And this is probably an idea that you're

在這影片中，我要討論一些加速度的問題

somewhat familiar with, or at least

你們可能對這個有點熟悉

you've heard the term used here or there.

你可能有聽過它

Acceleration is just the change in velocity over time.

或者你有聽過這個名詞

Probably one of the most typical examples of acceleration,

加速度就是隨時間改變的速度

if you're at all interested in cars,

如果你對車有興趣

is that many times they will give you

最典型的加速度就是

acceleration numbers, especially for sport cars, actually

它常常會告訴你加速度是多少

all cars if you look up in Consumer Reports,

它常常會告訴你加速度是多少

or wherever they give the stats on different cars.

特別是賽車，實際上

They'll tell you something like, I don't know, like a Porsche--

所有車如果看一下消費者指南

and I'm going to make up these numbers right over here.

或他給你不同車的統計

So let's say that we have a Porsche 911.

他們會告訴你像保時捷(我不知道)

They'll say that a Porsche 911, they'll literally measure it

我要編造一下數據

with a stopwatch, can go 0 to 60 miles per hour.

假設有一台保時捷911

And these aren't the exact numbers,

由碼表測量

although I think it's probably pretty close.

每小時可從0至60英里

0 to 60 miles per hour in, let's say, 3 seconds.

每小時可從0至60英里

So, although officially what they're giving you right here

這不是適確切的數字

are speeds, because they're only giving you

儘管我認為很接近

magnitude and no direction, you can

我們說三秒，每小時從0至60英里

assume that it's in the same direction.

我們說三秒，每小時從0至60英里

I mean, We could say, 0 miles per hour

儘管官方給的是速率

to the east to 60 miles per hour to the east in 3 seconds.

只有大小沒有方向

So what was the acceleration here?

只有大小沒有方向

So I just told you the definition of acceleration.

你可以假設方向不變

It's change in velocity over time.

我指的是我們可以說

So the acceleration-- and once again acceleration

向東三秒之內從0mph加速到60mph

is a vector quantity.

所以加速度是多少?

You want to know not only how much is velocity changing

我剛告訴你們加速度的定義

over time, you also care about the direction.

是單位時間內速度的變化量

It also makes sense because velocity

所以加速度同樣是向量

itself is a vector quantity.

所以加速度同樣是向量

It needs magnitude and direction.

你不只要知道速度的變化

So the acceleration here-- and we're

你也要在乎方向

just going to assume that we're going

因為速度

to the right, 0 miles per hour and 60 miles per hour

本身是向量

to the right-- so it's going to be change in velocity.

需要大小和方向

So let me just write it down with different notation

所以這裡的加速度

just so you could familiarize yourself

我們假設向右行駛

if you see it in the textbook this way.

從0mph至60mph

So change in velocity.

速度將會變化

This delta symbol right here just means "change in."

讓我寫不同的符號

Change in velocity over time.

這樣你可能比較習慣

It's really, as I've mentioned in previous videos,

當你在教科書看到

it's really time is really a change in time.

速度的變化

But we could just write time here.

△是"變化"

This 3 seconds is really change in time.

隨時間改變的速度

It might have been, if you looked at your second hand,

我在影片前提到

it might have been 5 seconds when it started, and then my 8

在時間內改變

seconds when it stopped, so it took a total of 3 seconds.

但我們在這裡寫成時間

So time is really a change in seconds.

在三秒內改變

But we'll just go with time right here, or just with a t.

如果你看一下秒針

So what's our change in velocity?

從五秒開始到第八秒停止

So our final velocity is 60 miles per hour.

總共花費三秒

And our original velocity was 0 miles per hour.

所以時間實際上是過多少秒

So it's 60 minus 0 miles per hour.

讓我們用"t"來代替時間

And then, what is our time?

所以速度變化量是多少

What is our time over here?

