# 查看完整版本 : A shortcut to E=mc2？

LT3648 2017-8-14 09:15 PM

## A shortcut to E=mc2？

[url=https://www.fotonowy.pl/news/emc2-shortcut/?lang=en]https://www.fotonowy.pl/news/emc2-shortcut/?lang=en[/url]

Have you ever considered how to actually derive the famous [b][i]E=mc2[/i][/b]  formula? Does it require knowledge of Special Relativity or even worse: an understanding of the theory?

Well, no!

The only skill needed is some high-school physics and a bit of curiosity.

There is a box of mass [b][i]M[/i][/b] and horizontal length [i][b]L[/b][/i] in the picture below.The center of mass of the box in rest hangs somewhere above point [b][i]×[/i][i]c[/i][/b].

To derive the formula it is enough to answer the question: [i]What will happen to the center of mass position if a photon is emitted from the left wall of the box, picture a), and caught by the right wall, as in the picture b)?[/i]

Lets assume that the photon has energy [b][i]E[/i][/b]. It is known1, that photons propagate with the speed of light [b][i]c[/i][/b] and carry momentum

From the momentum conservation principle, after the photon emission, the box should move leftwards with some velocity [i][b]v[/b][/i]. After the photon is absorbed the box must stop due to the momentum conservation.

The velocity [b][i]v[/i][/b] is calculated in the following manner.The box moves the distance [b][i]Δx[/i][/b] to the left before the photon hits the right wall. Time of flight of the photon [b][i]Δt[/i][/b] is given by

Thus, the mean velocity [b][i]v[/i][/b] of the box

So far, so good. But what has happened to the box’s center of mass? Initially it has been positioned over the point [b][i]×c[/i][/b], then the box moved leftwards and stopped. Did the center of mass move as well? Of course, not. There was no external force acting on the system, so the center of mass remained at the initial position.

Summarizing, there are two undeniable facts
1. The box moved,
2. Its center of mass did not.

The only way out of the apparent contradiction is to admit, that besides the energy and momentum the photon carried some mass [b][i]m[/i][/b] from the left to the right wall.[color=#ff0000][b] (Notice, that [i]m[/i] is not the rest mass of the photon. Photons have zero rest mass.)[/b][/color]

From the momentum conservation, the magnitudes of momenta of the box and photon are equal

Substituting the previously evaluated velocity [b][i]v[/i][/b] in the expression above one gets

For the center of mass to remain still, the mass displacement of the box must be balanced by displacement of the mass carried by the photon.

From the two last equations one obtains

Or in more recognizable form:

:smile_o12:

[[i] 本帖最後由 LT3648 於 2017-9-19 01:46 PM 編輯 [/i]]

sylim 2017-8-15 12:15 AM

macaupro893 2017-8-15 12:54 AM

【There was no external force acting on the system, so the center of mass remained at the initial position】
photon在撞墻之前，center of mass 係咪會變？

LT3648 2017-8-15 12:49 PM

[quote]原帖由 [i]sylim[/i] 於 2017-8-15 12:15 AM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466150683&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

[[i] 本帖最後由 LT3648 於 2017-8-15 10:06 PM 編輯 [/i]]

Ron~ 2017-8-15 01:05 PM

[quote]原帖由 [i]macaupro893[/i] 於 2017-8-15 12:54 AM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466152008&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]
【There was no external force acting on the system, so the center of mass remained at the initial position】
photon在撞墻之前，center of mass 係咪會變？

LT3648 2017-8-15 01:16 PM

[quote]原帖由 [i]macaupro893[/i] 於 2017-8-15 12:54 AM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466152008&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]
【There was no external force acting on the system, so the center of mass remained at the initial position】
photon在撞墻之前，center of mass 係咪會變？

[[i] 本帖最後由 LT3648 於 2017-8-15 01:34 PM 編輯 [/i]]

LT3648 2017-8-15 01:31 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 01:05 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466174793&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 01:54 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 01:16 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466175372&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 02:05 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 01:31 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466176239&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

LT3648 2017-8-15 02:25 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 01:54 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466177507&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 02:30 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 02:25 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466179146&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

LT3648 2017-8-15 02:42 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 02:30 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466179382&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 03:19 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 02:42 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466179961&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

LT3648 2017-8-15 03:26 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 03:19 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466181844&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

[[i] 本帖最後由 LT3648 於 2017-8-15 03:29 PM 編輯 [/i]]

Ron~ 2017-8-15 04:33 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 03:26 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466182180&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

「條數話無變，你用的比喻不正確，光子無電荷，但電子有電荷，你的質疑不成立。」

LT3648 2017-8-15 05:19 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 04:33 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466185806&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

「條數話無變，你用的比喻不正確，光子無電荷，但電子有電荷，你的質疑不成立。」

「條數話無變，你用的比喻不正確，[color=#ff0000]光子無電荷，但電子有電荷[/color]，你的質疑不成立。」<-------紅字不是我說的，所以你的質疑不成立。:smile_03:

Ron~ 2017-8-15 05:28 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 05:19 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466188232&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

「條數話無變，你用的比喻不正確，[color=#ff0000]光子無電荷，但電子有電荷[/color]，你的質疑不成立。」<-------紅字不是我說的，所以你的質疑不成立。:smile_03:
[/quote]

[[i] 本帖最後由 Ron~ 於 2017-8-15 05:30 PM 編輯 [/i]]

LT3648 2017-8-15 05:35 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 05:28 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466188751&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 05:44 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 05:35 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466189069&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

LT3648 2017-8-15 05:54 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 05:44 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466189526&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 06:07 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 05:54 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466190085&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

LT3648 2017-8-15 06:12 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 06:07 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466190824&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

"可否答答，中子由盒的左牆飛去右牆，即左牆減一個中子質量，右牆加一個中子質量，盒的center of mass是否仍不變呢？:smile_04:"

macaupro893 2017-8-15 06:24 PM

【There was no external force acting on the system, so the center of mass remained at the initial position】

https://en.wikipedia.org/wiki/Center_of_mass

Ron~ 2017-8-15 06:25 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 06:12 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466191080&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

"可否答答，中子由盒的左牆飛去右牆，即左牆減一個中子質量，右牆加一個中子質量，盒的center of mass是否仍不變呢？:smile_04:" [/quote]

Ron~ 2017-8-15 06:29 PM

[quote]原帖由 [i]macaupro893[/i] 於 2017-8-15 06:24 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466191720&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]
【There was no external force acting on the system, so the center of mass remained at the initial position】

[url=https://en.wikipedia.org/wiki/Center_of_mass]https://en.wikipedia.org/wiki/Center_of_mass[/url]

LT3648 2017-8-15 06:33 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 06:25 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466191751&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

macaupro893 2017-8-15 06:38 PM

[quote]原帖由 [i]Ron~[/i] 於 2017-8-15 06:29 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466191973&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

macaupro893 2017-8-15 06:44 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 06:33 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466192140&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ron~ 2017-8-15 06:44 PM

[quote]原帖由 [i]LT3648[/i] 於 2017-8-15 06:33 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=466192140&ptid=26857989][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

macaupro893 2017-8-15 06:47 PM