Integrals: Crash Course Physics #3YouTube↗  IA↗(long shot)
Education  802.8K views 11.5K likes 387 dislikes   14 Apr 2016
CrashCourse 17.1M subscribers United States YouTube↗


Please select the problem
You must specify your email and describe the problem in detail
Describe your problem in detail if it's not obvious

Click to Play Video Click to Play Video
If you watched our last episode -- and really, if you haven’t, you should.
Ako ste pogledali našu prošlu epizodu -- i stvarno, ako niste, trebali biste.
...you now know all about derivatives, and how to use them, to describe the way an equation is changing.
...sada znate sve o derivacijama i kako ih koristiti za opis načina na koji se jednadžba mijenja.
Which means that now we can talk about the other main part of calculus -- basically,
Što znači da sada možemo pričati o drugom glavnom dijelu računa -- u biti
the inverse of derivatives, called integrals.
o onome što je suprotno od derivacija, a to se zovu integrali.
Integrals are useful because they also tell you a lot about an equation: if you plotted
Integrali su korisni jer vam također govore mnogo o jednadžbi: ako prikažete
an equation on a graph, the integral is equal to the area between the curve and the horizontal axis.
jednadžbu grafom, integral je jednak površini između krivulje i horizontalne osi.
Finding an integral is a little less straightforward than finding a derivative, but, as with derivatives,
Nalaženje integrala je malo kompliciranije nego naći derivaciju, ali, kao i kod derivacija,
there are shortcuts we can use to make things a little easier.
postoje prečaci koje možemo koristiti da si malo olakšamo stvari.
We’ll even be able to use integrals to talk about how things move -- specifically, the
Čak ćemo moći koristiti integrale kako bismo pričali o tome kako se stvari kreću -- konkretno ćemo govoriti o
equation we’ve been calling the displacement curve, and why it looks the way it does.
jednadžbi koju smo zvali krivulja pomaka i zašto ona izgleda tako kako izgleda.
So let’s get started.
Počnimo onda.
[Theme Music]
[Glazba]
Say you want to know how high your bedroom window is above the ground outside below.
Recimo da želite znati na kojoj je visini od tla vaš prozor.
But you don’t have anything to measure it with --
Ali nemate ga čime mjeriti --
just a ball, the stopwatch app on your phone, and your impressive knowledge of physics.
imate samo lopticu, štopericu na mobitelu i impresivno znanje fizike.
The force of gravity is what makes the ball fall,
Sila gravitacije je ono što tjera lopticu da pada,
so you know that its acceleration is small g 9.81 ms^2, downward.
tako da znate da je njena akceleracija malo g, 9.81 m/s^2 prema dolje.
But you’re trying to find its change in position -- how FAR it falls.
Ali pokušavate naći njenu promjenu u položaju -- koliko daleko pada.
We’ve spent a lot of time talking about the connection between the qualities of motion:
Puno smo pričali o povezanosti između sastavnica gibanja:
position, velocity, and acceleration.
o položaju, brzini i akceleraciji.
But so far, we’ve been describing that connection in a particular direction: velocity is the derivative of position
Ali do sada smo opisivali tu povezanost u određenom smjeru: brzina je derivacija položaja
-- that is, a measure of its change -- and acceleration is the derivative of velocity.
-- odnosno mjera njegove promjene -- i akceleracija je derivacija brzine.
To figure out how far the ball falls, you need to use the reverse of that connection.
Da biste shvatili koju je udaljenost loptica prešla, morate koristiti suprotno od te povezanosti.
Expressed mathematically, that means that velocity is the integral of acceleration,
Izraženo matematički, brzina je integral akceleracije,
and position is the integral of velocity.
a položaj je integral brzine.
In other words: if you draw these equation on a graph, velocity is equal to the area
Drugim riječima, ako nacrtate te jednadžbe na grafu, brzina je jednaka površini
under the acceleration curve, and position is equal to the area under the velocity curve.
ispod krivulje akceleracije, a položaj je jednak površini ispod krivulje akceleracije.
Finding that area is the tricky part.
Nalaženje te površine je ono što je nezgodno.
There are simple ways to find the area of pretty much any shape, as long as that shape