我們最終速度是60mph

Well, our time is, or we could even say our change in time,

我們最終速度是60mph

our change in time is 3 seconds.

起始速度是0mph

So this gives us 20 miles per hour, per second.

所以是(60-0)mph

Let me write this down.

那我們的時間呢

So this becomes, this top part is 60.

這裡的時間是多少

60 divided by 3 is 20.

或者說 改變的時間是

So we get 20.

改變的時間是3秒

But then the units are little bit strange.

改變的時間是3秒

We have miles.

所以得到每秒20mph

Instead of writing MPH, I'm going to write miles per hour.

我把這寫下來

That's the same thing as MPH.

這變成上面的60

And then we also, in the denominator, right over here,

60/3=20

have seconds.

所以得到20

Which is a little bit strange.

但單位有一點奇怪

And as you'll see, the units for acceleration

我們有英里

do seem a little bit strange.

代替成mph我要寫成每小時英里

But if we think it through, it actually

這和mph是相同的

might make a little bit of sense.

在分母的地方

So miles per hour.

有秒

And then we could either put seconds like this,

這有一點奇怪

or we could write per second.

就你所看到的 加速的單位

And let's just think about what this is saying,

看起來有點奇怪

and then we could get it all into seconds,

但如果想一想他的運動過程

or we could all get into hours, whatever we like.

這其實說得通

This is saying that every second, this Porsche 911

所以mph

can increase its velocity by 20 miles per hour.

我們也可以把秒寫成這樣

So its acceleration is 20 miles per hour, per second.

或是寫成每秒

And actually, we should include the direction,

我們來想剛才說的東西

because we're talking about vector quantities.

我們都可以用秒

So this is to the east.

或者小時哪個都行

So this is east, and then this is east right over here--

保時捷911每秒

just so we make sure that we're dealing with vectors.

可以將速度增加20mph

You're giving it a direction, due east.

所以加速度是20miles/hr*s

So every second it can increase in velocity

實際上 我們要加上方向

by 20 miles per hour.

因為我們討論的是向量

So hopefully, with the way I'm saying it,

所以這是向東

it makes a little bit of sense.

所以這是向東

20 miles per hour, per second.

我們可以確定的是我們是在講向量

That's exactly what this is talking about.

你給定一個方向 向東

Now we could also write it like this.

所以每秒他的速度增加

This is the same thing as 20 miles per hour,

20mph

because if you take something and you

希望我所說的可以

divide by seconds, that's the same thing as multiplying it

讓你有感覺

by 1 over seconds.

20miles/hr*s

So that's miles per hour-seconds.

這就是我們討論的

And although this is correct, to me

我們也可以寫下

this makes a little less intuitive sense.

每小時20英里

This one literally says it.

因為當你有一些東西然後你

Every second, it's increasing in velocity by 20 miles per hour.

除以秒 相當於每1秒

20 miles per hour increase in velocity per second.

增加多少

So that kind of makes sense to me.

那是 miles/hr*s

Here it's saying 20 miles per hour-seconds.

那是 miles/hr*s

So once again, it's not as intuitive.

雖然是正確的 但對我來說

But we can make this so it's all in one unit of time,

少一點直覺的感覺

although you don't really have to.

從字面上來說

You can change this so that you get rid

每秒速度增加20mph

of maybe the hours in the denominator.

每秒速度增加20mph

And the best way to get rid of an hour in the denominator,

這對我來說比較有感覺

is by multiplying it by something

在這裡說20miles/hr*s

that has hours in the numerator.

同樣的 我沒感覺

So hour and seconds.

我們可以讓時間的單位一致

And here, the smaller unit is the seconds.

雖然你不必

So it's 3,600 seconds for every 1 hour.

你可以把

Or 1 hour is equal to 3,600 seconds.

分母上的小時消掉

Or 1/3600 of an hour per second.

最好去掉分母上小時的方法是

All of those are legitimate ways to interpret

乘以一個分子上有小時的數

this thing in magenta right over here.

