
Bulgarian: 
Нека сега приложим
формулата,
която намерихме
в предишното видео,
понякога позната като формула (правило)
на Айнщайн за събиране на скоростите.
И ще видим, че това
е доста хубаво нещо.
Това, отново, съм аз,
носещ се в Космоса.
Отправната ми система
е просто s отправната система.
Да кажем, че приятелката ми тук
се носи надалеч от мен,
нейната скорост –
нека дадем
някои числа.
Да кажем, че тя се носи
надалеч от мен
със скорост от, да кажем,
0,7с,
7 десети от
скоростта на светлината,
или 70% от
скоростта на светлината.
И да кажем, че този
трети човек тук
се носи към мен –
нека променя посоката,
вектора на скоростта.
Нека изтрия това.
Това не изглежда
чисто черно,
но изглежда добре изтрито,
така че ето.
И нека поставя вектора на скоростта
в другата посока.
Векторът на скоростта –
този човек
се носи към мен
с, да кажем, u ще е
равно на –
той лети към мен,

English: 
- [Voiceover] Now let's apply the formula
we came up with the last video
sometimes known as the Einstein
velocity addition formula.
And we'll see that it's
a pretty neat thing.
So let's say, so this is once
again me floating in space,
my frame of reference is just
the s frame of reference,
Let's say my friend over here,
she's flying away from me, her velocity,
let's give it some numbers.
Let's say she's flying away from me
with a velocity of, let's say,
0.7c, so seven tenths
of the speed of light
or 70% of the speed of light.
And let's say this third
character right over here,
let's say they're flying towards me,
so let me change the
direction, the velocity vector.
So let me erase that.
Gee, that doesn't look like a pure black
but it looks fairly
erased, so there you go.
And let me give the velocity
vector in the other direction.
So the velocity vector,
this character is flying towards me
at let's say, so u is
going to be equal to,
so it's flying towards me

Thai: 
ตอนนี้ลองใช้สูตร
ที่เราได้จากวิดีโอที่แล้ว
บางครั้งมันเรียกว่า 
สูตรการบวกความเร็วของไอน์สไตน์
และเราจะเห็นว่ามันเท่ดี
สมมุติว่านี่คือผมลอยอยู่ในอวกาศ
กรอบอ้างอิงของผมคือกรอบอ้างอิง S
สมมุติว่าเพื่อนผมตรงนี้
เธอกำลังบินจากผมไป ความเร็วของเธอ
ลองกำหนดตัวเลขขึ้นมา
สมมุติว่าเธอบินจากผม
ด้วยความเร็ว สมมุติว่า
0.7c เป็น 7/10 ของอัตราเร็วแสง
หรือ 70% ของอัตราเร็วแสง
และสมมุติว่าคนที่สามอยู่ตรงนี้
สมมุติว่าเขาเดินทางเข้าหาผม
ขอผมเปลี่ยนทิศ เวกเตอร์ความเร็ว
ขอผมลบนะ
มันดูไม่เหมือนสีดำสนิทเลย
มันดูเหมือนมีรอยลบ ได้แล้ว
ขอผมกำหนดเวกเตอร์ความเร็วอีกทิศหนึ่ง
เวกเตอร์ความเร็ว
นายคนนี้จะบินเข้าหาผม
ด้วยความเร็ว สมมุติว่า u จะเท่ากับ
เขาเข้าหาผม

Korean: 
전 영상에서 유도한 공식인
아인슈타인 속도 합 공식을
실제로 적용해봅시다
해보면 이 공식이 꽤 쓸만하다는 걸 알 수 있습니다
제가 다시 한번 우주에 떠 있다고 가정하면
저의 좌표계는 이 s 좌표계겠죠
여기 제 친구가 있는데
그녀는 저로부터 멀리 날아가고 있습니다
그 속도에 특정한 값을 지정해 줍시다
그녀가 저로부터
0.7c, 즉
빛의 속도의 10분의 7배 또는
빛의 속도의 70%로 날아가고 있다고 합시다
그리고 여기 세번째 인물이 있고
저를 향해서 날아온다고 합시다
속도 벡터, 즉 방향을 바꿔 봅시다
이걸 지워보죠
완벽히 까맣진 않아 보이지만
어느 정도 지워진 것 같으니 그냥 진행할게요
이제 이 속도 벡터의 방향을 바꿔 봅시다
여기 속도 벡터가 있고
이 캐릭터는 저를 향해서 오니까
u는 이것과 동일할 것이고
저를 향해서 오니까