Postoje jednostavni načini nalaženja površine više manje bilo kojeg lika, dokle god je taj oblik
is made of nothing but straight lines and corners.
sačinjen samo od ravnih linija i kuteva.
And when you think about it, a curve is just a set of infinitely tiny lines.
A ako malo razmislite o tome, krivulja je samo niz beskonačno malih linija.
So the area under a curve can be divided into a set of infinitely tiny rectangles.
Tako da se površina ispod krivulje može podijeliti u niz beskonačno sitnih pravokutnika.
Integrals tell you what happens when you divide the area under a curve into those infinitely
Integrali vam govore što se dogodi kada podijelite površinu ispod krivulje u te beskonačno
tiny rectangles, take the area of each of them, and add up those areas.
sitne pravokutnike i uzmete ih sve i zbrojite.
So, how do you find that integral?
Dakle, kako naći taj integral?
Well, you start by using the fact that integrals are basically the OPPOSITE of derivatives.
Pa, počnete tako da iskoristite činjenicu da su integrali u biti suprotno od derivacija.
If you know that your velocity is equal to twice time, for example, then you know that’s the derivative of the position.
Ako znate da je vaša brzina jednaka dvaput vrijeme, na primjer, onda znate da je to derivacija položaja.
So, to find the equation for your position, you just have to look for an equation whose DERIVATIVE is 2t … like x = t^2, for example.
Pa ćete kako bi našli jednadžbu položaja, samo morati ptražiti jednadžbu čija je derivacija 2t ... to je na primjer x = t^2.
It’s kind of a roundabout way of doing things, compared to the neat equation we were able to use to find derivatives.
To je malo zaobilazni način računanja, u usporedbi sa zgodnim jednadžbama koje smo mogli koristiti kako bi našli derivacije.
But there is no tidy equation that we can use to calculate any integral we want.
Ali ne postoji uređena jednadžba za izračunati bilo koji integral.
But! As with derivatives, there ARE shortcuts for finding certain, useful ones.
Ali stanite, nemojte paničariti! Kao i kod derivacija, postoje prečaci za naći neke korisne.
For instance, you can take the power rule that we used for derivatives, and run it backward.
Na primjer, možete uzeti pravilo potencije koje smo koristili za derivacije i obrnuti ga.
Basically: you add one to the exponent, then divide the variable by that number.
U biti: dodate jedan eksponentu i zatim podijelite varijablu tim brojem.
So the integral of 2t -- which is written like this -- becomes t^2.
Dakle integral od 2t -- što pišemo ovako -- postaje t^2.
In the same way, the integral of 42t^5 is 7t^6.
Na isti način, integral od 42t^5 je 7t^6.
You can take the trigonometric derivatives that we talked about, and do those backward, too.
Možete i uzeti trigonometrijske derivacije i njih isto napraviti obratno.
The integral of cos(x) is sin(x), and so on. And the integral of e^x is just e^x.
Integral od cos(x) je sin(x) itd. I integral od e^x je samo e^x.
But there’s one complication that we haven’t talked about yet -- maybe you’ve already spotted it: constants.
Ali postoji jedna komplikacija o kojoj još nismo pričali -- možda ste je već primjetili -- konstante.
A constant is basically just a number. It can LITERALLY be a number -- like 2, or a half, or negative 4.
Konstanta je u biti samo broj. Može doslovno biti broj -- na primjer 2 ili polovina ili negativno 4.
Or it can be a placeholder for a number, like the small g, we’ve been using to represent the acceleration caused by gravity.
Ili može predstavljati broj, kao malo g koje smo koristili za akceleraciju koja je uzrokovana gravitacijom.
And constants pose a problem when it comes to integrals because: the derivative of a constant is just 0.
Konstante predstavljaju problem kod integrala jer je derivacija konstante samo 0.
A derivative is a rate of change, after all, so a constant, which by definition DOESN'T
Uostalom, derivacija je samo stopa promjene pa će derivacija pošto se po definiciji ne
change, will always have a derivative of zero.
mijenja, uvijek imati derivaciju nula.
Which means that lots of different equations - an infinite number, in fact - can all have the same derivative.
Što znači da mnogo različitih jednadžbi -- zapravo beskonačno njih -- može imati istu derivaciju.
Like, the derivative of t^2 is 2t. But you can add ANY number -- or a letter representing
