Introduction to the atom

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Kind: captions
Language: en

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In most topics you have to get
pretty advanced before you

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start addressing the
philosophically interesting

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things, but in chemistry it
just starts right from the

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get-go with what's arguably
the most philosophically

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interesting part of the whole
topic, and that's the atom.

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And the idea of the atom, as
philosophers long ago, and you

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could look it up on the
different philosophers who

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first philosophized about it,
they said, hey, you know, if I

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started off with, I don't know,
if I started off with an

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apple, and I just kept cutting
the apple -- let me draw a

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nice looking apple just so
it doesn't look just

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like a heart .

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There you go.

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You have a nice looking apple,
And you just kept cutting it,

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smaller and smaller pieces.

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So eventually, you get a piece
so small, so tiny, that you

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can't cut it anymore.

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And I'm sure some of these
philosophers went out there

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with a knife and tried to do
it and they just felt that,

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oh, if I could just get my knife
a little bit sharper, I

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could cut it again and again.

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So it's a completely
philosophical construct, which

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frankly, in a lot of ways, isn't
too different to how the

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atom is today.

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It's really just a mental
abstraction that allows us to

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describe a lot of observations
we see in the universe.

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But anyway, these philosophers
said, well, at some point we

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think that there's going to be
some little part of an apple

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that they won't be able
to divide anymore.

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And they called that an atom.

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And it doesn't just have to just
be for an apple they said

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this is true for any substance
or any element to that you

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encounter in the universe.

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And so the word atom is really
Greek for uncuttable.

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Uncuttable or indivisible.

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Now we know that it actually is
cuttable and even though it

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is not a trivial thing, it's
not the smallest form of

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matter we know.

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We now know that an atom is
made up of other more

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fundamental particles.

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And let me write that.

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So the we have the neutron.

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And I'll draw in a second how
they all fit together and the

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structure of an atom.

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We have a neutron.

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We have a proton.

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And we have electrons.

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Electrons.

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And you might already be
familiar with this if you look

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at old videos about atomic
projects, you'll see a drawing

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that looks something
like this.

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Let me see if I can draw one.

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So you'll have something
like that.

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And you'll have these things
spinning around

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that look like this.

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They have orbits that
look like that.

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And maybe something that
looks like that.

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And the general notion behind
these kind of nuclear drawings

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-- and I'm sure that they
still show up at some

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government defense labs or
something like that -- is that

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you have a nucleus at the
center of an atom.

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You have a nucleus at the
center of an atom.

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And we know that a nucleus
has neutrons and protons.

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Neutrons and protons.

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And we'll talk a little bit more
about which elements have

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how many neutrons and
how many protons.

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And then orbiting, and I'm going
to use the word orbit

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right now, although we'll learn
in about two minutes

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that the word orbit is actually
the incorrect or even

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the mentally incorrect way
of visualizing what

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an electron is doing.

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But the old idea was that you
have these electrons that are

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orbiting around the nucleus very
similar to the way the

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Earth orbits around the
Sun or the moon

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orbits around the Earth.

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And it's been shown that
that's actually

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a very wrong way.

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And when we cover quantum
mechanics we'll learn why this

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doesn't work, what are the
contradictions that emerge

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when you try to model an
electron like a planet going

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around the Sun.

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But this was kind of the
original idea, and frankly I

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think this is kind of the idea
that is the most mainstream

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way of viewing an atom.

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Now, I said an atom is
philosophically interesting.

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Why is it philosophically
interesting?

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Because what we now view as the
accepted way of viewing an

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atom really starts to blur the
line between our physical

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reality and everything in the
world is just information, and

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there really isn't any such
thing as true matter or true

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particles as the way we define
them in our everyday life.

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You know, for me a particle,
oh, it looks

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like a grain of sand.

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I can pick it up, touch it.

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While a wave, that could be like
a soundwave. It could be

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just this change in
energy over time.

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But we'll learn, especially when
we do quantum mechanics,

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that it all gets jumbled up as
we start approaching the

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scales or the size of an atom.

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Anyway, I said this was an
incorrect way of doing it.

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What's the correct way?

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So it turns out-- this is a
picture, not a picture really,

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this is also a depiction.

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So it's an interesting question,
what I just said.

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How can you have a picture
of an atom?

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Because is actually turns out
that most wavelengths of

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light, especially the visible
wavelengths of light, are much

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larger than the size
of an atom.

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Everything else we
quote-unquote, observe in

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life, it's by reflected light.

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But all of a sudden when you're
dealing with an atom,

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reflected light you could almost
view it as too big, or

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too blunt of an instrument with
which to observe an atom.

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Anyway, this is a depiction
of a helium atom.

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A helium atom has two protons
and two neutrons.

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Or at least this helium
atom has two

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protons and two neutrons.

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And the way they depict it here
in the nucleus, right

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there, maybe these are the two--
I'm assuming they're

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using red for proton and
purple for neutron.

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Purple seems like more
of a neutral color.

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And they're sitting at the
center of this atom.

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And then this whole haze around
there, those are the

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two electrons that helium
has, or that at least

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this helium atom has.

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Maybe you could gain or
lose an electron.

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But these are the
two electrons.

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And you say, hey, Sal, how can
two electrons be this blur

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that's kind of smeared
around this atom.

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And that's where it gets
philosophically interesting.

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So you cannot describe an
electron's path around a

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nucleus with the traditional
orbit idea that we've

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encountered when we look at
planets or if we just imagine

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things at kind of
a larger scale.

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It turns out that an electron,
you cannot know exactly its

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momentum and location at any
given point in time.

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All you can know is a
probability distribution of

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where it is likely to be.

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And the way they depicted
this, black is a higher

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probability, so you're much
more likely to find the

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electron here than
you are here.

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But the electron really
could be anywhere.

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It could even to be here, even
though it's completely white

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there, with some very, very,
very, very, very low

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probability.

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And so this function of where an
electron is, this is called

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an orbital.

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Orbital.

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Not to be confused with orbit.

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Orbital.

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Remember, an orbit was
something like this.

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It's like Venus going
around the Sun.

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So it's very physically easy
for us to imagine.

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While an orbital is actually
a mathematical probability

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function that tells
us where we're

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likely to find an electron.

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We'll deal a lot more with that
when we cover quantum

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mechanics, but that's not going
to be in the scope of

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this kind of introductory set
of chemistry lectures.

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But it's interesting, right?

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An electron's behavior is so
bizarre at that scale that you

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can't-- I mean, to call it a
particle is almost misleading.

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It is called a particle, but
it's not a particle in the

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sense that we're used to
in our everyday life.

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It's this thing that you can't
even say exactly where it is.

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It can be anywhere
in this haze.

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And we'll learn later that there
are different shapes of

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the hazes is as we add more and
more electrons to an atom.

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But to me, it starts to address
philosophical issues

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of what matter even is, or do
the things we look at, how

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real are they?

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Or how real are they, at least
as we've defined reality?

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Anyway I don't want to get
too philosophical on you.

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But the whole notion of
electrons, protons, they're

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all kind of predicated on
this notion of charge.

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And we've talked about it before
when we learned about

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Coulomb's law.

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You could review Coulomb's laws
videos in the physics

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playlist. But the idea
is that an electron

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has a negative charge.

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A proton, sometimes
written like that,

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has a positive charge.

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And a neutron has no charge.

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And so that's what was tempting
about the original

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model of an electron.

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If they say, OK, if this thing
has positive charges, right?

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So let's say this is two
neutrons and two protons.

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Let's say it's a helium atom.

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Then we'll have some positive
charges here.

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We have some negative
charges out here.

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Opposite charges attract.

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And so if these things had
some velocity, enough

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velocity, they would orbit
around this, just the way a

00:09:08.350 --> 00:09:10.310
planet will orbit
around the Sun.

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But now we learn, even though
this is partially true, that

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the further away an electron is
from the nucleus, it does

00:09:23.760 --> 00:09:27.700
have more, it's true,
potential energy.

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In that it will want to move
towards the nucleus, but

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because of all the mechanics at
the quantum level, it won't

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just do something simple like
move in a path like that, like

00:09:37.950 --> 00:09:41.140
a comet would do around the Sun,
it actually has this kind

00:09:41.140 --> 00:09:43.690
of wave-like behavior, where it
just has this probability

00:09:43.690 --> 00:09:45.170
function that describes it.

00:09:45.170 --> 00:09:48.680
But the further away
an orbital, it

00:09:48.680 --> 00:09:50.280
does have more potential.

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We're going to go a lot more
into that in future videos.

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But anyway, how do you recognize
what an element is?

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I've talked a lot about the
philosophy and all of that,

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but how do I know that
this is helium?

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Is it by the number of
neutrons it has?

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Is it by the number
of protons it has?

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Is it by the number
of electrons?

00:10:07.070 --> 00:10:09.960
Well the answer is, it's by
the number of protons.

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So if you know the number of
protons in an element, you

00:10:13.090 --> 00:10:14.660
know what that element is.

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And the number of protons,
this is defined

00:10:21.760 --> 00:10:23.010
as the atomic number.

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Now, so let's say I said
something has four protons.

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How do we know what it is?

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Well if we haven't memorized it,
we could look it up on the

00:10:37.070 --> 00:10:39.560
periodic table of elements,
which we'll be dealing with a

00:10:39.560 --> 00:10:43.210
lot in this playlist. And you'd
say, oh, four protons,

00:10:43.210 --> 00:10:45.680
that is beryllium.

00:10:45.680 --> 00:10:46.540
Right there.

00:10:46.540 --> 00:10:50.150
And the atomic number is the
number that you see up there.

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And that' s literally the
number of protons.

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And that is what differentiates

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one atom from another.

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If you have fifteen protons,
you're dealing with

00:10:59.340 --> 00:11:00.315
phosphorus.

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And all of a sudden, if you
have seven protons, you're

00:11:06.340 --> 00:11:07.150
dealing with nitrogen.

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If you have eight, you're
dealing with oxygen.

00:11:09.060 --> 00:11:11.680
That is what defines
the element.

00:11:11.680 --> 00:11:15.940
Now, we'll talk in the future
about what happens with charge

00:11:15.940 --> 00:11:17.140
and all of that.

