Name: 能量守恆的簡單證明 (A Simple Proof of Conservation of Energy)
Uploaded: 2021-10-08T17:40:36.000Z
Duration: 3 min 58 s

One of the fundamental principles in modern physics is that there's no absolute time.

現代物理學的一個基本原則是，沒有絕對的時間。

程度副詞

And I'm not even talking about relativity making time go at different rates if you're

我甚至不是在談論相對論使時間以不同的速度流逝，如果你是

進階形容詞比較級句型

going near light speed – I just mean that any time is as good as any other to set your

接近光速--我的意思是，任何時候都可以設置你的

The predictions of physics work all the same, and it's not like there's some “absolute

物理學的預測都是一樣的，並不像有什麼 "絕對的"。

starting time” – I mean, if there were, time zones wouldn't work!

我的意思是，如果有的話，時區就不起作用了！"。

In addition to allowing time zones, the fact that there's no absolute time also implies

除了允許時區之外，沒有絕對時間的事實也意味著

Here's a simple proof to show that if a force doesn't depend explicitly on time,

這裡有一個簡單的證明，表明如果一個力不明確地依賴於時間。

First, conserving energy just means that there's a certain number, called the energy, which

首先，節約能源只是意味著有一個一定的數字，稱為能量，這

doesn't change as time passes – if you compare the energy at two different times

不會隨著時間的推移而改變--如果你比較兩個不同時間的能量

And the total energy of a system is the sum of its energy of motion, or kinetic energy,

而一個系統的總能量是其運動能量，或動能的總和。

and its energy due to position – the potential energy.

和它由於位置而產生的能量--勢能。

So all we need is to find the change in kinetic and potential energies over time, and add

是以，我們所需要的是找到動能和勢能隨時間的變化，並加上

Ok, we know the kinetic energy of an object is half of its mass times its velocity squared.

好吧，我們知道一個物體的動能是其品質的一半乘以其速度的平方。

So the change in kinetic energy over time is just one half m times the difference of

是以，動能隨時間的變化只是半米的差異，即

Some clever algebra can rearrange this expression to become the average velocity times m times

一些巧妙的代數可以重新排列這個表達式，成為平均速度乘以m乘以

But a change of velocity over time is just an acceleration, and mass times acceleration

但速度隨時間的變化只是一個加速度，而品質乘以加速度

So the change in kinetic energy of an object over time is just velocity times force.

是以，一個物體的動能隨時間的變化只是速度乘以力。

On the other hand, the change in the potential energy of an object is negative the amount

另一方面，一個物體的勢能的變化是負數。

of work it takes to get the object to its current position from its previous position

將物體從之前的位置移至當前位置所需的工作量

independent of the path taken – that is, potential energy is the negative of the force

與所走的路徑無關，也就是說，勢能是力的負值。

This is where the “there's no absolute time” part comes into play – you can't

這就是 "沒有絕對的時間 "這一部分發揮作用的地方--你不能

have potential energy for a force that changes over time.

對一個隨時間變化的力有勢能。

And just to be clear, “no changing over time” doesn't mean that an object can't

為了明確起見，"不隨時間變化 "並不意味著一個物體不能

experience a changing force along the course of its journey: the force might be different

在其旅程中經歷了一個不斷變化的力量：這個力量可能是不同的

at different places, but at any particular place, the force must remain the same.

在不同的地方，但在任何特定的地方，力必須保持相同。

Anyway, this is all just to say that the change in potential energy is negative F times the

總之，這一切只是為了說明，勢能的變化是負的F倍。

The negative comes from the fact that if you let the force push you along your potential

負面因素來自於這樣一個事實：如果你讓力量推著你走，你的潛力

energy decreases, while if you fight in opposition to the force your potential energy increases.

勢能會減少，而如果你與力對抗，你的勢能會增加。

So the change in potential energy over time is the negative of the force times the change

是以，隨著時間的推移，勢能的變化是力的負值乘以變化量

in position over time, but change in position over time is velocity!

在時間上的位置變化，但在時間上的位置變化是速度!

Which means change in potential energy of an object over time is negative velocity times

這意味著物體隨時間變化的勢能是負速度乘以

And thus the change in the total energy over time, which is the sum of the changes in the

是以，隨著時間的推移，總能量的變化，也就是以下變化的總和

kinetic and potential energies over time, is v*F plus negative v*F, which equals zero!

隨著時間的推移，動能和勢能，是v*F加上負v*F，等於零!

And zero change in energy over time is precisely conservation of energy!

而能量隨時間的變化為零，正是能量守恆!

All for the same reason that time zones work.

所有的原因都與時區的作用相同。

This video is brought to you in part by Audible.com, the leading provider of audiobooks across

本視頻部分內容由Audible.com提供，Audible.com是全球領先的有聲讀物提供商。

all types of literature, including fiction, non-fiction and periodicals.

所有類型的文學作品，包括小說、非小說和期刊。

If you go to audible.com/minutephysics, you can try audible out by downloading a free

如果你進入 audible.com/minutephysics，你可以通過下載免費的audible來嘗試。

I'd like to recommend the book Unruly Places: Lost Spaces, Secret Cities, and Other Inscrutable

我想推薦《不守規矩的地方》這本書。失落的空間，祕密的城市，和其他不可告人的地方。

Geographies by Alastair Bonnett [called Off the Map in the UK] – it's a tour of all

阿拉斯泰爾-博內特（Alastair Bonnett）的《地理學》[在英國稱為《地圖之外》]--它是對所有國家的巡禮。

sorts of weird, abandoned or disregarded places around the world, from the unrecognized self-declared

世界各地各種奇怪的、被遺棄的或被忽視的地方，從未被承認的自稱的

island nation of Sealand to traffic islands, and beyond.

從海島國到交通島，甚至更遠。

Again, you can download your free audiobook at audible.com/minutephysics and thanks to

同樣，你可以在audible.com/minutephysics下載你的免費有聲書，並感謝

audible for helping me continue to make these videos.

有聲有色，幫助我繼續製作這些視頻。

As a followup, I'll admit that I skimmed over some details in this derivation and it's

作為後續，我承認我略過了這個推導中的一些細節，它是

You just have to know that the time derivative of kinetic energy is m times v times a which

你只需知道，動能的時間導數是m乘以v乘以a，即

is F*v, and the time derivative of potential energy is -F times dx/dt, or -F*v, and so

是F*v，而勢能的時間導數是-F乘以dx/dt，或-F*v，是以

the time derivative of the total energy, dE/dt, is F*v-F*v = 0, which means energy doesn't

總能量的時間導數，dE/dt，是F*v-F*v=0，這意味著能量沒有

And as a final followup, the most legit and robust version of this proof as it applies

作為最後的跟進，這個證明的最合法和最有力的版本，因為它適用於

to ALL OF PHYSICS is called Noether's theorem, discovered by Emmy Noether in 1915.

在所有的物理學中，諾特定理被稱為諾特定理，由埃米-諾特在1915年發現。