Primjerice, derivacija od t^2 je 2t. Ali možete tome dodati bilo koji broj, ili slovo koje predstavlja
a number -- to it, and the derivative will STILL be 2t. So the derivative of t^2 + 1
broj, i derivacija će i dalje biti 2t. Dakle derivacija od t^2 + 1
is also 2t. And the same is true for t^2 - 7.
je isto 2t. I isto je istinito za t^2 - 7.
Which means: If you’re looking for the INTERGRAL of an equation like x = 2t, you have INFINITE
Što znači da ako tražite integral jednadžbe kao što je x = 2t, imate beskonačno mnogo
CHOICES, all of which are equally correct.
izbora i svi su jednako točni.
t^2 would work, but so would t^2 + 1 … or t^2 - 7 … or t^2 + 0.256.
t^2 bi odgovaralo, ali odgovaralo bi i t^2 + 1 ... ili t^2 - 7 ... ili t^2 + 0.256.
In these cases, we might know what the SHAPE of the integral should look like on a graph
U ovim slučajevima možemo znati kako bi oblik integrala trebao izgledati na grafu
-- like, whether it’s a straight line, or how it curves -- but we don’t know where to put it along the vertical axis.
-- tipa je li to ravna linija, ili kako je zakrivljen -- ali ne znamo gdje da ga stavimo duž vertikalne osi.
So we need to know what the constant is, in order to know where to start drawing its integral.
Dakle trebamo znati kolika je konstanta kako bismo znali gdje da počnemo crtati integral.
Whatever the constant is equal to, that’s where the curve will intersect with the vertical axis.
Čemu god je konstanta jednaka je mjesto gdje će krivulja sjeći vertikalnu os.
So t^2 would intersect at 0, but t^2 - 7 would intersect at -7. You get the idea.
Dakle t^2 će je sjeći u 0, ali t^2 - 7 će je sjeći u -7. Shvatili ste.
Mathematicians had to figure out how to get around the problem of having infinite integrals
Matematičari su morali shvatiti kako da riješe problem beskonačno mnogo integrala
to choose from, so they came up with a way to represent ALL of them: just add a C at the end of the integral.
između kojih se može birati pa su smislili način da ih sve predstave: samo se doda C na kraj integrala.
That C stands for all of the constants that we know we COULD put there.
To C predstavlja sve konstante koje znamo da tamo možemo staviti.
So if we say that the integral of 2t is t^2+ C,
Tako da ako kažemo da je integral od 2t = t^2 + C,
then we’re including t^2 + 1 and t^2 - 7 and all those other infinite options --
onda smo uključili t^2 + 1, t^2 - 7 i sve druge beskonačne opcije --
every equation whose derivative is 2t.
svaku jednadžbu čija je derivacija 2t.
But sometimes you don’t need the C at all, because you CAN figure out where your integral
Ali nekada uopće ne trebate C jer možete shvatiti gdje bi vaš integral
is supposed to be on the y axis. Like if you have what’s known as the initial value.
trebao biti na y osi. Na primjer ako imate ono što se zove početna vrijednost.
In the case of a position graph, for instance, the initial value would be where you started out,
U slučaju grafa položaja na primjer, početna vrijednost je tamo gddje ste počeli
so you’d know to draw the rest of the graph’s shape from there.
pa biste znali nacrtati ostatak grafa od tamo nadalje.
If you started at the 2 meter mark, say, and moved one meter every second, you’d put
Ako ste počeli na dva metra, recimo, i ako ste se kretali jedan metar svake sekunde, stavili bi
the graph here. But if you started at the 4 meter mark, you’d shift it up a little.
graf ovdje. Ali ako ste počeli na 4 metra, pomaknuli biste ga malo prema gore.
Basically, it gives you the point where your integral intersects the vertical axis -- which is the value of C.
U biti, to vam daje točku gdje vaš integral sjeće vertikalnu os -- to je vrijednost C.
Let’s try it, and at the same time, we might as well figure out the height of your bedroom window.
Iskušajmo to, i u isto vrijeme bismo mogli odrediti visinu na kojoj je prozor vaše sobe.
You’re standing in your room, holding a tennis ball out the window with your arm resting on the sill.
Stojite u svojoj sobi, držite lopticu za tenis vani tako da vam je ruka na prozorskoj dasci.
Now you drop the ball and start your stopwatch app at the same time.
Sada ispustite lopticu i pokrenete štopericu u isto vrijeme.
Turns out that the ball takes 1.7 seconds to hit the ground.
Ispada da loptici treba 1.7 sekundi da padne na tlo.