00:11:17.140 --> 00:11:19.010
Or what happens when you
gain or lose electrons.

00:11:19.010 --> 00:11:21.760
But that does not change what
element you're dealing with.

00:11:21.760 --> 00:11:26.230
And likewise, when you change
the number of neutrons, that

00:11:26.230 --> 00:11:30.290
also does not change the element
you're dealing with.

00:11:30.290 --> 00:11:32.980
But that leads to an obvious
question of, well, how many

00:11:32.980 --> 00:11:35.900
neutrons and electrons
do you have?

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Well, if an atom is
charge-neutral, that means it

00:11:43.370 --> 00:11:46.100
has the same number
of electrons.

00:11:46.100 --> 00:11:47.835
So let's say that
I have carbon.

00:11:52.030 --> 00:11:53.620
Its atomic number is six.

00:11:53.620 --> 00:12:00.240
And let's say its mass
number is twelve.

00:12:00.240 --> 00:12:01.450
Now what does this mean?

00:12:01.450 --> 00:12:05.870
And let me say further that this
is a neutral particle.

00:12:05.870 --> 00:12:07.460
This is a neutral atom.

00:12:07.460 --> 00:12:10.570
So the atomic number
for carbon is six.

00:12:10.570 --> 00:12:13.860
That tells us exactly how
many protons it has.

00:12:13.860 --> 00:12:16.280
So if I were to draw a little
model here, and this is in no

00:12:16.280 --> 00:12:17.330
way an accurate model.

00:12:17.330 --> 00:12:21.000
I'll draw six-- two, three,
four, five, six

00:12:21.000 --> 00:12:23.380
protons in the center.

00:12:23.380 --> 00:12:26.100
And the weight of these protons,
each proton is one

00:12:26.100 --> 00:12:28.870
atomic mass unit, and we'll
talk more about how that

00:12:28.870 --> 00:12:31.120
relates to kilograms.
It's a very small

00:12:31.120 --> 00:12:32.580
fraction of a kilogram.

00:12:32.580 --> 00:12:36.230
Roughly I think it's
1.6 times 10 to the

00:12:36.230 --> 00:12:38.160
minus 27th of a kilogram.

00:12:38.160 --> 00:12:45.080
So let's say each of these are
one atomic mass unit, and

00:12:45.080 --> 00:12:48.990
that's approximately equal to,
I think, 1.67 times 10 to the

00:12:48.990 --> 00:12:53.320
minus 27 kilograms. This
is a very small number.

00:12:53.320 --> 00:12:57.290
It's actually almost impossible
to visualize.

00:12:57.290 --> 00:12:58.750
At least it is for me.

00:12:58.750 --> 00:13:03.240
This tells me the mass of the
entire carbon atom, of this

00:13:03.240 --> 00:13:04.300
particular carbon atom.

00:13:04.300 --> 00:13:05.790
And this can actually
change from carbon

00:13:05.790 --> 00:13:07.100
atom to carbon atom.

00:13:07.100 --> 00:13:10.720
And this is essentially the
mass of all of the protons

00:13:10.720 --> 00:13:12.410
plus all of the neutrons.

00:13:12.410 --> 00:13:17.540
And each proton has an atomic
mass of one, in atomic mass

00:13:17.540 --> 00:13:20.760
units, and each neutron
has an atomic mass of

00:13:20.760 --> 00:13:22.530
one atomic mass unit.

00:13:22.530 --> 00:13:28.660
So this is really the number
of protons plus

00:13:28.660 --> 00:13:29.910
the number of neutrons.

00:13:33.830 --> 00:13:37.280
So in this case we have six
protons, so we must also have

00:13:37.280 --> 00:13:39.110
six neutrons.

00:13:39.110 --> 00:13:42.920
Six neutrons plus six protons.

00:13:42.920 --> 00:13:44.490
Now, where are the electrons?

00:13:44.490 --> 00:13:49.130
Well, I said it's neutral, so
the proton has an equal

00:13:49.130 --> 00:13:51.360
positive charge as the
electron's negative charge.

00:13:51.360 --> 00:13:55.340
So this is a neutral atom, and
it has six protons, so it also

00:13:55.340 --> 00:13:56.380
has six electrons.

00:13:56.380 --> 00:13:57.000
Let me draw that.

00:13:57.000 --> 00:13:59.180
So we said it has six
neutrons in here.

00:13:59.180 --> 00:14:01.900
One, two, three, four,
five, six.

00:14:01.900 --> 00:14:04.863
So that's the nucleus
right there.

00:14:04.863 --> 00:14:09.300
And then if we were to draw the
electons-- well, I could

00:14:09.300 --> 00:14:12.110
draw it as a smear, but if we
want to kind of visualize it a

00:14:12.110 --> 00:14:13.670
little better, we could say,
OK, there's going to be six

00:14:13.670 --> 00:14:15.290
electrons orbiting.

00:14:15.290 --> 00:14:18.460
One, two, three, four,
five, six.

00:14:18.460 --> 00:14:21.280
And they're going to be moving
around in this unpredictable

00:14:21.280 --> 00:14:22.830
way that we would have
to describe with

00:14:22.830 --> 00:14:24.770
a probability function.

00:14:24.770 --> 00:14:30.170
And so the interesting thing
about it is, most of the mass

00:14:30.170 --> 00:14:32.890
of an atom is sitting
right in here.

00:14:32.890 --> 00:14:34.860
I mean, you might notice that
when people care about the

00:14:34.860 --> 00:14:40.480
mass, when they care about the
atomic mass number of an atom,

00:14:40.480 --> 00:14:41.610
they ignore the electrons.

00:14:41.610 --> 00:14:47.660
And that's because the mass
of a proton, one proton

00:14:47.660 --> 00:14:52.530
mass-wise, is equal
to 1,836 electons.

00:14:52.530 --> 00:14:58.250
So for thinking about the mass
of an atom, for all basic

00:14:58.250 --> 00:15:01.920
purposes, you can ignore the
mass of an electron.

00:15:01.920 --> 00:15:08.570
It's really the mass of the
nucleus that counts as the

00:15:08.570 --> 00:15:10.200
mass of the atom.

00:15:10.200 --> 00:15:12.400
Now, you might see this periodic
table here, and you

00:15:12.400 --> 00:15:15.680
say, OK, they gave us the
atomic number up there.

00:15:15.680 --> 00:15:18.030
The atomic number of
oxygen is eight.

00:15:18.030 --> 00:15:20.190
It means it has eight protons.

00:15:20.190 --> 00:15:22.470
The atomic number of
silicon is 14.

00:15:22.470 --> 00:15:24.660
It has 14 protons.

00:15:24.660 --> 00:15:27.230
Now what is this right here?

00:15:27.230 --> 00:15:28.505
Let's see, in carbon.

00:15:28.505 --> 00:15:34.150
In carbon they have
this 12.0107.

00:15:34.150 --> 00:15:36.980
That is the atomic
weight of carbon.

00:15:36.980 --> 00:15:38.230
Let me write this.

00:15:41.660 --> 00:15:47.390
Atomic weight of carbon.

00:15:47.390 --> 00:15:56.730
The atomic weight of
carbon is 12.0107.

00:15:56.730 --> 00:15:57.710
Now, what does that mean?

00:15:57.710 --> 00:16:03.200
Does that mean that carbon has
six protons and then the

00:16:03.200 --> 00:16:09.480
remainder, the remaining 6.0107
neutrons, it has kind

00:16:09.480 --> 00:16:11.710
of this fraction of a neutron?

00:16:11.710 --> 00:16:12.130
No.

00:16:12.130 --> 00:16:16.790
It means if you were to average
all the different

00:16:16.790 --> 00:16:19.470
versions of carbon you find on
the planet and you were to

00:16:19.470 --> 00:16:25.490
average the number of neutrons
based on the quantity of the

00:16:25.490 --> 00:16:27.360
different types of carbon,
this is the

00:16:27.360 --> 00:16:28.730
average you would get.

00:16:28.730 --> 00:16:32.980
So it turns out that carbon, the
two major forms, the main

00:16:32.980 --> 00:16:35.130
one you'll find is carbon-12.

00:16:35.130 --> 00:16:36.460
So that's like this.

00:16:36.460 --> 00:16:39.210
So that has six protons
and six neutrons.

00:16:39.210 --> 00:16:42.710
And then another isotope
of carbon.

00:16:42.710 --> 00:16:45.420
Now an isotope is the same
element with a different

00:16:45.420 --> 00:16:46.370
number of neutrons.

00:16:46.370 --> 00:16:50.170
Another isotope of carbon is
carbon-14, which is much more

00:16:50.170 --> 00:16:52.240
scarce on the planet.

00:16:52.240 --> 00:16:55.920
We don't know how much in the
universe, but on the planet.

00:16:55.920 --> 00:16:58.750
Now, if you were to average
these, not just a straight-up

00:16:58.750 --> 00:17:01.480
average, then you would get
carbon-13 and then the atomic

00:17:01.480 --> 00:17:04.130
weight would be 13, but you
weight this one much higher

00:17:04.130 --> 00:17:07.160
because this exists in much
larger quantities on Earth.

00:17:07.160 --> 00:17:08.589
I mean, this is pretty
much all of the

00:17:08.589 --> 00:17:09.270
carbon that you see.

00:17:09.270 --> 00:17:10.619
But there's a little
bit of this.

00:17:10.619 --> 00:17:13.530
So if you weight them
appropriately, the average

00:17:13.530 --> 00:17:14.420
becomes this.

00:17:14.420 --> 00:17:17.200
So most of the carbon you'll
find-- if you just found

00:17:17.200 --> 00:17:22.569
carbon someplace, on average
its weight in atomic mass

00:17:22.569 --> 00:17:27.040
units is going to be 12.0107.

00:17:27.040 --> 00:17:29.190
But that idea of an isotope
is an interesting one.

00:17:29.190 --> 00:17:32.180
Remember, when you change the
neutrons, you're not changing

00:17:32.180 --> 00:17:33.820
the actual, fundamental
element.

00:17:33.820 --> 00:17:36.210
You're just getting a different
isotope, a different

00:17:36.210 --> 00:17:37.590
version, of the element.

00:17:37.590 --> 00:17:41.500
So these two versions of carbon
are both isotopes.

