the second case?
>
> Luigi.

I think it would be more helpful for you to explain what you think will
happen in the two cases, and why. Then we could look at your analysis.

Sylvia.

### Luigi Fortunati

Nov 12, 2022, 4:30:10 PM11/12/22
to
I am on board of trolley A and, during acceleration, I see trolley B
moving away from me by twice the initial distance (as Bell says in his
explanation).

At the same time, I see the distances between the track sleepers (the
rail) shrinking.

The effect of both causes increases the number of rails between me and
trolley B from 1 to 4, as is the case at the top of my animation.

But 4 rails between me and B contrast with my speed over the ground
(v=0.866c range=2), for which there should only be 2 rails (twice as
much) between me and B (as is the case at the Bottom of my animation)
and not 4 rails.

Hence my doubt that I was hoping someone could clarify, because if one
solution is right, the other is wrong (and vice versa).

Luigi

### Sylvia Else

Nov 12, 2022, 10:08:54 PM11/12/22
to
On 13/11/2022 8:30 am, Luigi Fortunati wrote:
> Sylvia Else venerd=EC 11/11/2022 alle ore 14:57:28 ha scritto:
>>> Consider both cases individually.
>>>
>>> In the first case, which of the two conditions is correct?
>>>
>>> And in the second case?
>>>
>>> Luigi.
>>
>> I think it would be more helpful for you to explain what you think will
>> happen in the two cases, and why. Then we could look at your analysis.
>>
>> Sylvia.
>
> I am on board of trolley A and, during acceleration, I see trolley B
> moving away from me by twice the initial distance (as Bell says in his
> explanation).
>
> At the same time, I see the distances between the track sleepers (the
> rail) shrinking.
>
> The effect of both causes increases the number of rails between me and
> trolley B from 1 to 4, as is the case at the top of my animation.
>
> But 4 rails between me and B contrast with my speed over the ground
> (v=0.866c range=2), for which there should only be 2 rails (twice as
> much) between me and B (as is the case at the Bottom of my animation)
> and not 4 rails.
>
> Hence my doubt that I was hoping someone could clarify, because if one
> solution is right, the other is wrong (and vice versa).
>
> Luigi

In this scenario, the choice you've made is that the two trolleys
accelerate at the same rate as determined by the observer at rest
relative to the initial state.

They accelerate at different rates in their respective frames (which
implies that differing, and indeed changing, forces are being applied to
them). If the trolleys were connected by an inextensible tether, then it
would break.

Sylvia

### Luigi Fortunati

Nov 13, 2022, 5:10:38 AM11/13/22
to
Sylvia Else sabato 12/11/2022 alle ore 12:08:50 ha scritto:
>> I am on board of trolley A and, during acceleration, I see trolley B
>> moving away from me by twice the initial distance (as Bell says in his
>> explanation).
>>
>> At the same time, I see the distances between the track sleepers (the
>> rail) shrinking.
>>
>> The effect of both causes increases the number of rails between me and
>> trolley B from 1 to 4, as is the case at the top of my animation.
>>
>> But 4 rails between me and B contrast with my speed over the ground
>> (v=0.866c range=2), for which there should only be 2 rails (twice as
>> much) between me and B (as is the case at the Bottom of my animation)
>> and not 4 rails.
>>
>> Hence my doubt that I was hoping someone could clarify, because if one
>> solution is right, the other is wrong (and vice versa).
>>
>> Luigi
>
> In this scenario, the choice you've made is that the two trolleys
> accelerate at the same rate as determined by the observer at rest
> relative to the initial state.
>
> They accelerate at different rates in their respective frames (which
> implies that differing, and indeed changing, forces are being applied to
> them). If the trolleys were connected by an inextensible tether, then it
> would break.
>
> Sylvia

So the scenario of the top of my animation
https://www.geogebra.org/m/ytws9kbr
where the ribbon breaks is correct.

But how do you justify the fact that reaching the speed v=0.866c
(gamma=2) the distance between A and B (in the reference of the trolley=

A) is 4 sleepers (quadruple) and not 2 (double)?