Thai: 
x ลดลงเมื่อเวลาผ่านไป
สมมุติว่ามันคือลบ
ลบ 0.5c
บินเข้าผมด้วยอัตราครึ่งหนึ่งของแสง
ย้ำอีกครั้ง ความเร็วทั้งสองนี้
ให้มาในกรอบอ้างอิงของผม
ทีนี้ ถ้าเราอยู่ในโลกแบบนิวตัน
หรือถ้าเราใช้การแปลงกาลิเลียน
และวัตถุนี้ปล่อยออกมาบนทางด่วน
ถ้าผมอยู่บนทางด่วน และถ้าผมวิ่งไปทิศหนึ่ง
อัตรา 70 ไมล์ต่อชั่วโมง และมีคนเข้ามาหาผม
ที่ 50 ไมล์ต่อชั่วโมง มันจะดูเหมือนว่า
เขาเข้าหาผมด้วยอัตรา 120 ไมล์ต่อชั่วโมง
เราแค่บวกสองค่านี้เข้าด้วยกัน
แล้ว ถ้าเราอยู่ในโลกแบบนิวตัน
คนนี้บินด้วยอัตรา 0.7c
และคนนี้บินในทิศนี้ที่ 0.5c
ในโลกแบบนิวตัน คุณก็บอกว่า โอเค
จากมุมมองนี้ มันดูเหมือนว่า
เจ้าตัวสีฟ้านี้กำลังเข้ามาด้วย
อัตรา 0.7c บวก 0.5c
หรือ 1.2 คูณอัตราเร็วแสง
ซึ่งเรารู้ว่ามันขัดกับกฎของเอกภพ

Bulgarian: 
х намалява
с времето,
да кажем, че е
отрицателно,
-0,5с –
лети към мен с половината
от скоростта на светлината.
И, отново, тези
две скорости
са дадени в моята
отправна система.
Ако бяхме в един
Нютонов свят
или ако използвахме
Галилеевата трансформация
и това беше като
нещо на магистралата...
Ако съм на една магистрала
и се движа в една посока
със 70 мили в час,
а някой идва към мен с 50 мили в час,
на мен ще ми изглежда
сякаш той се движи към мен
със 120 мили в час.
Бихме събрали тези двете.
И ако бяхме в
Нютонов свят,
този човек лети
с 0,7с,
а този човек
лети насам с 0,5с,
в един Нютонов свят
ще кажеш:
"От тази гледна точка
ще изглежда сякаш
този син човек
се доближава с
0,7с + 0,5с,
тоест 1,2 пъти
скоростта на светлината,
което, както знаем, нарушава
законите на Вселената."

English: 
x is decreasing with time,
let's say it's negative,
- 0.5c,
so flying towards me at
half the speed of light.
And once again, both of these velocities
are given in my frame of reference.
Now if we were in a Newtonian world
or if we used the
Galilean transformations,
and this is released as
something on your highway.
If I'm on a highway and if
I'm going in one direction
at 70 miles an hour, and
someone is coming towards me
at 50 miles an hour, it'll look to me
like they are coming towards
me at 120 miles per hour.
We would essentially
add these two together.
And so, if we were in a Newtonian world,
this person is flying at 0.7c
and this person is
flying this way at 0.5c,
you would, in a Newtonian world, say okay
from this point of view it will look like
this blue character is approaching
at 0.7c plus 0.5c
or at 1.2 times the speed of light
which we know violates
the laws of the universe.