00:17:41.500 --> 00:17:43.500
Now, I want to leave this video
with what I think is

00:17:43.500 --> 00:17:46.850
kind of the neatest idea behind
atoms. And it's the

00:17:46.850 --> 00:17:49.780
most philosophically interesting
things about them.

00:17:49.780 --> 00:17:54.190
It's that the relative size--
so, we have these electrons,

00:17:54.190 --> 00:17:58.340
which represent very little
of the mass of an atom.

00:17:58.340 --> 00:18:01.470
It's 1/2000 of the mass of an
atom are the electrons.

00:18:01.470 --> 00:18:04.560
And even those, it's hard
to even describe them as

00:18:04.560 --> 00:18:08.790
particles, because you can't
even tell me exactly where and

00:18:08.790 --> 00:18:11.170
how fast one of these
particles is moving.

00:18:11.170 --> 00:18:12.880
They just have a probability
function.

00:18:12.880 --> 00:18:16.280
So most of the atom is sitting
inside the nucleus.

00:18:16.280 --> 00:18:17.880
And this is the interesting
thing.

00:18:17.880 --> 00:18:21.080
If you look at an atom
on average, if you

00:18:21.080 --> 00:18:23.040
say this is my atom.

00:18:23.040 --> 00:18:26.300
Let's say I had two atoms that
are bonded to each other.

00:18:26.300 --> 00:18:29.065
And I were to say, how much
of this is actual stuff?

00:18:29.065 --> 00:18:32.120
And when I say stuff, that's a
very abstract concept, because

00:18:32.120 --> 00:18:33.670
we're talking about the
nucleus, right?

00:18:33.670 --> 00:18:34.740
Because the nucleus
is where all the

00:18:34.740 --> 00:18:36.470
mass is, all the stuff.

00:18:36.470 --> 00:18:39.750
It turns out that it's actually
an infinitesimally

00:18:39.750 --> 00:18:44.160
small fraction of the volume of
the atom where-- the volume

00:18:44.160 --> 00:18:45.920
of the atom is hard to define,
because the electron can

00:18:45.920 --> 00:18:48.870
pretty much be anywhere, but
if you view the volume as

00:18:48.870 --> 00:18:52.580
where you're most likely to find
the electron, or with 90%

00:18:52.580 --> 00:18:55.710
probability you're likely to
find the electron, then the

00:18:55.710 --> 00:18:59.150
nucleus is, in a lot of cases
and the way I think about it,

00:18:59.150 --> 00:19:01.740
it's about 1/10,000
of the volume.

00:19:01.740 --> 00:19:03.590
So if you think about it, when
you look at something, if you

00:19:03.590 --> 00:19:06.460
look at your hand or if you
look at the wall or if you

00:19:06.460 --> 00:19:14.830
look at your computer, 99.999%
of it is free space.

00:19:14.830 --> 00:19:15.470
It's nothing.

00:19:15.470 --> 00:19:18.320
It's vacuum.

00:19:18.320 --> 00:19:20.600
If you had ultra-small-- I
guess we could call them

00:19:20.600 --> 00:19:23.020
particles or something-- most
of them would pass straight

00:19:23.020 --> 00:19:24.690
through whatever you look at.

00:19:24.690 --> 00:19:26.530
So it already starts to kind of

00:19:26.530 --> 00:19:28.210
question our hold on reality.

00:19:28.210 --> 00:19:32.120
What is there when, if-- and
this is fact, this isn't

00:19:32.120 --> 00:19:35.350
theory right here-- that if you
take anything down to the

00:19:35.350 --> 00:19:39.580
building blocks, down to the
atomic level, most of the

00:19:39.580 --> 00:19:43.450
space of that kind of,
quote-unquote object, is free

00:19:43.450 --> 00:19:44.470
vacuum space.

00:19:44.470 --> 00:19:46.490
You could go straight through
it if you could get down to

00:19:46.490 --> 00:19:47.940
that scale.

00:19:47.940 --> 00:19:50.880
This image of a helium atom,
they say right here this is

00:19:50.880 --> 00:19:52.520
one femtometer.

00:19:52.520 --> 00:19:52.810
Right?

00:19:52.810 --> 00:19:54.060
One femtometer.

00:19:57.610 --> 00:20:00.180
This is the scale of
the nucleus of a

00:20:00.180 --> 00:20:01.780
helium atom, right?

00:20:01.780 --> 00:20:03.090
One femtometer.

00:20:03.090 --> 00:20:04.310
This is one angstrom, right?

00:20:04.310 --> 00:20:06.820
And they say that equals
100,000 femtometers.

00:20:06.820 --> 00:20:10.360
And just to get a sense of
scale, one angstrom is 1 times

00:20:10.360 --> 00:20:13.420
10 to the negative
10 meters, right?

00:20:13.420 --> 00:20:15.830
So the atom is roughly on the
scale of an angstrom.

00:20:15.830 --> 00:20:17.780
In the case of helium,
the nucleus is

00:20:17.780 --> 00:20:18.990
even a smaller fraction.

00:20:18.990 --> 00:20:20.930
It's 1/100,000.

00:20:20.930 --> 00:20:23.700
So if you had-- let's say you
had liquid helium, which you'd

00:20:23.700 --> 00:20:25.190
have to get very cold to get.

00:20:25.190 --> 00:20:27.610
If you're looking at that,
most of it is free space.

00:20:27.610 --> 00:20:30.550
If you're looking at an iron
bar, the great, great, great,

00:20:30.550 --> 00:20:33.310
great, great, great majority
of it is free space.

00:20:33.310 --> 00:20:35.500
And we're not even talking
about, maybe there's some free

00:20:35.500 --> 00:20:37.240
space inside the nucleus
that we could talk

00:20:37.240 --> 00:20:38.210
about in the future.

00:20:38.210 --> 00:20:41.950
But to me, that just blows my
mind that most things we look

00:20:41.950 --> 00:20:44.870
at are not really solid.

00:20:44.870 --> 00:20:47.530
They're really just empty space,
but they look solid

00:20:47.530 --> 00:20:50.360
because of the way light
reflects on them or the forces

00:20:50.360 --> 00:20:51.130
that repel us.

00:20:51.130 --> 00:20:55.100
But there really isn't something
to touch there.

00:20:55.100 --> 00:20:57.800
That most of this right here
is all free space.

00:20:57.800 --> 00:21:00.260
I think I've said the word free
space now, and I think

00:21:00.260 --> 00:21:02.370
I'll leave further

00:21:02.370 --> 00:21:05.020
mind-blowing to the next video.


Two Passing Bicycles Word Problem

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.070
This is the last of the word
problems that Kortaggio had

00:00:04.070 --> 00:00:05.470
sent, and they're all
interesting because they're

00:00:05.470 --> 00:00:08.780
all involving distances
and rates and times.

00:00:08.780 --> 00:00:11.960
They all involve you having to
do a little bit of implicit

00:00:11.960 --> 00:00:15.000
algebra without having to
explicitly solve for values of

00:00:15.000 --> 00:00:16.320
variables, and that's
why they're fun.

00:00:16.320 --> 00:00:19.070
So this one says Ann and
Betty, two cyclists,

00:00:19.070 --> 00:00:20.330
start simultaneously.

00:00:20.330 --> 00:00:28.080
Ann at point A and
Betty at point B.

00:00:28.080 --> 00:00:31.510
They travel to each other
at constant speeds.

00:00:31.510 --> 00:00:36.730
So let's say that Ann is
traveling with velocity A

00:00:36.730 --> 00:00:42.130
and Betty is traveling
with velocity B.

00:00:42.130 --> 00:00:42.710
Constant speeds.

00:00:42.710 --> 00:00:45.020
They pass each other
in 30 minutes.

00:00:45.020 --> 00:00:46.900
OK, so they're going to
pass each other someplace

00:00:46.900 --> 00:00:48.230
in between A and B.

00:00:48.230 --> 00:00:51.300
I'll just arbitrarily
draw that.

00:00:51.300 --> 00:00:53.100
They're going to pass
each other there.

00:00:53.100 --> 00:00:54.380
And what are these distances?

00:00:54.380 --> 00:00:58.600
It took them 30 minutes to both
to get this point, right?

00:00:58.600 --> 00:01:02.620
They pass each other
in 30 minutes.

00:01:02.620 --> 00:01:04.740
And Ann, headed from point
B, arrives there in

00:01:04.740 --> 00:01:06.170
another 20 minutes.

00:01:06.170 --> 00:01:10.970
OK, so this distance
takes Ann 30 minutes.

00:01:10.970 --> 00:01:13.240
So what is the actual distance?

00:01:13.240 --> 00:01:15.900
Well, it's always useful to
write distance is equal

00:01:15.900 --> 00:01:17.540
to rate times time.

00:01:17.540 --> 00:01:20.040
So what's the actual
distance she traveled?

00:01:20.040 --> 00:01:22.990
Well, the rate is va.

00:01:22.990 --> 00:01:26.780
She traveled a rate of va and
her time -- they tell us -- it

00:01:26.780 --> 00:01:29.490
took her 30 minutes
to get there.

00:01:29.490 --> 00:01:30.830
So va times 30.

00:01:30.830 --> 00:01:33.910
Let's say this velocity
is expressed in minutes.

00:01:33.910 --> 00:01:35.970
Fair enough.

00:01:35.970 --> 00:01:38.055
And then Ann, headed for
point B, arrives there

00:01:38.055 --> 00:01:38.950
in another 20 minutes.

00:01:38.950 --> 00:01:41.390
So this is going to take
her another 20 minutes.

00:01:41.390 --> 00:01:44.670
So this distance
right here is what?

00:01:44.670 --> 00:01:46.366
She's going at the
same velocity, but it

00:01:46.366 --> 00:01:47.290
takes her 20 minutes.

00:01:47.290 --> 00:01:49.400
So the distance is
velocity times time.

00:01:49.400 --> 00:01:53.230
So it's va times 20, 20va.

00:01:53.230 --> 00:01:54.610
Fair enough.

00:01:54.610 --> 00:01:57.770
How many minutes after Ann
arrives at point B will

00:01:57.770 --> 00:01:59.860
Betty arrive at point A?

