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简单的能量守恒证明 – 译学馆
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简单的能量守恒证明

A Simple Proof of Conservation of Energy

现代物理的基本原理之一
One of the fundamental principles in modern physics
就是不存在绝对时间
is that there’s no absolute time.
我说的还不是相对论里面的
And I’m not even talking about relativity making time
接近光速 时间的速度会改变
go at different rates if you’re going near light speed.
我只是说你在任何时间都可以
I just mean that any time is as good as any other
把钟表调到0
to set your clock to zero.
物理学的预测也是一样
The predictions of physics work all the same,
“绝对开始时间”是不存在的
it’s not like there’s some “absolute starting time”
我是说 如果真存在的话 就不会有时区了
I mean, if there were, time zones wouldn’t work!
除了允许时区存在
In addition to allowing time zones,
不存在绝对时间还暗含着
the fact that there’s no absolute time also implies
能量守恒定律
the law of conservation of energy.
一个简单的证据就是
Here’s a simple proof to show that
如果力不直接依赖于时间
if a force doesn’t depend explicitly on time,
则此力可存储能量
then that force conserves energy!
首先 存储能量意味着
First, conserving energy just means that
有一定数量的能量
there’s a certain number, called the energy, which
大小不会随时间改变
doesn’t change as time passes.
将不同时间的能量进行对比
If you compare the energy at two different times,
你会发现它们完全一样
you’ll see no difference.
一个系统的能量包括其运动能量 也就是动能
And the total energy of a system is the sum of its energy of motion, or kinetic energy,
还有与位置有关的能量――势能
and its energy due to position – the potential energy.
我们只需要关注系统动能和势能的变化
So all we need is to find the change in kinetic and potential energies over time,
把这些变化加起来
and add them up!
好 大家都知道动能是
Ok, we know the kinetic energy of an object is
物体质量乘以速度的平方再除以二
half of its mass times its velocity squared.
所以 动能的变化量就是
So the change in kinetic energy over time is just
质量的一半乘以速度平方的差
one half m times the difference of the squares of the velocities.
通过巧妙的代数学方法 上式可化简为
Some clever algebra can rearrange this expression to become
平均速度乘以质量乘以速度的变化量
the average velocity times m times the CHANGE in velocity.
速度随时间的变化就是加速度
But a change of velocity over time is just an acceleration,
质量乘以加速度得到施加在物体上的力
and mass times acceleration is equal to the force on the object.
因此 动能的单位时间变化量
So the change in kinetic energy of an object over time
就等于速度乘以力
is just velocity times force.
我们再看势能 物体势能的变化
On the other hand, the change in the potential energy of an object
与将物体从之前位置
is negative the amount of work it takes to
移动到当前位置所做的功大小正好相反
get the object to its current position from its previous position,
和移动路径无关
independent of the path taken.
也就是说 势能是负的
That is, potential energy is the negative of the force applied
力乘以位移
times the change in position.
由此就可推出 “不存在绝对时间”
This is where the “there’s no absolute time” part comes into play,
随时间变化的力没有势能
you can’t have potential energy for a force that changes over time.
为了避免混淆
And just to be clear,
“不随时间变化”不是说
“no changing over time” doesn’t mean that an object
物体在运动过程中受力情况不变
can’t experience a changing force along the course of its journey.
在不同位置 力的大小可能不同
The force might be different at different places,
但是在任一特定地点 力一定是相同的
but at any particular place, the force must remain the same.
不过 这些只能说明
Anyway, this is all just to say that the change in potential energy
势能等于负的力乘以位移
is negative F times the change in position.
之所以有负号 是因为
The negative comes from the fact that
力推着你动时 你的势能会减少
if you let the force push you along, your potential energy decreases.
你向与力相反的方向运动
While if you fight in opposition to the force,
势能就会增加
your potential energy increases.
势能随时间的变化就是
So the change in potential energy over time is
负的力乘以位移随时间的变化量
the negative of the force times the change in position over time,
位移的变化量就是速度
but change in position over time is velocity!
意味着势能的时间变化量
Which means change in potential energy of an object over time
等于负的速度乘以力
is negative velocity times force.
由此 总能量的变化
And thus the change in the total energy over time,
也就是动能和势能变化量的总和
which is the sum of the changes in the kineticand potential energies over time,
等于 v×F + (-v×F) 结果为0
is v*F plus negative v*F, which equals zero!
能量的变化量为零就是能量守恒定律!
And zero change in energy over time is precisely conservation of energy!
时区有效也是同样的道理
All for the same reason that time zones work.
本视频由Audible.com协助制作
This video is brought to you in part by Audible.com,
Audible.com是有声书网站的一流供应商 提供所有文学类型的有声书
the leading provider of audiobooks across all types of literature,
包括小说 纪实文学 期刊
including fiction, non-fiction and periodicals.
点击网址audible.com/minutephysics
If you go to audible.com/minutephysics,
可以选择免费有声书进行下载
you can try audible out by downloading a free audiobook of your choice.
我想给大家推荐一本书
I’d like to recommend the book:
失控之地:遗落之处 秘密之城 神秘所在
Unruly Places — Lost Spaces, Secret Cities, and Other Inscrutable Geographies
作者为阿拉斯泰尔·邦尼特
by Alastair Bonnett.
这本书是关于一场旅行 途经世界上
It’s a tour of all sorts of weird, abandoned
各种怪异 废弃或不毛之地
or disregarded places around the world,
从自称建立而未被承认的岛国西兰公国之内
from the unrecognized self-declared island nation of Sealand
到安全岛之外
to traffic islands, and beyond.
再次提醒 你可以在audible.com/minutephysics上免费下载有声书
Again, you can download your free audiobook at audible.com/minutephysics and thanks to
给有声书好评 帮助我继续制作视频
audible for helping me continue to make these videos.
作为后续 我承认我略过了推导中的一些细节
As a followup, I’ll admit that I skimmed over some details in this derivation and it’s
计算也更容易
much easier to do with calculus.
你只要知道动能的时间导数是质量乘以速度乘以加速度
You just have to know that the time derivative of kinetic energy is m times v times a which
也就是力乘以速度 势能的时间导数是力乘以位移的导数 或负的力乘以速度
is F*v, and the time derivative of potential energy is -F times dx/dt, or -F*v, and so
因此 总能量的时间导数dE/dt就是F×v-F×v=0 意味着
the time derivative of the total energy, dE/dt, is F*v-F*v = 0, which means energy doesn’t
能量不随时间变化
change over time.
最最后最正规严谨的推理
And as a final followup, the most legit and robust version of this proof as it applies
叫做诺特定理 由艾米·诺特在1915年提出
to ALL OF PHYSICS is called Noether’s theorem, discovered by Emmy Noether in 1915.
这就是物理之美
It’s beautiful physics.

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视频概述

简单证明了能量守恒定律以及不存在绝对时间

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视频来源

https://www.youtube.com/watch?v=PplaBASQ_3M

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