Korean: 
x는 시간에 따라서 감소합니다
u는 그럼 음수이니
-0.5c정도로 하죠
빛의 속도의 반으로 저를 향해서 날아오는 것이죠
다시 한번 말하지만 이 속도들은
저의 좌표계에서 주어진 것입니다
만약 우리가 뉴턴역학적 세상에 살고 있거나
갈릴레이 변환을 사용하는 상태에서
비슷한 상황이 고속도로에서 펼쳐졌다면
제가 한 방향으로 70mph로 가고
누군가가 50mph로 저를 향해서 오게 되는데
그렇다면 저에게는
그가 저 쪽으로 120mph로 오는 것처럼 보일 것입니다
이 두 속도를 더하는 것이죠
원래 상황이 만약 뉴턴역학적 세상에서 일어났다면
즉 이 사람은 0.7c로 날고 있고
저 사람은 이쪽으로 0.5c로 날아오고 있다면
뉴턴역학적으로 보면
이 관점에서 파란색 물체가
0.7c + 0.5c의 속도
또는 빛의 속도의 1.2배로 접근하고 있다는 것에
동의하실 겁니다
물론 우리는 이게 우주의 법칙에 위배됨을 알고 있습니다

Korean: 
그리고 이때 이 공식이 정말로 빛을 발합니다
아예 이 우주선이
헷갈리지 않도록 왼쪽을 가리키도록 합시다
그럼 이 우주선은
왼쪽 방향으로 이렇게 진행하게 됩니다
되짚어 보자면 저를 향해서 오는 우주선의 속도는
빛의 속도의 절반입니다
마이너스 부호를 놓은 이유는
저를 향해서 온다는 것을 나타내기 위해섭니다
또는 x가 시간에 따라서 줄어든다는 것이죠
운이 좋게도 전 영상에서 아인슈타인 합공식을
유도했으니 이번에는 적용을 해 봅시다
이 친구의 좌표계 즉
s' 좌표계에서는
이 캐릭터의 속도 또는
t의 변화 분에 x'의 변화는
이렇게 될 것입니다
여기서 이 식은 무엇일까요?
u는 제 좌표계에서의 파랑이의 속도 또는
-0.5c 이고
v는 제 친구의 좌표계의 속도입니다
우리가 이 속에서 속도를 구하려고 하는
좌표계죠
그럼 이건 0.7c일텐데

English: 
And that's where this
formula is really handy.
Actually let me even
make the spaceship point
in this direction just
so we don't get confused.
So the spaceship is going
in that direction just like that.
And once again, the
magnitude of the velocity
coming towards me is
half the speed of light.
I put the negative there to show
that it's coming towards me,
that x is decreasing over time.
Well, lucky for us in the
last video, we came up with
this Einstein velocity addition
formula, so let's apply it.
From this friend's frame of reference,
the s-prime frame of reference,
the velocity of this character
which is change in
x-prime over change in t
is going to be this.
So what do we have here?
So u is the blue velocity
in my frame of reference,
So that is -0.5c.
v is my friend's frame of reference,
the frame of reference
that we're trying to get the velocity in.
So that's 0.7c,

Bulgarian: 
И тук тази формула
е полезна.
Нека направя космическия кораб
да сочи в тази посока,
за да не се
объркаме.
Космическият кораб
се движи в тази посока, ето така.
И, отново, големината
на скоростта му,
като идва към мен,
е половината от скоростта на светлината.
Поставям "-" тук, за да покажа,
че идва към мен,
че х намалява
с времето.
За наше щастие
в последното видео намерихме
формулата на Айнщай за събиране на скоростта,
така че нека я приложим.
От отправната система
на този приятел,
s' отправната система,
скоростта на този човек,
която е промяната в x'
върху промяната в t,
ще е това.
Какво имаме тук?
u е синята скорост в
моята отправна система.
Тоест това е -0,5с.
v е отправната система
на приятелката ми,
отправната система,
в която опитваме
да намерим скоростта.
Това е 0,7с –

Thai: 
และนั่นคือจุดที่สูตรนี้มีประโยชน์จริงๆ
ขอผมวาดยานชี้
ในทิศนี้ เราจะได้ไม่งง
ยานจะไป
ในทิศนั้นอย่างนั้น
ย้ำอีกครั้ง ขนาดความเร็ว
ที่เข้าหาผมเป็นครึ่งหนึ่งของอัตราเร็วแสง
ผมใส่ลบตรงนั้นเพื่อแสดงว่า
มันเคลื่อนที่เข้าหาผม
x ลดลงเมื่อเวลาผ่านไป
โชคดีของเราในวิดีโอที่แล้ว เราได้
ตั้งสูตรการบวกความเร็วไอน์สไตน์ขึ้น 
ลองใช้กัน
จากกรอบอ้างอิงของเพื่อนผม
กรอบอ้างอิง S ไพรม์
ความเร็วของตัวนี้
ซึ่งก็คือการเปลี่ยนแปลงของ x ไพรม์
ส่วนการเปลี่ยนแปลงของ t ไพรม์
จะเป็นแบบนี้
แล้วเรามีอะไรตรงนี้?
u คือความเร็วสีฟ้าในกรอบอ้างอิงของผม
นั่นก็คือ ลบ 0.5c
v คือกรอบอ้างอิงของเพื่อนผม
กรอบอ้างอิง
ที่เราพยายามเทียบความเร็ว
นั่นคือ 0.7c