00:01:59.860 --> 00:02:02.380
So Betty was going in this
direction, they meet right

00:02:02.380 --> 00:02:06.110
here, and then she keeps
going and hits point A.

00:02:06.110 --> 00:02:11.390
So the key realization here is
that this took Ann 30 minutes

00:02:11.390 --> 00:02:13.670
to get here, but it also took
Betty 30 minutes to get here.

00:02:13.670 --> 00:02:16.920
So what's this distance in
terms of Betty's velocity?

00:02:16.920 --> 00:02:17.980
What's this distance?

00:02:17.980 --> 00:02:20.150
Well, distance is equal
to rate times time.

00:02:20.150 --> 00:02:22.970
It took Betty 30 minutes
to go this distance

00:02:22.970 --> 00:02:25.660
at a velocity of vb.

00:02:25.660 --> 00:02:31.970
So it took her 30vb meters,
or whatever, actually.

00:02:31.970 --> 00:02:33.490
We don't even have to
specify the units.

00:02:33.490 --> 00:02:34.890
Whatever the units
the velocity is in.

00:02:34.890 --> 00:02:36.580
If it's in meters per
minute, then this is

00:02:36.580 --> 00:02:39.040
the distance in meters.

00:02:39.040 --> 00:02:41.320
This distance is 30vb.

00:02:41.320 --> 00:02:41.730
Right?

00:02:41.730 --> 00:02:44.860
So essentially, we just have to
figure out how long it takes

00:02:44.860 --> 00:02:46.920
her to travel this remaining
distance and we would

00:02:46.920 --> 00:02:47.850
have solved the problem.

00:02:47.850 --> 00:02:49.410
So how can we do that?

00:02:49.410 --> 00:02:52.030
Well, we can already set
up a relationship between

00:02:52.030 --> 00:02:54.660
vb and va, because this
distance is the same.

00:02:54.660 --> 00:02:56.370
It doesn't matter how
fast you travel it, the

00:02:56.370 --> 00:02:57.750
distance doesn't change.

00:02:57.750 --> 00:03:04.880
So we know that 30 times
Betty's velocity is equal to 20

00:03:04.880 --> 00:03:08.830
times Ann's velocity, 20va.

00:03:08.830 --> 00:03:13.230
And so if we wanted to express
Ann's velocity in terms of

00:03:13.230 --> 00:03:19.940
Betty's velocity -- let's just
do it that way -- if we divide

00:03:19.940 --> 00:03:27.420
both sides by 20 we get 3/2 of
Betty's velocity is equal

00:03:27.420 --> 00:03:29.730
to Ann's velocity.

00:03:29.730 --> 00:03:31.080
Fair enough.

00:03:31.080 --> 00:03:36.040
So what is this distance in
terms of Betty's velocity?

00:03:36.040 --> 00:03:38.070
We know it in terms
of Ann's velocity.

00:03:38.070 --> 00:03:43.630
In terms of Ann's velocity,
it's 30 times Ann's velocity.

00:03:43.630 --> 00:03:43.870
Right?

00:03:43.870 --> 00:03:46.280
So what is 30 times Ann's
velocity in terms of

00:03:46.280 --> 00:03:47.770
Betty's velocity?

00:03:47.770 --> 00:03:50.510
Well, if we multiply this times
30, that's 30 times Ann's

00:03:50.510 --> 00:03:53.270
velocity, so we can multiply
both sides of this times 30.

00:03:53.270 --> 00:03:57.280
So if you multiply 30 times
here and times thirty,

00:03:57.280 --> 00:03:58.980
this is the distance.

00:03:58.980 --> 00:04:00.150
This distance right here.

00:04:00.150 --> 00:04:02.360
Ann's velocity
times 30 minutes.

00:04:02.360 --> 00:04:04.620
It took her 30 minutes
to travel that far.

00:04:04.620 --> 00:04:09.620
So this is the same distance
expressed in Betty's velocity.

00:04:09.620 --> 00:04:14.600
So 30 times 3 is 90
divided by 2 is 45.

00:04:14.600 --> 00:04:15.450
So it's 45vb.

00:04:17.980 --> 00:04:20.694
So this distance
up here is 45vb.

00:04:25.340 --> 00:04:28.010
So distance is equal to
rate times -- so this

00:04:28.010 --> 00:04:30.490
is rate -- times time.

00:04:30.490 --> 00:04:33.730
So this is how many
minutes it took Betty to

00:04:33.730 --> 00:04:35.270
travel this distance.

00:04:35.270 --> 00:04:37.190
So what is the
question asking us?

00:04:37.190 --> 00:04:41.230
How many minutes after
Ann reaches point B will

00:04:41.230 --> 00:04:42.970
Betty arrive at point A.

00:04:42.970 --> 00:04:44.755
So they both reach right here.

00:04:44.755 --> 00:04:49.270
In another 20 minutes,
Ann gets here.

00:04:49.270 --> 00:04:53.990
And in another 45 minutes,
Betty will get over there.

00:04:53.990 --> 00:04:55.750
Another 45 minutes.

00:04:55.750 --> 00:04:57.820
So it takes Ann 20
minutes to get here.

00:04:57.820 --> 00:05:01.520
It takes Betty 45 minutes to
get here, after their meeting

00:05:01.520 --> 00:05:02.630
right at the center.

00:05:02.630 --> 00:05:06.000
So it's going to take
Betty another 25 minutes.

00:05:06.000 --> 00:05:06.190
Right?

00:05:06.190 --> 00:05:08.110
After 20 minutes Ann is here
and then we have to wait

00:05:08.110 --> 00:05:11.810
another 25 minutes for Betty
to get here, 45 minutes

00:05:11.810 --> 00:05:12.720
after this meeting.

00:05:12.720 --> 00:05:15.250
So the answer is 25 minutes.

00:05:18.740 --> 00:05:21.430
How many minutes after Ann
arrives at point B will

00:05:21.430 --> 00:05:22.980
Betty arrive at point A.

00:05:22.980 --> 00:05:24.060
Yes, 25 minutes.

00:05:24.060 --> 00:05:26.580
They could have just asked how
many minutes after the meeting

00:05:26.580 --> 00:05:28.250
will it take Betty
to reach point A.

00:05:28.250 --> 00:05:28.965
It would have been 45.

00:05:28.965 --> 00:05:30.700
But they want to know
how many more minutes

00:05:30.700 --> 00:05:32.920
after Ann gets there.

00:05:32.920 --> 00:05:35.310
Anyway, I thought you would
find that useful and, a final

00:05:35.310 --> 00:05:39.150
time, thanks to Kortaggio
for sending that problem.


Passed Bike Word Problem

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.740
Here's another interesting
word problem that was

00:00:02.740 --> 00:00:04.560
sent by [? Cortagio ?].

00:00:04.560 --> 00:00:07.720
A woman cycles to work
alongside a railroad track

00:00:07.720 --> 00:00:10.190
at 6 kilometers per hour.

00:00:10.190 --> 00:00:11.790
That's an interesting
piece of data.

00:00:11.790 --> 00:00:13.590
6 kilometers per hour.

00:00:13.590 --> 00:00:16.190
Every day she arrives
at a crossing the same

00:00:16.190 --> 00:00:18.360
time that a train does.

00:00:18.360 --> 00:00:19.070
OK.

00:00:19.070 --> 00:00:23.120
One day she was 50 minutes
late, and was overtaken by

00:00:23.120 --> 00:00:26.940
the train 6 kilometers
from the crossing.

00:00:26.940 --> 00:00:30.630
In how many minutes will the
train reach the crossing.

00:00:30.630 --> 00:00:34.390
Let's see if we can draw a
diagram here that can somewhat

00:00:34.390 --> 00:00:35.960
describe what's going
on in this problem.

00:00:35.960 --> 00:00:37.210
So what's on a normal day?

00:00:37.210 --> 00:00:40.780
On a normal day, she leaves
from home, she leaves from

00:00:40.780 --> 00:00:44.060
home, and she goes and she
travels alongside

00:00:44.060 --> 00:00:45.710
a railroad track.

00:00:45.710 --> 00:00:49.700
Not too different than my own
commute on my own bicycle.

00:00:49.700 --> 00:00:53.150
Actually, I actually do travel
on a bike path that's right

00:00:53.150 --> 00:00:54.370
along a railroad track.

00:00:54.370 --> 00:00:58.580
Anyway, let's say she leaves
here at time is equal to 0.

00:00:58.580 --> 00:01:00.270
Time is equal to 0.

00:01:00.270 --> 00:01:03.000
And right at this point right
here, maybe that's where the

00:01:03.000 --> 00:01:06.790
railroad crossing is, right
there, the train passes her up.

00:01:06.790 --> 00:01:10.420
And she's traveling at
6 kilometers per hour.

00:01:10.420 --> 00:01:13.610
6 kilometers per hour.

00:01:13.610 --> 00:01:16.060
6 kilometers per hour.

00:01:16.060 --> 00:01:17.150
And what else does it tell us?

00:01:17.150 --> 00:01:20.570
It doesn't tell us how
long it takes her, or

00:01:20.570 --> 00:01:22.026
how far she travels.

00:01:22.026 --> 00:01:28.470
So let's just say she arrives
here at t is equal to a hours.

00:01:28.470 --> 00:01:29.730
t is equal to a.

00:01:29.730 --> 00:01:32.800
We'll do time in hours, since
we're in kilometers per hour.

00:01:32.800 --> 00:01:36.600
So this path takes her
a hours to travel.

00:01:36.600 --> 00:01:39.410
So it takes her a hours.

00:01:39.410 --> 00:01:43.560
If we care about
the time, a hours.

00:01:43.560 --> 00:01:47.080
Now on this next day,
she's 50 minutes late.

00:01:47.080 --> 00:01:50.900
So on this day, she has to
travel the same distance in

00:01:50.900 --> 00:01:54.850
time, well in both
distance and time.

00:01:54.850 --> 00:01:57.250
But this is this day.

00:01:57.250 --> 00:01:58.970
She leaves not a t equals 0.

00:01:58.970 --> 00:02:00.320
She leaves 50 minutes late.

00:02:00.320 --> 00:02:01.770
Since we're dealing with
hours, let's write

00:02:01.770 --> 00:02:03.350
50 minutes in hours.