乘以一個分子上有小時的數

And then you multiply, do a little dimensional analysis.

所以小時和秒

Hour cancels with hour.

在這裡比較小的單位是秒

And then will be equal to 20/3600.

一小時有3600秒

20/3600 miles per seconds times seconds.

或一小時等於3600秒

Or we could say, miles-- let me write

或一小時的1/3600是1秒

it this way-- miles per seconds times seconds.

所以這些洋紅色的字都是相同的意思

Or we could say, miles per second-- I

所以這些洋紅色的字都是相同的意思

want to do that in another color-- miles

然後相乘 做一點單位的分析

per second squared.

小時跟小時消掉

And we can simplify this a little bit.

會等於20/3600

Divide the numerator and denominator by 10.

20/3600 miles/s*s

You get 2/360.

或者說 讓我寫下英里

Or you could get, this is the same thing

miles/s*s

as, 1/180 miles per second squared.

或者說每秒英里

And I'll just abbreviate it like that.

我換另一個顏色

And once again, this 1/180 of a mile, how much is that?

每秒平方分之英里

You might want to convert to feet.

每秒平方分之英里

But the whole point in here is, I just

我們可以化簡一下

wanted to show you that, well, one, how

讓分子和分母都除以10

do you calculate acceleration?

得到2/360

And give you a little bit of a sense of what it means.

或者和1/180一樣

And once again, this right here, when

1/180 miles/s^2

you have seconds squared in the bottom of your units,

我這樣縮寫

it doesn't make a ton of sense.

同樣的 1/180英里 是多少?

But we can rewrite it like this up here.

你可能想用英尺表示

This is 1/180 miles per second.

但重點是

And then we divide by seconds again, per second.

我只想給你們知道

Or maybe I can write like this, per second,

怎麼計算加速度

where this whole thing is the numerator.

讓你們體會它的意義

So this makes a little bit more sense

同樣的在這

from an acceleration point of view.

當你分母的單位是s*s

1/180 miles per second, per second.

那沒有感覺

Every second, this Porsche 911 is

我們寫成上面那樣

going to go 1/180 of a mile per second faster.

1/180 miles/s

And actually, it's probably more intuitive

再除以時間(每秒)

to stick to the miles per hour, because that's something

或者可以寫成這樣

that we have a little bit more sense on.

把整個寫成分數

And another way to visualize it.

這樣從加速度的觀點來看

If you were to be driving that Porsche,

比較有感覺

and you were to look at the speedometer for that Porsche,

1/180 miles/s/s

and if the acceleration was constant--

每秒 保時捷911

it's actually not going to be completely constant--

增加1/180 miles/s

and if you look at speedometer-- let me draw it.

實際上 如果用每小時英里

So this would be 10, 20, 30, 40, 50, 60.

會更直觀

This is probably not what the speedometer for a Porsche

因為我們對這個比較有感覺

looks like.

另一種方式去想像

This is probably more analogous to a small four cylinder car's

如果你們開著保時捷

spectrometer.

看著保時捷的車速表

I suspect the Porsche's speedometer goes much

如果加速度是固定的

beyond 60 miles per hour.

事實上不會完全不變

But what you would see for something

如果你看著車速表 讓我來畫一下

accelerating this fast is, right when you're starting,

應該是10 20 30 40 50 60

the speedometer would be right there.

這可能像是保時捷的車速表

And that every second it would be 20 miles per hour faster.

這可能像是保時捷的車速表

So after a second the speedometer

這可能類似四汽缸汽車

would have moved this far.

的車速表

After another second the speedometer

我猜保時捷的車速表

would have moved this far.

會超過60mph

And then after another second the speedometer

你可能會看到

would have moved that far.

如果一輛車加速度那麼大，當你開始時

And the entire time you would have kind of

車速表在這

been pasted to the back of your seat.

每過一秒應該就變快20mph