Like we said earlier, we know the ball’s acceleration -- 9.81 ms^2 -- and we know the time involved.
Kao što smo rekli ranije, znamo akceleraciju loptice -- 9.81 m/s^2 -- i znamo vrijeme.
Somehow, we have to get from there to the equation for the ball’s position.
Nekako trebamo odatle doći do jednadžbe položaja loptice.
So, first, let’s find its velocity -- the middle step -- by taking the integral of its acceleration.
Dakle idemo prvo naći njenu brzinu -- srednji korak -- tako što ćemo uzeti integral njene akceleracije.
Take a look at this graph of the ball’s acceleration over time.
Pogledajmo ovaj graf akceleracije loptice kroz vrijeme.
It’s just a flat line, which means that it should be pretty easy to find the area between it and the horizontal axis.
To je samo ravna linija, što znači da bi trebalo biti poprilično jednostavno naći površinu između linije i horizontalne osi.
It’s a rectangle! And the area of a rectangle is just its base times its height.
To je pravokutnik! A površina pravokutnika je samo baza puta visina.
In this case, the base is t, the amount of time the ball took to fall.
U ovom slučaju, baza je t, iznos vremena koje je trebalo loptici da padne.
And the height is a, the acceleration.
A visina je a, akceleracija.
So, the area between the acceleration graph and the horizontal axis is just (a) times (t).
Dakle, površina između grafa akceleracije i horizontalne osi je samo a puta t.
And the integral is (a) times (t), plus that constant we add -- C.
A integral je a putat plus ona konstanta koju dodajemo -- C.
For now, we need the C, because we know the general shape of the velocity graph:
Za sada trebamo C jer znamo osnovni oblik grafa brzine,
It’s a diagonal line slanted in such a way that every second, it rises by an amount equal to the acceleration.
to je dijagonalna linija takvog nagiba da se svake sekunde podigne za iznos akceleracije,
But still we don’t know where to PUT that line on the vertical axis. Not yet, anyway.
Ali ne znamo gdje da stavimo tu liniju na vertikalnoj osi. Bar ne još.
Now, we could have figured out the integral of acceleration just as easily by using the power rule:
Mogli smo naći integral akceleracije jednako lako koristeći pravilo potencije:
The acceleration, a, is a constant, but we could also say that it’s (a) x (t^0) --
Akceleracija, a, je konstanta, ali mogli smo reći i da je at^0
because anything raised to the power 0 is just 1.
jer je bilo što na 0 samo 1.
So, according to the power rule, the integral of acceleration -- which is the velocity -- would
Dakle po pravilu potencije, integral akceleracije, brzina, bi
be equal to the acceleration, multiplied by time, plus C.
bio jednak akceleraciji pomnoženoj vremenom plus C.
That’s the same answer we got earlier!
To je isti odgovor koji smo dobili ranije!
Now, here’s where the initial value comes in. The velocity graph tells you what the
Sada evo gdje početna vrijednost nastupa. Graf brzine vam govori kolika
velocity is for each moment in time. But we had to add the C, because we didn’t know
je brzina za svaki trenutak u vremenu. Ali morali smo dodati C jer nismo znali
where to place it on the vertical axis -- when time equals zero.
gdje da ga smjestimo na vertikalnoj osi -- kada je vrijeme jednako nula.
So, the integral of the acceleration COULD have just been (acceleration) x (time), or
Dakle integral akceleracije bi mogao biti samo akceleracija puta vrijeme, odnosno
(a)(t). But it could also have been (at) + 4. Or (at) - 6.
at, ali također je mogao biti at + 4, ili at - 6.
So we put a C in the integral instead, to represent all those options.
Zato stavljamo C u integral umjesto toga, kako bismo predstavili sve te opcije.
But we can get rid of that C if we can figure out the velocity, when time equals zero --
Ali možemo se riješiti tog C ako saznamo brzinu kada je vrijeme jednako nula --
what we’ve been calling v(0).
ono što smo zvali v0.
And if we write our equation with that v(0) in it, as a placeholder for the velocity when
I ako napišemo našu jednadžbu sa v0 što je brzina kada je
time equals zero, we end up with the the full equation for velocity.
vrijeme jednako nula, dobijemo punu jednadžbu za brzinu.
That should look familiar, because it’s one of our kinematic equations -- the definition of acceleration!
To bi vam trebalo izgledati poznato jer je to jedna od naših jednadžbi gibanja -- definicija akceleracije!