00:02:03.350 --> 00:02:06.500
So she leaves at-- if this was
time equal 0, that's when she

00:02:06.500 --> 00:02:09.260
normally leaves --she's now
leaving at that time is equal

00:02:09.260 --> 00:02:13.400
to 50 minutes, or time
is equal to 5/6 hours.

00:02:13.400 --> 00:02:13.670
Right?

00:02:13.670 --> 00:02:19.410
50 minutes is just 50/60 hours
or 5/6 hours of time is

00:02:19.410 --> 00:02:23.490
equal to 5 over 6 hours.

00:02:23.490 --> 00:02:25.940
This time is equal to 0 hours.

00:02:25.940 --> 00:02:31.050
And let's see, it says that she
was overtaken by the train 6

00:02:31.050 --> 00:02:33.550
kilometers from the crossing.

00:02:33.550 --> 00:02:36.410
So she was overtaken by
the train some place over

00:02:36.410 --> 00:02:39.170
here, and this distance.

00:02:39.170 --> 00:02:40.890
Right this is where the
crossing is, that's

00:02:40.890 --> 00:02:42.090
that crossing.

00:02:42.090 --> 00:02:44.680
This is 6 kilometers.

00:02:44.680 --> 00:02:46.530
6 kilometers.

00:02:46.530 --> 00:02:50.190
In how many minutes will the
train reach the crossing?

00:02:50.190 --> 00:02:53.800
So if we can figure out some
way of expressing this time, if

00:02:53.800 --> 00:02:56.410
we can figure out with this
time is, we know that the train

00:02:56.410 --> 00:02:59.010
reaches the crossing
a t is equal to a.

00:02:59.010 --> 00:03:01.810
So if we know this time, we
can just find the difference

00:03:01.810 --> 00:03:04.000
between these two times and we
would have solved the problem.

00:03:04.000 --> 00:03:05.940
So let's see if we can
figure out this time.

00:03:05.940 --> 00:03:08.600
So we know that it normally
takes her a hours to travel

00:03:08.600 --> 00:03:09.280
this whole distance.

00:03:09.280 --> 00:03:12.240
It doesn't matter when she
leaves it takes her a hours.

00:03:12.240 --> 00:03:15.290
So in this case it should
also take her a hours.

00:03:15.290 --> 00:03:18.570
And how long does she have to
go when the train passes up on

00:03:18.570 --> 00:03:20.400
this day that she was late?

00:03:20.400 --> 00:03:23.530
How long will it take her to
go this distance right there?

00:03:23.530 --> 00:03:25.885
Well she's traveling at 6
kilometers per hour, she

00:03:25.885 --> 00:03:27.690
has to go 6 kilometers.

00:03:27.690 --> 00:03:31.820
So in time, this is going
to take her 1 hour.

00:03:31.820 --> 00:03:34.400
To get-- you know, the train
will have long passed her --but

00:03:34.400 --> 00:03:36.220
to get back to the crossing,
she's going to have to

00:03:36.220 --> 00:03:37.850
travel for another hour.

00:03:37.850 --> 00:03:43.480
So if her whole route takes her
a hours, this whole length

00:03:43.480 --> 00:03:45.410
in time is a hours.

00:03:45.410 --> 00:03:47.650
Then this distance right
here is going to have

00:03:47.650 --> 00:03:48.810
to take her, what?

00:03:48.810 --> 00:03:52.380
She has an hour left, her whole
trip is going to be a hours, so

00:03:52.380 --> 00:03:53.970
this is going to be
a minus 1 hours.

00:03:53.970 --> 00:03:56.280
Let me scroll down
a little bit.

00:03:56.280 --> 00:04:02.330
This is going to take
her a minus 1 hours.

00:04:02.330 --> 00:04:05.680
So if on the day she was late
it took her a minus 1 hours to

00:04:05.680 --> 00:04:08.500
get passed by the train, what
time are we talking

00:04:08.500 --> 00:04:10.310
about right here?

00:04:10.310 --> 00:04:11.330
What time is this?

00:04:11.330 --> 00:04:17.080
This is t is equal to--
well we started at 5/6

00:04:17.080 --> 00:04:18.380
--t is equal to 5/6.

00:04:18.380 --> 00:04:19.350
That's where we started.

00:04:19.350 --> 00:04:24.040
Plus the amount of hours she
traveled plus a minus 1.

00:04:24.040 --> 00:04:26.580
So there we, I think, we now
have all the information we

00:04:26.580 --> 00:04:28.330
need to solve this problem.

00:04:28.330 --> 00:04:31.490
On the day she is late, the
train passes or at time is

00:04:31.490 --> 00:04:37.390
equal to 5/6 plus a minus 1
hours, where a is the amount of

00:04:37.390 --> 00:04:39.690
time it normally takes her to
travel her entire route

00:04:39.690 --> 00:04:40.660
to the crossing.

00:04:40.660 --> 00:04:45.720
We know that the train passes
this point at t is equal to a.

00:04:45.720 --> 00:04:48.550
So if we want to know how many
minutes does it take the train

00:04:48.550 --> 00:04:52.130
to go from this point to
this point, we just have to

00:04:52.130 --> 00:04:54.080
subtract the two times.

00:04:54.080 --> 00:04:57.990
So the minutes it takes,
well the hours because

00:04:57.990 --> 00:04:58.890
we're doing it hours.

00:04:58.890 --> 00:05:02.260
And remember I said 50 minutes
was 5/6 of an hour, so the

00:05:02.260 --> 00:05:09.080
minutes it takes is a,
that a, minus this.

00:05:09.080 --> 00:05:14.050
Minus 5/6 plus a minus 1.

00:05:14.050 --> 00:05:14.300
Right?

00:05:14.300 --> 00:05:16.630
That's just the difference in
time for the train, in hours.

00:05:16.630 --> 00:05:22.760
So this is a minus
5/6 minus a plus 1.

00:05:22.760 --> 00:05:25.150
Luckily the a's cancel out.

00:05:25.150 --> 00:05:29.600
a minus a, and you're left with
1 minus 5/6 hours, which is

00:05:29.600 --> 00:05:32.060
just equal to 1/6 hours.

00:05:32.060 --> 00:05:34.390
There are 60 minutes
in an hour, so 1/6 of

00:05:34.390 --> 00:05:36.220
that is 10 minutes.

00:05:36.220 --> 00:05:39.040
So it takes the train 10
minutes to get to the

00:05:39.040 --> 00:05:41.160
crossing from the point
that it passed her.

00:05:41.160 --> 00:05:43.260
Another neat problem, and
I thank [? Cortagio ?].

00:05:43.260 --> 00:05:43.740
for that.

00:05:43.740 --> 00:05:45.720
And then you can actually use
this information if you want

00:05:45.720 --> 00:05:48.280
figure out how fast the train
is going, or you could even

00:05:48.280 --> 00:05:50.850
think of other types of
derivative problems

00:05:50.850 --> 00:05:51.580
off of this one.

00:05:51.580 --> 00:05:53.710
Anyway, another good problem.


Passing Trains

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.630
This is the fourth of
the five problems that

00:00:02.630 --> 00:00:04.020
Kortaggio sent me today.

00:00:04.020 --> 00:00:05.430
And I've been doing these
problems because I think

00:00:05.430 --> 00:00:08.190
they're really neat
applications of essentially

00:00:08.190 --> 00:00:12.520
fairly basic, even you could
call it physics, rate problems,

00:00:12.520 --> 00:00:13.750
and a little bit of algebra.

00:00:13.750 --> 00:00:16.280
But you're able to figure out
pretty neat things, where

00:00:16.280 --> 00:00:18.850
you're not sure if there's
enough information at first.

00:00:18.850 --> 00:00:21.710
So, here we have train A,
represented by these two dots

00:00:21.710 --> 00:00:24.000
and arrow, is 200 meters long.

00:00:24.000 --> 00:00:26.810
And I did this because I think
we want to be specific about

00:00:26.810 --> 00:00:28.480
the configurations we're
talking to later

00:00:28.480 --> 00:00:29.230
in the problems.

00:00:29.230 --> 00:00:30.390
Let's just read it.

00:00:30.390 --> 00:00:32.440
Train A is 200 meters long.

00:00:32.440 --> 00:00:35.410
Train B is 400 meters long.

00:00:35.410 --> 00:00:38.620
They run on parallel tracks
at constant speeds.

00:00:38.620 --> 00:00:41.250
When moving in the same
direction, A passes

00:00:41.250 --> 00:00:42.770
B in 15 seconds.

00:00:42.770 --> 00:00:44.460
From, so these are
the configurations.

00:00:44.460 --> 00:00:47.760
From A is right behind B, to
A is right in front of B.

00:00:47.760 --> 00:00:48.720
So let me draw that.

00:00:48.720 --> 00:00:52.050
So, train A, I'll do
in this blue color.

00:00:52.050 --> 00:00:53.670
So train a is like that.

00:00:53.670 --> 00:00:56.380
And it's 200 meters long.

00:00:56.380 --> 00:01:00.210
And then train B, I'll
do in this green color.

00:01:00.210 --> 00:01:02.430
Train B is double the length.

00:01:02.430 --> 00:01:05.360
So it's 400 meters.

00:01:05.360 --> 00:01:07.390
And so this is a
starting configuration.

00:01:07.390 --> 00:01:09.090
The outline right here.

00:01:09.090 --> 00:01:11.400
And it says, it takes 15
seconds to pass it, to go

00:01:11.400 --> 00:01:12.670
to this configuration.

00:01:12.670 --> 00:01:16.080
So the ending configuration
looks like this.

00:01:16.080 --> 00:01:17.340
Ending configuration.

00:01:17.340 --> 00:01:18.600
Train B here.

00:01:18.600 --> 00:01:22.830
And then train A has
passed train B.

00:01:22.830 --> 00:01:26.600
200 meters.

00:01:26.600 --> 00:01:30.070
Once again, this is
400 meters here.

00:01:30.070 --> 00:01:33.410
And this situation
takes 15 seconds.

00:01:33.410 --> 00:01:36.850
So this takes 15
seconds to happen.

00:01:36.850 --> 00:01:39.010
15 seconds.

00:01:39.010 --> 00:01:41.380
So, A passes B in 15 seconds.

