# A shortcut to E=mc2？

## A shortcut to E=mc2？

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

Have you ever considered how to actually derive the famous E=mc2  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 M and horizontal length L in the picture below.The center of mass of the box in rest hangs somewhere above point ×c.

To derive the formula it is enough to answer the question: 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)?

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

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

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

Thus, the mean velocity v 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 ×c, 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 m from the left to the right wall. (Notice, that m is not the rest mass of the photon. Photons have zero rest mass.)

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

Substituting the previously evaluated velocity v 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:

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

 sylim VIP 帖子8312 積分6925 金幣0  註冊時間2013-7-12  發短消息 加為好友 2# 大 中 小 發表於 2017-8-15 12:15 AM  只看該作者     用E=MC^2來derive E=MC^2 真係值得驚訝? .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 被引用(1) TOP
 macaupro893 高級會員 帖子993 積分746 金幣0  註冊時間2016-4-20  發短消息 加為好友 3# 大 中 小 發表於 2017-8-15 12:54 AM  只看該作者     【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？ 就好似有隻老鼠在box內走來走去咁。 .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 被引用(2) TOP
##### 引用:

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

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##### 引用:

【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？

##### 引用:

【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？

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

##### 引用:

 Ron~ 尊貴會員 帖子1671 積分1510 金幣0  註冊時間2012-1-25  發短消息 加為好友 9# 大 中 小 發表於 2017-8-15 02:05 PM  只看該作者     [quote]原帖由 LT3648 於 2017-8-15 01:31 PM 發表 回應主題，這個證明是成立的（雖然與愛因斯坦最始的證明不同）。 <-------聽講愛因斯坦最始的證明(類似這例子)不正確，請問他錯在那裏？[\quote] 你看過愛因斯坦那份最原始的論文嗎？ .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 天地不仁，以萬物為芻狗； 聖人不仁，以百姓為芻狗。 回覆 引用 被引用(1) TOP
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##### 引用:

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

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「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。」

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