Exponential growth: How folding paper can get you to the Moon

Kako možemo doći do Mjeseca savijajući papir

6,284,737 views

2012-04-19 ・ TED-Ed


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Exponential growth: How folding paper can get you to the Moon

Kako možemo doći do Mjeseca savijajući papir

6,284,737 views ・ 2012-04-19

TED-Ed


Dvaput kliknite na engleske titlove ispod za reprodukciju videozapisa.

Prevoditelj: Ivan Stamenković Recezent: Tilen Pigac - EFZG
(Glazba)
Koliko puta možete presaviti komad papira?
00:14
How many times can you fold a piece of paper?
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Pretpostavimo da imamo jako fini papir,
00:17
Assume that one had a piece of paper that was very fine,
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kao onaj na kojem se tiska Biblija.
00:20
like the kind they typically use to print the Bible.
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U stvarnosti, čini se kao komad svile.
00:24
In reality, it seems like a piece of silk.
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Da bi kvalificirali ove ideje,
00:28
To qualify these ideas,
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00:30
let's say you have a paper
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recimo da imate papir koji je debeo tisućinu centimetra.
00:31
that's one-thousandth of a centimeter in thickness.
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To je 10 na minus treću centimetara,
00:35
That is 10 to the power of minus three centimeters,
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00:38
which equals .001 centimeters.
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što je jednako .001 centimetara.
Pretpostavimo da imate veliki komad papira,
00:43
Let's also assume that you have a big piece of paper,
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kao što je stranica iz novina.
00:46
like a page out of the newspaper.
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00:48
Now we begin to fold it in half.
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Sad ćemo ga presavinuti na pola.
Što mislite koliko ga puta tako možemo presavinuti?
00:52
How many times do you think it could be folded like that?
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00:55
And another question:
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I još jedno pitanje:
00:57
If you could fold the paper over and over, as many times as you wish,
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ako biste mogli presaviti papir uvijek iznova, koliko god puta želite,
01:01
say 30 times,
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recimo 30 puta, što mislite koja bi debljina takvog papira bila?
01:03
what would you imagine the thickness of the paper would be then?
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Prije nego nastavite,
01:07
Before you move on,
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01:08
I encourage you to actually think about a possible answer to this question.
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ohrabrujem vas da razmislite o mogućem odgovoru na ovo pitanje.
OK. Kad smo savili papir jednom,
01:14
OK.
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01:15
After we have folded the paper once,
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sada je debeo dvije tisućine centimetra.
01:17
it is now two thousandths of a centimeter in thickness.
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Ako ga još jednom presavijemo po pola,
01:21
If we fold it in half once again,
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01:23
the paper will become four thousandths of a centimeter.
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postat će četiri tisućine centimetra.
01:27
With every fold we make, the paper doubles in thickness.
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Sa svakim savijanjem, papir postaje dvostruko deblji.
I ako ga nastavimo savijati uvijek iznova,
01:32
And if we continue to fold it again and again,
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uvijek na pola, mogli bi se suočiti sa sljedećom situacijom
01:35
always in half, we would confront the following situation
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01:38
after 10 folds.
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nakon 10 presavijanja.
01:40
Two to the power of 10,
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Dva na desetu potenciju,
01:42
meaning that you multiply two by itself 10 times,
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što znači da pomnožite 2 sa samim sobom 10 puta,
je tisuću i 24 tisućine centimetra,
01:46
is one thousand and 24 thousandths of a centimeter,
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01:50
which is a little bit over one centimeter.
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što je nešto malo preko jednog centimetra.
Pretpostavimo da nastavljamo presavijati papir na pola.
01:54
Assume we continue folding the paper in half.
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Što će se onda dogoditi?
01:57
What will happen then?
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01:59
If we fold it 17 times,
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Ako ga presavijemo 17 puta,
02:01
we'll get a thickness of two to the power of 17,
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dobit ćemo debljinu od 2 na sedamnaestu,
02:04
which is 131 centimeters,
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što je 131 centimetar,
02:07
and that equals just over four feet.
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i to je nešto preko četiri stope.
02:10
If we were able to fold it 25 times,
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Ako ga uspijemo presaviti 25 puta,
02:13
then we would get two to the power of 25,
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dobili bismo dva na dvadesetpetu,
02:16
which is 33,554 centimeters,
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što je 33,554 centimetra,
nešto preko 1.100 stopa.
02:21
just over 1,100 feet.
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Tako bi postao visok otprilike kao Empire State Building.
02:24
That would make it almost as tall as the Empire State Building.
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Vrijedi ovdje zastati i razmisliti na trenutak.
02:29
It's worthwhile to stop here and reflect for a moment.
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02:32
Folding a paper in half, even a paper as fine as that of the Bible,
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Presaviti papir na pola, makar i papir tako fin kao onaj iz Bibilije,
02:37
25 times would give us a paper almost a quarter of a mile.
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25 puta bi nam dao četvrt milje papira.
02:42
What do we learn?
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Što smo naučili?
02:43
This type of growth is called exponential growth,
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Ovakav rast zove se eksponencijalni rast,
02:47
and as you see, just by folding a paper
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i kao što vidite, slagajući papir
02:50
we can go very far, but very fast too.
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možemo otići vrlo daleko, ali i vrlo brzo.
Da sažmemo, ako savijemo papir
02:54
Summarizing, if we fold a paper 25 times,
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25 puta, debljina je četvrt milje.
02:58
the thickness is almost a quarter of a mile.
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Trideset puta, debljina je 6,5 milja,
03:02
30 times, the thickness reaches 6.5 miles,
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što je otprilike visina na kojoj lete avioni.
03:06
which is about the average height that planes fly.
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03:08
40 times, the thickness is nearly 7,000 miles,
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40 puta, gotovo 7.000 milja,
03:13
or the average GPS satellite's orbit.
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ili prosječna orbita GPS satelita.
03:15
48 times, the thickness is way over one million miles.
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Četrdesetosam puta, debljina je daleko preko milijun milja.
Ako zamislite udaljenost Zemlje i Mjeseca
03:20
Now, if you think that the distance between the Earth and the Moon
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koja je manja od 250.000 milja,
03:24
is less than 250,000 miles,
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ako počnemo sa komadom papira iz Biblije
03:27
then starting with a piece of Bible paper
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03:29
and folding it 45 times, we get to the Moon.
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i savijemo ga 45 puta, možemo doći do Mjeseca.
I ako ga presavijemo još jednom,
03:34
And if we double it one more time,
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vratimo se natrag na Zemlju.
03:36
we get back to Earth.
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Lekcija: Adrian Paenza Naracija: Adrian Paenza Animacija: TED-ED Tim
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