00:01:41.380 --> 00:01:46.370
So this whole sentence right
here, is this right here.

00:01:46.370 --> 00:01:48.820
We go from the situation
to that situation.

00:01:48.820 --> 00:01:50.540
And then in opposite
directions, they pass

00:01:50.540 --> 00:01:52.390
each other in 5 seconds.

00:01:52.390 --> 00:01:54.260
So from this
configuration to that.

00:01:54.260 --> 00:01:55.480
So let me draw that.

00:01:55.480 --> 00:01:59.540
So, in the opposite direction
example, they start like this.

00:01:59.540 --> 00:02:02.260
Train A is 200 meters.

00:02:02.260 --> 00:02:06.540
Train B is facing in
the other direction.

00:02:06.540 --> 00:02:09.660
I'm doing it probably little
bit too long, 400 meters.

00:02:09.660 --> 00:02:13.160
And then in 5 seconds they
get from this configuration

00:02:13.160 --> 00:02:17.160
to this configuration.

00:02:17.160 --> 00:02:19.890
To that configuration,
right there.

00:02:19.890 --> 00:02:20.900
200.

00:02:20.900 --> 00:02:25.850
And, of course, this
right here is 400.

00:02:25.850 --> 00:02:30.540
And this takes-- this right
here takes 5 seconds.

00:02:30.540 --> 00:02:34.890
So in opposite directions they
pass each other in 5 seconds.

00:02:34.890 --> 00:02:38.620
Now, their question is how
fast is each train moving

00:02:38.620 --> 00:02:39.980
in meters per second?

00:02:39.980 --> 00:02:42.270
So, once again they gave us--
they didn't really give us a

00:02:42.270 --> 00:02:43.490
lot of velocity information.

00:02:43.490 --> 00:02:45.140
They just tell us how
long it takes to pass.

00:02:45.140 --> 00:02:47.830
But maybe using both of these
pieces of information,

00:02:47.830 --> 00:02:49.110
we can solve it.

00:02:49.110 --> 00:02:51.370
So let's say that--
well, you have velocity

00:02:51.370 --> 00:02:54.900
of train A, so vA.

00:02:54.900 --> 00:02:56.840
Velocity of train A, and
then of course you have

00:02:56.840 --> 00:02:58.960
the velocity of train B.

00:02:58.960 --> 00:03:00.820
And regardless of which
direction they're facing, it

00:03:00.820 --> 00:03:04.470
assumes that they're always
going at the same velocity.

00:03:04.470 --> 00:03:06.330
So in this situation, and we
always have to just

00:03:06.330 --> 00:03:08.970
remember, distance is equal
to rate times time.

00:03:08.970 --> 00:03:11.900
So relative, if we assume,
because when you take a

00:03:11.900 --> 00:03:14.110
velocity, you can always
take a velocity relative

00:03:14.110 --> 00:03:15.160
to something else.

00:03:15.160 --> 00:03:21.090
So, first of all, relative to
this train right here, to train

00:03:21.090 --> 00:03:23.510
B, how far does train A travel?

00:03:23.510 --> 00:03:25.510
Well, it goes from this point.

00:03:25.510 --> 00:03:27.710
It goes from right here.

00:03:27.710 --> 00:03:30.880
To, not just 400 meters, it
gets to the point where the

00:03:30.880 --> 00:03:34.930
front of the train is out here.

00:03:34.930 --> 00:03:37.560
So it has to travel 400 meters.

00:03:37.560 --> 00:03:39.650
And another 200 meters.

00:03:39.650 --> 00:03:42.136
So it travels 600 meters.

00:03:42.136 --> 00:03:44.660
It travels 600 meters
in this situation.

00:03:44.660 --> 00:03:47.530
And how far does it travel
in this situation?

00:03:47.530 --> 00:03:50.680
Well, once again, if we assume
that this train is stationary,

00:03:50.680 --> 00:03:53.420
and that's, I guess, the key
assumption we have to make.

00:03:53.420 --> 00:03:55.090
We're going to do
everything relative.

00:03:55.090 --> 00:03:57.660
We're going to assume that,
even though it is moving at a

00:03:57.660 --> 00:04:00.800
velocity, the position we're
going to make relative.

00:04:00.800 --> 00:04:04.330
So, relative to this train,
we move from right here,

00:04:04.330 --> 00:04:06.630
we move 400 meters.

00:04:06.630 --> 00:04:08.010
And then we move 200 more.

00:04:08.010 --> 00:04:10.990
So, again, in both situations,
we move 600 meters.

00:04:10.990 --> 00:04:14.310
In this situation we move
600 meters, relative.

00:04:14.310 --> 00:04:18.020
I guess you could say, we move
600 meters relative to the back

00:04:18.020 --> 00:04:20.190
of train B in this situation.

00:04:20.190 --> 00:04:23.640
And we move 600 meters relative
to the front of train

00:04:23.640 --> 00:04:25.560
B in this situation.

00:04:25.560 --> 00:04:27.330
In this situation, we
do it in 15 seconds.

00:04:27.330 --> 00:04:31.890
That's because we're, kind of,
where the velocity of train A

00:04:31.890 --> 00:04:34.510
is being eaten away by
the velocity of B.

00:04:34.510 --> 00:04:38.050
If the velocity of B was 0, if
this green train was really

00:04:38.050 --> 00:04:40.870
stationary, then we'd be moving
relative to this train

00:04:40.870 --> 00:04:42.010
with velocity A.

00:04:42.010 --> 00:04:45.400
But now this is moving
at some velocity.

00:04:45.400 --> 00:04:48.135
So our relative velocity
to this train is going

00:04:48.135 --> 00:04:48.970
to be something lower.

00:04:48.970 --> 00:04:51.150
And what is the
relative velocity?

00:04:51.150 --> 00:04:54.390
If you're a passenger sitting
in train B, if you're a

00:04:54.390 --> 00:04:58.830
passenger sitting in train B,
right there, how fast will it

00:04:58.830 --> 00:05:01.180
look like train A is going?

00:05:01.180 --> 00:05:03.290
What will be the
relative velocity?

00:05:03.290 --> 00:05:05.720
Well, it's going to be the
difference between the two.

00:05:05.720 --> 00:05:06.930
Right?

00:05:06.930 --> 00:05:11.710
So it's going to
be vA minus vB.

00:05:11.710 --> 00:05:14.910
If you're sitting in
this train right there.

00:05:14.910 --> 00:05:18.380
And you would say, it
takes 15 seconds.

00:05:18.380 --> 00:05:20.270
So, rate times time.

00:05:20.270 --> 00:05:21.700
Times 15 seconds.

00:05:21.700 --> 00:05:23.860
Is equal to a distance
that it traveled.

00:05:23.860 --> 00:05:26.480
And, once again, if you're a
passenger sitting in this green

00:05:26.480 --> 00:05:29.120
train right here, you would
say, OK, it went

00:05:29.120 --> 00:05:29.830
from this point.

00:05:29.830 --> 00:05:31.420
It crossed this entire
train then it went

00:05:31.420 --> 00:05:32.980
another 200 meters.

00:05:32.980 --> 00:05:35.500
So it went 600 meters.

00:05:35.500 --> 00:05:37.360
Now, and this is of
course in seconds.

00:05:37.360 --> 00:05:39.220
Now, obviously both of
these trains in this have

00:05:39.220 --> 00:05:40.680
some positive velocity.

00:05:40.680 --> 00:05:43.470
This train would have moved
even more than 600 meters.

00:05:43.470 --> 00:05:45.880
This train moved 100 and
this train would've

00:05:45.880 --> 00:05:47.050
moved 700 meters.

00:05:47.050 --> 00:05:49.830
But I'm doing everything
relative to what this passenger

00:05:49.830 --> 00:05:51.870
in the green train sees.

00:05:51.870 --> 00:05:52.800
Likewise.

00:05:52.800 --> 00:05:56.350
The passenger in the green
train here, let's say the

00:05:56.350 --> 00:05:59.820
passenger in the green
train right here.

00:05:59.820 --> 00:06:02.040
Oh, I'm doing his arms
coming out of his head.

00:06:02.040 --> 00:06:06.990
But what velocity does he see
this blue train coming in at?

00:06:06.990 --> 00:06:08.970
Well, he's going in
this direction at 400

00:06:08.970 --> 00:06:09.990
meters per second.

00:06:09.990 --> 00:06:13.150
The other train is coming in--
no, sorry he's going in this

00:06:13.150 --> 00:06:15.640
direction at velocity
of train B.

00:06:15.640 --> 00:06:16.470
We don't know what that is.

00:06:16.470 --> 00:06:18.450
400 is how long it is.

00:06:18.450 --> 00:06:21.250
And this train is coming with
velocity, want to do it

00:06:21.250 --> 00:06:24.350
in blue, with velocity a.

00:06:24.350 --> 00:06:25.920
So you would add the
two velocities.

00:06:25.920 --> 00:06:28.380
If this is coming at 60 miles
per hour and this is going in

00:06:28.380 --> 00:06:31.040
that direction at 60 miles per
hour, to this guy who's

00:06:31.040 --> 00:06:35.350
stationery in train B, he would
just feel like this train

00:06:35.350 --> 00:06:37.720
is approaching him at
120 miles per hour.

00:06:37.720 --> 00:06:39.920
Or the addition of those two.

00:06:39.920 --> 00:06:43.940
So from this guy's point of
view, this train is approaching

00:06:43.940 --> 00:06:46.970
with the velocity-- let me do,
is approaching with the

00:06:46.970 --> 00:06:49.690
velocity vA plus vB.

00:06:52.850 --> 00:06:55.985
And in 5 seconds-- and they
give us that information.

00:06:55.985 --> 00:06:58.895
In 5 seconds-- so velocity
times time, or rate times

00:06:58.895 --> 00:07:02.900
time, is equal to distance--
it travels 600 meters.

00:07:02.900 --> 00:07:06.150
Remember, this is all relative
to the guy, or the gal,

00:07:06.150 --> 00:07:07.660
sitting on this green train.

00:07:07.660 --> 00:07:09.540
And that's kind of the key
assumption you have to make to

00:07:09.540 --> 00:07:11.330
make this problem solvable.