Thai: 
ขอผมใช้สีนั้นนะ
นั่นคือ 0.7c
แล้ว ข้างบน uv ส่วน c กำลังสอง
เหมือนเดิม u คือ
ลบ 0.5c
ผมรู้ว่าผมเขียนเล็กจริงๆ
แล้ว v คือ
0.7c
ค่านี้จะเท่ากับอะไร?
ตัวเศษคือลบ 0.5c
ลบ 0.7c
มันจะเป็นลบ
ตัวเศษ ตรงนี้
จะเท่ากับ
ลบ 1.2c
และนี่คือความร้อนที่คุณคาด
ถ้าเราคิดในโลกแบบนิวตัน
คนคนนี้จะบอกว่า เฮ้ คนนี้ดูเหมือนว่า
จะเข้าหาผมด้วยอัตรา
1.2 คูณอัตราเร็วแสง
นั่นคือสิ่งที่การแปลงแบบกาลิเลียนบอกเรา
แต่โชคดี เรามีตัวนี้ข้างล่าง

English: 
so let me do it in that color.
So that is 0.7c.
And then, on top of uv over c-squared,
so once again, u is
- 0.5c.
I know I wrote it really small.
And then, v is
0.7c.
So what is this going to be equal to?
Well, the numerator -0.5c
minus 0.7c,
that's going to be negative.
Our numerator, right over here
is going to be
- 1.2c.
And this is the velocity
that you would expect
if we were dealing in a Newtonian world.
This person says, "Hey, this person looks
"like they're coming towards me
"at 1.2 times the speed of light."
That's what our Galilean
transformations would give us.
But luckily, we have all
this business at the bottom

Korean: 
색깔로 하도록 하죠
0.7c가 됩니다
다음으로 c제곱 분에 uv를 봅시다
다시 한번 u는
-0.5c입니다
너무 조그만하게 썼네요
v는
0.7c겠죠
그럼 이게 총 얼마가 될까요?
분자인 -0.5c 빼기
0.7c는
음수가 되겠네요
여기 분자의 값은
바로
-1.2c가 될 것입니다
이건 우리가 뉴턴역학적 세상에서
문제를 다루고 있을 때 예상되는 속도입니다
이 사람은
'오 저사람이 나에게 빛의 속도의
1.2배로 오는 것 같아!' 라고 하겠죠
갈릴레이 변환을 사용하면 그렇게 됩니다
운이 좋게도 우리는 빛의 속도의 절대성을

Bulgarian: 
нека го направя
в този цвят.
Това е 0,7с.
И после 
uv/c^2 –
отново, u е -0,5с.
Знам, че го записах
много мъничко.
А после v е 0,7с.
На какво ще е
равно това?
Числителят -0,5с - 0,7с –
това ще е
отрицателно.
Числителят ни тук
ще е -1,2с.
Това е скоростта,
която ще очакваш,
ако си имаме работа
с Нютонов свят.
Този човек казва:
"Хей, изглежда този човек
идва към мен с
1,2 пъти скоростта на светлината."
Това ще ни даде нашата
Галилеева трансформация.
Но за щастие имаме
всичко това отдолу,

Korean: 
위반하지 않도록 도와주는
분자가 있습니다
빛의 속도의 절대성이란
빛의 속도보다 빠르게 움직일 수 있는 물체는 없다는 것이죠
분자를 보면
마이너스 0.5c 곱하기
0.7c가 있죠
0.5 곱하게 0.7은
0.3이 될 겁니다
이제 0.3, 또는 0.30이라고 할 수도 있지만
그냥 0.3 c제곱이라고 쓸게요
물론 음수겠죠
음수 곱하기 양수이니
여기 마이너스 부호가 올텐데
음수를 빼는 상황이니
양수가 될 것입니다
또 c제곱으로 나누어 주니까
이 역시 약분됩니다
여기서 보면 지금 우리는 -1.2를
1.2보다 약간 큰 수로 나누고 있습니다
정리를 해 보면
1.3 분의
-1.2가 됩니다
1.3입니다

