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#### 简单的能量守恒证明

A Simple Proof of Conservation of Energy

One of the fundamental principles in modern physics

is that there’s no absolute time.

And I’m not even talking about relativity making time

go at different rates if you’re going near light speed.

I just mean that any time is as good as any other

to set your clock to zero.

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”

I mean, if there were, time zones wouldn’t work!

In addition to allowing time zones,

the fact that there’s no absolute time also implies

the law of conservation of energy.

Here’s a simple proof to show that

if a force doesn’t depend explicitly on time,

then that force conserves energy!

First, conserving energy just means that

there’s a certain number, called the energy, which

doesn’t change as time passes.

If you compare the energy at two different times,

you’ll see no difference.

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

and its energy due to position – the potential energy.

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

Ok, we know the kinetic energy of an object is

half of its mass times its velocity squared.

So the change in kinetic energy over time is just

one half m times the difference of the squares of the velocities.

Some clever algebra can rearrange this expression to become

the average velocity times m times the CHANGE in velocity.

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

and mass times acceleration is equal to the force on the object.

So the change in kinetic energy of an object over time

is just velocity times force.

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

is negative the amount of work it takes to

get the object to its current position from its previous position,

independent of the path taken.

That is, potential energy is the negative of the force applied

times the change in position.

This is where the “there’s no absolute time” part comes into play,

you can’t have potential energy for a force that changes over time.

And just to be clear,
“不随时间变化”不是说
“no changing over time” doesn’t mean that an object

can’t experience a changing force along the course of its journey.

The force might be different at different places,

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

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

is negative F times the change in position.

The negative comes from the fact that

if you let the force push you along, your potential energy decreases.

While if you fight in opposition to the force,

So the change in potential energy over time is

the negative of the force times the change in position over time,

but change in position over time is velocity!

Which means change in potential energy of an object over time

is negative velocity times force.

And thus the change in the total energy over time,

which is the sum of the changes in the kineticand potential energies over time,

is v*F plus negative v*F, which equals zero!

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

All for the same reason that time zones work.

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,

including fiction, non-fiction and periodicals.

If you go to audible.com/minutephysics,

I’d like to recommend the book:

Unruly Places — Lost Spaces, Secret Cities, and Other Inscrutable Geographies

by Alastair Bonnett.

It’s a tour of all sorts of weird, abandoned

or disregarded places around the world,

from the unrecognized self-declared island nation of Sealand

to traffic islands, and beyond.

audible for helping me continue to make these videos.

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

much easier to do with calculus.

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

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

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

change over time.

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

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

It’s beautiful physics.