00:07:11.330 --> 00:07:13.530
Well, now we have two equations
and two unknowns, we should

00:07:13.530 --> 00:07:14.560
be able to solve this.

00:07:14.560 --> 00:07:17.450
For the velocity of
the two trains.

00:07:17.450 --> 00:07:19.490
So, just to simplify,
let's divide both sides

00:07:19.490 --> 00:07:21.510
of this one by 15.

00:07:21.510 --> 00:07:25.190
So we have the velocity of A
minus the velocity of B, is

00:07:25.190 --> 00:07:29.510
equal to 40 meters per second.

00:07:29.510 --> 00:07:29.960
Right?

00:07:29.960 --> 00:07:32.050
60 divided by 15 is 4.

00:07:32.050 --> 00:07:33.080
Yep, 40.

00:07:33.080 --> 00:07:36.900
And then here we have the
velocity of A plus the velocity

00:07:36.900 --> 00:07:43.190
of B, that's an A, is equal
to 120 meters per second.

00:07:43.190 --> 00:07:45.140
And, see, we could just
take this equation.

00:07:45.140 --> 00:07:46.060
Put it down here.

00:07:46.060 --> 00:07:47.730
We could add the two
equations to each other.

00:07:47.730 --> 00:07:51.870
So if the velocity of A minus
the velocity of B is equal

00:07:51.870 --> 00:07:54.980
to 40 meters per second.

00:07:54.980 --> 00:07:57.210
Add the two equations.

00:07:57.210 --> 00:08:00.030
We get 2 times the
velocity of A.

00:08:00.030 --> 00:08:01.770
These two cancel out.

00:08:01.770 --> 00:08:05.200
Is equal to 160
meters per second.

00:08:05.200 --> 00:08:09.230
Or the velocity of A is equal
to 80 meters per second.

00:08:09.230 --> 00:08:10.780
And then we can just
back-substitute here.

00:08:10.780 --> 00:08:13.020
The difference between
the two is 40.

00:08:13.020 --> 00:08:16.490
So it's the 80 minus the
velocity of B is equal to

00:08:16.490 --> 00:08:18.360
40 meters per second.

00:08:18.360 --> 00:08:20.290
So what's the velocity of B?

00:08:20.290 --> 00:08:23.540
Well, you could subtract
80 from both sides.

00:08:23.540 --> 00:08:27.100
You get minus velocity of
B is equal to minus 40.

00:08:27.100 --> 00:08:30.940
Or the velocity of B is equal
to 40 meters per second.

00:08:30.940 --> 00:08:32.410
And we've done the problem.

00:08:32.410 --> 00:08:34.960
And the key assumption there is
to do everything relative to

00:08:34.960 --> 00:08:38.270
the green guy sitting
inside of train B.

00:08:38.270 --> 00:08:39.260
You could have done
it the other way.

00:08:39.260 --> 00:08:41.780
You could have picked
other relative positions.

00:08:41.780 --> 00:08:45.480
But that, in my brain, is the
easiest way to figure it out.

00:08:45.480 --> 00:08:48.330
So, this guy, in this case,
is going at the speed at

00:08:48.330 --> 00:08:50.530
40 meters per second.

00:08:50.530 --> 00:08:53.930
And this guy is going at
80 meters per second.

00:08:53.930 --> 00:08:57.890
In this situation, this guy is
traveling at 80 meters per

00:08:57.890 --> 00:09:01.240
second and this guy going in
this direction at 40

00:09:01.240 --> 00:09:02.330
meters per second.

00:09:02.330 --> 00:09:03.030
Anyway.

00:09:03.030 --> 00:09:06.000
Thanks again to Kortaggio
for that problem.


Overtaking Word Problem

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.160
Here's another problem that
Kortaggio sent me and,

00:00:04.160 --> 00:00:06.030
like the other two, this
is quite interesting.

00:00:06.030 --> 00:00:08.550
Although, I think you'll start
to see a pattern forming

00:00:08.550 --> 00:00:09.500
in how these are solved.

00:00:09.500 --> 00:00:11.880
And that's the whole point, not
just to show you neat problems,

00:00:11.880 --> 00:00:14.240
but also to give you an
intuition of how to solve it.

00:00:14.240 --> 00:00:15.200
So what does it say?

00:00:15.200 --> 00:00:19.470
It says Alice is 100 meters
from Bill and Bill is 300

00:00:19.470 --> 00:00:22.020
meters away from Chelsea.

00:00:22.020 --> 00:00:24.910
Maybe this is a bit of a
Clintonian reference, but who

00:00:24.910 --> 00:00:26.660
knows, or maybe they just
wanted people with starting

00:00:26.660 --> 00:00:28.350
names a, b, and c.

00:00:28.350 --> 00:00:29.750
So let's draw that out.

00:00:29.750 --> 00:00:34.520
So Alice is 100 meters
from Bill-- all right.

00:00:34.520 --> 00:00:36.830
Alice is 100 meters from Bill--
and actually he drew a little

00:00:36.830 --> 00:00:39.450
diagram, so we know that also
Chelsea is to the

00:00:39.450 --> 00:00:40.280
right of Bill.

00:00:40.280 --> 00:00:43.415
So let's see, the diagram he
sent me looks like this.

00:00:43.415 --> 00:00:45.830
So you have Alice.

00:00:45.830 --> 00:00:49.940
And then 100 meters
away you have Bill.

00:00:49.940 --> 00:00:54.450
And then 300 meters to the
right of that you have Chelsea.

00:00:54.450 --> 00:01:00.820
This distance right
here is 100 meters.

00:01:00.820 --> 00:01:05.510
And then this distance
right there is 300 meters.

00:01:05.510 --> 00:01:11.460
And the problem tells us they
are all facing east and are

00:01:11.460 --> 00:01:13.030
standing on the same line.

00:01:13.030 --> 00:01:13.600
Fair enough.

00:01:13.600 --> 00:01:14.960
We'll say that's east.

00:01:14.960 --> 00:01:17.690
So they're all facing in
that direction standing

00:01:17.690 --> 00:01:18.650
in the same line.

00:01:18.650 --> 00:01:21.890
They all travel to the
east at constant speeds.

00:01:21.890 --> 00:01:24.220
OK, but they don't say that
they're all traveling

00:01:24.220 --> 00:01:26.175
at the same speed.

00:01:26.175 --> 00:01:27.810
Each individual
speed is constant.

00:01:27.810 --> 00:01:32.815
So this could be velocity of
Alice, lowercase a, velocity

00:01:32.815 --> 00:01:38.210
of Bill, and then this is
the velocity of Chelsea.

00:01:38.210 --> 00:01:40.820
Alright, they all travel
east at constant speeds.

00:01:40.820 --> 00:01:44.940
In 6 minutes, Alice
overtakes Bill.

00:01:44.940 --> 00:01:45.980
So what has to happen?

00:01:45.980 --> 00:01:48.390
Let me do this in another color
and write down what that

00:01:48.390 --> 00:01:50.310
equation would look like.

00:01:50.310 --> 00:01:51.630
So I'll do it in green.

00:01:51.630 --> 00:01:56.700
In 6 minutes, Alice
overtakes Bill.

00:01:56.700 --> 00:01:59.290
This tells the distance that
Alice travels in 6 minutes.

00:01:59.290 --> 00:02:01.890
So what's the distance that
Alice travels in 6 minutes?

00:02:01.890 --> 00:02:06.350
It would be the velocity
of Alice times 6.

00:02:06.350 --> 00:02:08.170
And she overtakes Bill.

00:02:08.170 --> 00:02:12.430
So that means she goes 100
meters more than Bill went.

00:02:12.430 --> 00:02:15.290
So that means she went 100
meters plus the distance that

00:02:15.290 --> 00:02:16.790
Bill went in that 6 minutes.

00:02:16.790 --> 00:02:19.700
And the distance that Bill
went in 6 minutes is

00:02:19.700 --> 00:02:23.380
velocity of Bill times 6.

00:02:23.380 --> 00:02:23.960
Right?

00:02:23.960 --> 00:02:27.810
This tell us in 6 minutes,
Alice went 100 meters more

00:02:27.810 --> 00:02:30.260
than however far Bill
went in 6 minutes.

00:02:30.260 --> 00:02:31.310
That's that equation there.

00:02:31.310 --> 00:02:33.270
We got it from
this information.

00:02:33.270 --> 00:02:36.480
And then the next thing,
this problem tells us, in

00:02:36.480 --> 00:02:40.830
another 6 minutes Alice
overtakes Chelsea.

00:02:40.830 --> 00:02:43.200
So from the beginning of
our time, it essentially

00:02:43.200 --> 00:02:44.190
takes 12 minutes.

00:02:44.190 --> 00:02:44.730
Right?

00:02:44.730 --> 00:02:47.070
6 minutes to overtake Bill
and then another 6 minutes

00:02:47.070 --> 00:02:47.910
to overtake Chelsea.

00:02:47.910 --> 00:02:50.080
So 12 minutes to
overtake Chelsea.

00:02:50.080 --> 00:02:54.620
So we could write that as the
distance that Alice travels

00:02:54.620 --> 00:02:57.950
in 12 minutes is equal to--

00:02:57.950 --> 00:02:59.450
Well, she has to
overtake Chelsea.

00:02:59.450 --> 00:03:02.340
That means she made up 400
meters of distance, so that

00:03:02.340 --> 00:03:04.560
she traveled 400 more meters.

00:03:04.560 --> 00:03:07.870
So the distance that Alice
travels is going to be 400

00:03:07.870 --> 00:03:12.260
meters more than the distance
that Chelsea travels

00:03:12.260 --> 00:03:13.300
in 12 minutes.

00:03:13.300 --> 00:03:15.990
And remember, distance is just
equal to rate times time.

00:03:15.990 --> 00:03:16.280
Right?

00:03:16.280 --> 00:03:18.400
So in 12 minutes this is how
far-- oh sorry --this is how

00:03:18.400 --> 00:03:20.650
far Chelsea travels,
that's a c.

00:03:20.650 --> 00:03:22.860
And this is how far Alice
travels, and she is going

00:03:22.860 --> 00:03:24.670
to travel 400 meters more.

00:03:24.670 --> 00:03:27.130
OK, that's what those
two equations tell us.