Bulgarian: 
което не ни позволява
да нарушим абсолютността
на скоростта на светлината,
тази идея, че нищо не може да се движи
със скорост по-голяма
от скоростта
на светлината.
Понеже в знаменателя
ще получим 1 минус
0,5с по 0,7с.
Да видим, 0,5*0,7
ще е 0,3.
0,3 или мога да кажа 0,30,
ако исках,
но просто ще запиша
0,3с^2.
Това ще е отрицателно.
Имаме "-" по "+".
Мога да поставя
този "-" тук,
но ако изваждам
отрицателна стойност,
това просто
ще е положително.
И просто ще разделя
на c^2.
Това се съкращава.
И, забележи,
ще делим -1,2с на нещо,
което е малко
по-голямо от 1,2.
Това ще е -1,2с
върху 1,3.

English: 
that keeps us from
violating the absoluteness
of the velocity of light,
this notion that nothing can travel faster
than the speed of light.
Because in our denominator,
we're going to get 1 minus 0.5c
times 0.7c.
Let's see, 0.5 times 0.7
is going to be 0.3.
So 0.3, I could say 0.30 if I liked,
but I'll just write 0.3c squared.
It'll be a negative.
I have a negative times a positive.
So I could put the negative there
but if I'm subtracting a negative,
that's just going to be a positive.
And I'm just going to divide by c-squared.
And so, that cancels out.
And so notice, we're going to be dividing
- 1.2c by something that's
slightly larger than 1.2.
So this is going to be
- 1.2c
over 1.3.
1.3

Thai: 
ที่ทำให้เราไม่ขัดกับ
ความเร็วแสง
หลักการที่ว่าไม่มีอะไรเคลื่อนที่เร็ว
กว่าอัตราเร็วแสง
เพราะในตัวส่วน
เราจะได้ 1 ลบ 0.5c
คูณ  0.7c
ลองดู 0.5 คูณ 0.7
จะเท่ากับ 0.3
0.3 ผมบอกว่า 0.30 ก็ได้ถ้าต้องการ
ผมจะเขียนว่า 0.3c กำลังสองแล้วกัน
มันจะเป็นลบ
ผมมีลบคูณบวก
ผมใส่ลบตรงนั้นได้
แต่ถ้าผมลบค่าลบ
มันจะเป็นบวก
และผมจะหารด้วย c กำลังสอง
แล้วพวกมันตัดกัน
สังเกตว่า เราจะหาร
ลบ 1.2c ด้วยค่าที่มากกว่า 1.2 เล็กน้อย
มันจะเท่ากับ
ลบ 1.2c
ส่วน 1.3
1.3

Thai: 
โชคดีของเรา มันจะน้อยกว่า c
ค่าสัมบูรณ์นี้จะน้อยกว่า c
มันจะเท่ากับ ลองเอาเครื่องคิดเลขออกมา
ถ้าเรามี
1.2 หารด้วย 1.3
ค่านี้จะประมาณได้ 0.92
ค่านี้จะประมาณได้
ลบ 0.92c
ซึ่งเจ๋งดี
มันได้
ขอผมเขียนตรงนี้นะ
เดลต้า x ไพรม์ส่วนเดลต้า t ไพรม์นี้เท่ากับ
ลบ 0.92c
มันสมเหตุสมผลจากมุมมองของเพื่อนผม
ยานนี้ดูเหมือนจะเข้าหาเธอ
เร็วกว่าจากกรอบอ้างอิงของผม
ดูเหมือนว่ามันเข้าหาผมในกรอบอ้างอิงของผม
มันดูเหมือนว่า ยานเข้าหาผม
ด้วยอัตราครึ่งหนึ่งของแสง
ในกรอบอ้างอิงของเธอ
ดูเหมือนยานมาด้วยอัตรา 
0.92 คูณอัตราเร็วแสง