Neat how everything works out like that.
Lijepo je kako se sve tako riješi.
This equation tells us that the final velocity of our falling tennis ball, when it hit the
Ova nam jednadžba govori da je konačna brzina naše padajuće loptice za tenis pri kontaktu s
ground, was 16.7 ms downward.
tlom bila 16.7 m/s prema dolje.
But we aren’t done yet. We’re looking to link acceleration and POSITION,
Ali još nismo gotovi. Želimo povezati akceleraciju i položaj
so we’ll need to go one step further by integrating again.
pa ćemo morati otići korak dalje opet integrirajući.
There are a couple of different ways we could do it, but let’s just use the power rule again.
Postoji nekoliko različitih načina na koje bi to mogli učiniti, ali idemo samo opet koristiti pravilo potencije.
The integral of (a * t) is (half)(a t)(squared),
Integral od at je pola at na kvadrat,
and the integral of v_0 is just v_0 * t.
a integral od v0 je samo v0 puta t.
Put ‘em together, and you end up with this, which is starting to look a whole lot like
Stavimo do zajedno i dobijemo ovo, što počinje izgledati kao
ANOTHER kinematic equation -- the one we’ve been calling the displacement curve.
još jedna jednadžba gibanja -- ona koju smo zvali krivulja pomaka.
Now, what about that C?
Sada, što je sa onim C?
Well, just like before with the starting velocity, the starting position will tell us where to
Pa kao ranije sa početnom brzinom, početni položaj će nam reći gdje da
stick this equation on the vertical axis. So we’ll just make C equal to the starting
stavimo ovu jednadžbu na vertikalnoj osi. Zato ćemo za C samo uzeti da je jednak početnom
position, which we’ll write as x_0.
položaju, što ćemo zapisati kao x0.
And that’s our integral -- the displacement curve equation.
I to je naš integral -- krivulja pomaka.
Which means that now, we have everything we need to figure out, how high your window is.
Što znači da sada imamo sve što trebamo za otkriti koliko je visoko naš prozor.
The starting velocity is zero, because you just dropped the ball without throwing it.
Početna brzina je nula jer ste samo ispustili lopticu bez bacanja,
The acceleration is 9.81 ms^2. And it took 1.7 seconds to land.
akceleracija je 9.81 m/s^2 i pad je trajao 1.7 sekundi.
And now you know everything there is to know about calculus. (long pause) No you don’t.
I sada znate sve što trebate znati o računu... Ma ne znate.
As you can probably imagine, we’ve barely scratched the surface here -- there’s a
Kao što vjerojatno možete i zamisliti, jedva smo zagrebali površinu ovdje -- postoji
reason it normally takes two semesters of university, just to cover the basics.
razlog zašto obično trebaju dva semestra sveučilišta da bi se samo prošle osnove.
And, you know some people spend their whole lives studying this stuff.
I znate, neki ljudi provedu cijeli život proučavajući ove stvari.
But we’ve at least established enough background, that when those things do come up in this
Ali barem smo uspostavili dosta osnovnih stvari da kada se ovakve stvari pojave u ovom
course, we’ll be able to use what we’ve covered here, to talk about them.
tečaju možemo koristiti što smo ovdje naučili kako bismo pričali o njima.
Today, you learned that integrals are the area between an equation on a graph and the horizontal axis.
Danas ste naučili da su integrali površina između jeandžbe na grafu i horizontalne osi.
You also learned a few shortcuts to help find them, and how to find C using an initial value.
Također ste naučili nekoliko prečaca koji vam mogu pomoći da ih nađete te kako naći C koristeći početnu vrijednost.
Crash Course Philosophy is produced in association with PBS Digital Studios. You can head over
Crash Course Physics se proizvodi u suradnji sa PBS Digital Studios. Možete otići
to their channel to check out amazing shows like Shanks FX, Gross Science, and PBS Game Show.
na njihov kanal kako biste vidjeli nevjerojatne emisije kao što su Shanks FX, Gross Science i PBS Game Show.
This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio
Ova epizoda Crash Coursa je snimana u Doctor Cheryl C. Kinney Crash Course Studiju
with the help of these amazing people and our equally amazing Graphics Team is Thought Cafe.
uz pomoć ovih nevjerojatnih ljudi i naš jednako toliko nevjerojatan tim za grafiku je Thought Cafe.
Similar videos:

Thermodynamics: Crash Course Physics #23
CrashCourse
1.1M views 15K likes 525 dislikes 15 Sep 201660% match

A Planetary Physics Crash Course
enoch solano
762 views 15 likes 1 dislikes 04 Feb 201860% match

Friction: Crash Course Physics #6
CrashCourse
1.2M views 13.4K likes 535 dislikes 05 May 201660% match

Derivatives: Crash Course Physics #2
CrashCourse
1.6M views 22.8K likes 1K dislikes 07 Apr 201660% match


Sound: Crash Course Physics #18
CrashCourse
1.2M views 13.9K likes 563 dislikes 04 Aug 201660% match

Magnetism: Crash Course Physics #32
CrashCourse
1.2M views 15.9K likes 759 dislikes 01 Dec 201660% match

Quantum Physics Crash Course
IntroBooks Education
1.2K views 23 likes 1 dislikes 20 Jul 201760% match

Electric Charge: Crash Course Physics #25
CrashCourse
1.7M views 22.6K likes 687 dislikes 29 Sep 201650% match

Newton's Laws: Crash Course Physics #5
CrashCourse
3.4M views 40.3K likes 1.8K dislikes 28 Apr 201645% match


Work, Energy, and Power: Crash Course Physics #9
CrashCourse
1.9M views 22.6K likes 819 dislikes 26 May 201644% match

What is Philosophy?: Crash Course Philosophy #1
CrashCourse
5.1M views 94.3K likes 1.4K dislikes 08 Feb 201643% match




Utilitarianism: Crash Course Philosophy #36
CrashCourse
3.2M views 56.8K likes 865 dislikes 21 Nov 201643% match

🍌 What is Geography? Crash Course Geography #1
CrashCourse
338.7K views 10.3K likes 242 dislikes 30 Nov 202041% match

Crash Course Geography Preview
CrashCourse
226.3K views 9K likes 75 dislikes 17 Nov 202041% match

What is a Map? Crash Course Geography #2
CrashCourse
248.4K views 6.2K likes 182 dislikes 14 Dec 202041% match

Digestive System, Part 2: Crash Course A&P #34
CrashCourse
2M views 18.3K likes 329 dislikes 14 Sep 201540% match

Digestive System, Part 1: Crash Course A&P #33
CrashCourse
3.7M views 34.8K likes 1K dislikes 07 Sep 201540% match

Crash Course Linguistics Preview
CrashCourse
264.8K views 13.5K likes 95 dislikes 04 Sep 202039% match

What is space and how do we study it? Crash Course Geography #3
CrashCourse
151.6K views 4.3K likes 77 dislikes 22 Dec 202039% match

What is Physical Geography? Crash Course Geography #4
CrashCourse
147.4K views 3.8K likes 76 dislikes 12 Jan 202139% match

Karl Popper, Science, & Pseudoscience: Crash Course Philosophy #8
CrashCourse
1.9M views 38.3K likes 501 dislikes 28 Mar 201639% match



Deep Time: Crash Course Astronomy #45
CrashCourse
2M views 35.3K likes 751 dislikes 14 Jan 201638% match

World War II: Crash Course European History #38
CrashCourse
824.2K views 15.7K likes -1 dislikes 17 Mar 202036% match

Motion in a Straight Line: Crash Course Physics #1
CrashCourse
3.8M views 50.8K likes 1.7K dislikes 31 Mar 201636% match




What is Linguistics?: Crash Course Linguistics #1
CrashCourse
399K views 19.6K likes 150 dislikes 11 Sep 202034% match

Acidity: Crash Course Organic Chemistry #11
CrashCourse
138.1K views 3.7K likes 84 dislikes 03 Sep 202032% match

More Stereochemical Relationships: Crash Course Organic Chemistry #9
CrashCourse
94K views 2.1K likes 25 dislikes 05 Aug 202031% match

Officer on Horseback
Khan Academy
60.1K views 125 likes 5 dislikes 21 Aug 200930% match


Intro to Substitution Reactions: Crash Course Organic Chemistry #20
CrashCourse
41.9K views 1.1K likes 22 dislikes 21 Jan 202129% match

The Mystery at the Bottom of Physics
exurb1a
4.2M views 252.6K likes -1 dislikes 21 Jan 201929% match



Settings Privacy About Contact Donate