English: 
And so, lucky for us, this
is going to be less than c,
or this absolute value is
going to be less than c.
So it's going to be, let's
get a calculator out.
So, if we have
1.2 divided by 1.3,
this is equal to approximately 0.92.
So this is going to be approximately
- 0.92c
which is cool.
It kind of goes with our,
let me write it over here.
So this delta x prime over delta t prime
is -0.92c.
So it makes sense that from our
friend's frame of reference,
this ship will look like
it's approaching her
faster than from my frame of reference.
It looks like it's approaching
me in my frame of reference,
it looks like it's coming to
me at half the speed of light.
In her frame of reference,
it looks like it's coming at
0.92 times the speed of light.

Korean: 
운이 좋게도 이 값은 c보다 작습니다
또는 절댓값이 c보다 작다고 할 수 있습니다
계산기를 사용해서 계산을 해 봅시다
만약 여기에
1.2 나누기 1.3이 오게 되면
약 0.92가 되겠네요
그럼 이 식의 값이 대략
-0.92가 된다는 의미입니다
꽤 멋집니다
뭔가 말이 되네요
여기에 써보겠습니다
그럼 이 델타t' 분의 델타x' 의 값이
-0.92c가 된다는 겁니다
즉 우리 친구의 좌표계에서는
이 우주선이 제 좌표계에서 보이는 것보다
빨리 접근한다는 것이 말이 됩니다
내 좌표계에서는
빛의 속도의 반으로 접근하는 것으로 보이지만
그녀의 좌표계에서는
빛의 속도의 0.92배로 오는 것으로 보입니다

Bulgarian: 
И, за наше щастие, това ще е
по-малко от с,
или абсолютната стойност
ще е по-малка от с.
Нека извадим
калкулатора.
Ако имаме
1,2 делено на 1,3
това е равно на
приблизително 0,92.
Това ще е
приблизително -0,92с,
което е добре.
Съвпада с –
нека го запиша тук.
Това делта x'
върху делта t'
е -0,92с.
Логично е, че от отправната система
на приятелката ни
космическият кораб ще изглежда
все едно я приближава по-бързо,
отколкото от моята
отправна система.
В моята отправна система
изглежда сякаш ме приближава
с половината от скоростта на светлината.
В нейната отправна система
изглежда сякаш се движи с
0,92 пъти скоростта на светлината.

Bulgarian: 
Минусът просто
уточнява посоката.
Но не нарушихме фундаменталния постулат
на специалната теория на относителността,
че нищо не може да се движи
със скорост по-голяма
от скоростта
на светлината,
че скоростта на светлината
е абсолютна.
Това е доста хубаво.
Често дори съм мислил
да направя видео игра,
където използваш това,
където нещата летят с
различни относителни скорости,
но се прилага специалната
теория на относителността.

English: 
The negative is just
specifying the direction.
But we didn't violate
the fundamental postulate
of special relativity,
that nothing can go faster
than the speed of light.
The speed of light is absolute.
So this is pretty cool.
I've often even thought about
making a video game somehow
where you leverage this,
where things are flying in
different relative velocities
but special relativity applies.

Korean: 
마이너스 부호는 방향을 나타냅니다
또한 우리는 특수상대성 이론의
근본적인 가정인 
빛의 속도보다 빠른 물체는 없다는 것을
위반하지 않았습니다
빛의 속도는 절대적입니다
꽤 멋진 결과입니다
전 이 효과를 고려해서
여러 물체들이 서로 다른
상대속도로 날고 있는데 특수상대론이 적용되는
게임을 만드는 것도 생각해봤습니다

Thai: 
เครื่องหมายลบแค่ระบุทิศ
แต่เราไม่ขัดสมมติฐานหลัก
ของสัมพัทธภาพพิเศษ ว่าไม่มีอะไรเร็ว
กว่าอัตราเร็วแสงได้
อัตราเร็วแสงเป็นปริมาณสัมบูรณ์
มันเจ๋งดี
ผมมักคิดถึงการสร้างวิดีโอเกม
ที่ใช้ประโยชน์จากเรื่องนี้
ว่าสิ่งต่างๆ บินด้วยความเร็วสัมพัทธ์ต่างๆ กัน
และนำสัมพัทธภาพพิเศษมาใช้