00:03:27.130 --> 00:03:28.890
And what is their question?

00:03:28.890 --> 00:03:33.510
They're asking us, how many
minutes did it take Bill

00:03:33.510 --> 00:03:35.210
to overtake Chelsea?

00:03:35.210 --> 00:03:40.550
How many minutes did it take
Bill to overtake Chelsea?

00:03:40.550 --> 00:03:46.950
All right, so let's say we
want to know time in minutes.

00:03:46.950 --> 00:03:50.450
How long does it take Bill
to overtake Chelsea?

00:03:50.450 --> 00:03:52.600
Let's say we know
that it's time, t.

00:03:52.600 --> 00:03:58.250
So that tells us that the
velocity of Bill times this

00:03:58.250 --> 00:04:03.300
time is going to be equal to
--for him to overtake Chelsea,

00:04:03.300 --> 00:04:06.370
he has to travel 300 meters
more than her in that

00:04:06.370 --> 00:04:07.390
same amount of time.

00:04:07.390 --> 00:04:12.080
So he has to go 300 meters
more than how far Chelsea

00:04:12.080 --> 00:04:14.360
travels in that time.

00:04:14.360 --> 00:04:17.270
So that's what this is, and
then we essentially are just

00:04:17.270 --> 00:04:19.200
trying to solve for this time.

00:04:19.200 --> 00:04:20.150
We want to be able
to solve for this.

00:04:20.150 --> 00:04:22.190
Let's solve this equation.

00:04:22.190 --> 00:04:24.620
Remember, all these equations
really have the same form.

00:04:24.620 --> 00:04:26.410
In order for this person
to overtake this

00:04:26.410 --> 00:04:28.710
person in 6 minutes.

00:04:28.710 --> 00:04:31.770
That means in 6 minutes she
would have to travel 100

00:04:31.770 --> 00:04:33.100
meters more than him.

00:04:33.100 --> 00:04:36.360
In 12 minutes she would
have to travel 400 meters

00:04:36.360 --> 00:04:38.020
more than Chelsea.

00:04:38.020 --> 00:04:40.090
And then in t minutes-- and
that's what we're going to have

00:04:40.090 --> 00:04:42.850
to solve the problem for --b is
going to have to travel

00:04:42.850 --> 00:04:46.150
300 meters more than
Chelsea right there.

00:04:46.150 --> 00:04:48.800
We just solve for t, so
let's just solve for t.

00:04:48.800 --> 00:04:49.440
So what do you get?

00:04:49.440 --> 00:04:57.750
You get vbt minus vct
is equal to 300.

00:04:57.750 --> 00:05:01.240
I just subtracted Chelsea's
velocity times time from

00:05:01.240 --> 00:05:03.060
both sides of the equation.

00:05:03.060 --> 00:05:06.380
We can factor out the
time, so let's do that.

00:05:06.380 --> 00:05:13.590
So you get vb minus vc times
time-- I'm just factoring out

00:05:13.590 --> 00:05:16.180
the time --is equal to 300.

00:05:16.180 --> 00:05:25.310
Or time is equal to
300/vb minus vc.

00:05:25.310 --> 00:05:27.580
So if we could somehow figure
out what this is equal

00:05:27.580 --> 00:05:29.680
to, then we would have
solved our problem.

00:05:29.680 --> 00:05:32.450
Let's see if we can use
the other information.

00:05:32.450 --> 00:05:36.750
Let's see, if we solve for vb
here and solved for vc here,

00:05:36.750 --> 00:05:38.330
hopefully the va's
will cancel out.

00:05:38.330 --> 00:05:40.900
And I suspect they will,
otherwise this problem

00:05:40.900 --> 00:05:42.440
would not be solvable.

00:05:42.440 --> 00:05:43.330
So let's do that.

00:05:43.330 --> 00:05:45.510
Let's rewrite this
equation up here.

00:05:45.510 --> 00:05:47.845
Let's solve for vb,
for Bill's velocity.

00:05:51.305 --> 00:05:54.920
Let's see, if we divide both
sides by 6-- well, let's

00:05:54.920 --> 00:05:56.780
subtract 100 from both sides.

00:05:56.780 --> 00:06:01.180
So you have Bill's velocity
times 6-- that's just that

00:06:01.180 --> 00:06:07.250
--is equal to 6 times
Alice's velocity minus 100.

00:06:07.250 --> 00:06:09.470
All I did is I subtracted
100 from both sides of

00:06:09.470 --> 00:06:12.090
this equation and I
swapped the sides.

00:06:12.090 --> 00:06:13.530
Divide both sides by 6.

00:06:13.530 --> 00:06:17.890
You got the velocity of Bill
is equal to the velocity

00:06:17.890 --> 00:06:23.360
of Alice minus 100/6.

00:06:23.360 --> 00:06:27.160
100/6 or 50/3-- I'll just
keep it as a 100/6.

00:06:27.160 --> 00:06:28.700
Fair enough.

00:06:28.700 --> 00:06:30.480
Let's see if we can do the same
thing with this equation.

00:06:33.110 --> 00:06:37.670
If we subtract 400 from both
sides, we get 12 times

00:06:37.670 --> 00:06:44.490
Chelsea's velocity is equal
to 12 times Alice's

00:06:44.490 --> 00:06:47.580
velocity minus 400.

00:06:47.580 --> 00:06:49.780
Divide both sides of
this equation by 12.

00:06:49.780 --> 00:06:52.790
You get Chelsea's velocity
is equal to Alice's

00:06:52.790 --> 00:06:57.170
velocity minus 400/12.

00:06:57.170 --> 00:06:58.150
All right.

00:06:58.150 --> 00:07:04.170
So let's if we can substitute
this back in for vb, and

00:07:04.170 --> 00:07:05.680
substitute this back in for vc.

00:07:05.680 --> 00:07:07.530
And hopefully the
va's cancel out.

00:07:07.530 --> 00:07:09.780
It already looks
like they should.

00:07:09.780 --> 00:07:16.050
So our time, the time that it
takes for Bill to overtake

00:07:16.050 --> 00:07:18.920
Chelsea-- remember that's what
t was, it's always good to

00:07:18.920 --> 00:07:21.250
remind yourself what this whole
problem was about to begin

00:07:21.250 --> 00:07:26.340
with --is equal to 300/vb.

00:07:26.340 --> 00:07:29.460
Well instead of vb
let's write this.

00:07:29.460 --> 00:07:38.900
Va, the velocity of Alice,
minus 100/6 minus vc.

00:07:38.900 --> 00:07:41.890
So minus this right here.

00:07:41.890 --> 00:07:47.730
Minus va minus 400/12.

00:07:47.730 --> 00:07:51.710
This should simplify to
t-- I'll just arbitrarily

00:07:51.710 --> 00:07:52.880
switch colors.

00:07:52.880 --> 00:08:00.370
t is equal to 300/va
minus-- what's 100/6?

00:08:00.370 --> 00:08:05.290
100/6 is the same
thing as 50/3.

00:08:05.290 --> 00:08:06.420
Distribute the minus sign.

00:08:06.420 --> 00:08:09.400
Minus va-- we can already see
these are going to cancel out

00:08:09.400 --> 00:08:13.610
--minus times a minus plus
400/12 is the same thing

00:08:13.610 --> 00:08:17.560
is 200/6 or 100/3.

00:08:17.560 --> 00:08:18.460
Right?

00:08:18.460 --> 00:08:21.950
Plus 100/3.

00:08:21.950 --> 00:08:25.300
These two cancel
out, va minus va.

00:08:25.300 --> 00:08:29.640
So then we're left with
t is equal to 300 over

00:08:29.640 --> 00:08:33.300
100/3 minus 50/3.

00:08:33.300 --> 00:08:33.640
Right?

00:08:33.640 --> 00:08:38.030
Or I could say minus 50/3 plus
100/3, but either way, minus

00:08:38.030 --> 00:08:42.480
50/3 plus 100/3 is just 50/3.

00:08:42.480 --> 00:08:43.380
Did I do that right?

00:08:43.380 --> 00:08:46.990
If I divide the top and the
bottom by 4, I get this.

00:08:46.990 --> 00:08:50.400
And then sure, this is
a 100/3 and than 50/3.

00:08:50.400 --> 00:08:52.790
And so when you divide by a
fraction that's the same thing

00:08:52.790 --> 00:08:54.210
is multiplying by its inverse.

00:08:54.210 --> 00:08:59.550
300 times 3/50 over 1.

00:08:59.550 --> 00:09:02.800
And then we can cancel out
some terms, so we don't have

00:09:02.800 --> 00:09:04.010
to multiply big numbers.

00:09:04.010 --> 00:09:07.470
So if we divide the top and the
bottom by 50 that becomes 1,

00:09:07.470 --> 00:09:13.550
this becomes 6, and then we're
left with t is equal to 6 times

00:09:13.550 --> 00:09:17.340
3/1, which is 18, and all the
time units we were working with

00:09:17.340 --> 00:09:20.070
the whole time were in minutes.

00:09:20.070 --> 00:09:20.795
And we're done.

00:09:20.795 --> 00:09:23.330
That was a little hairier than
the other two, but notice we

00:09:23.330 --> 00:09:27.190
were able to solve it by just
algebraically writing down the

00:09:27.190 --> 00:09:29.670
information that they actually
gave us in the problem and just

00:09:29.670 --> 00:09:32.580
seeing what we have to solve
for and then substituting back

00:09:32.580 --> 00:09:36.250
in and then miraculously
things canceled out.

00:09:36.250 --> 00:09:38.700
And in general, if you're given
a nice problem like this you

00:09:38.700 --> 00:09:42.480
can normally just, you know,
march forward knowing that if

00:09:42.480 --> 00:09:44.650
you do things correctly
probably nice things will

00:09:44.650 --> 00:09:46.220
happen and terms
will cancel out.

00:09:46.220 --> 00:09:47.830
But I thought you would find
that interesting and once

00:09:47.830 --> 00:09:51.230
again, thanks to Kortaggio
for that problem.


Early train word problem

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:04.430
This is another problem that
Kortaggio sent me, and I like

00:00:04.430 --> 00:00:07.750
these problems a lot because
they seem fairly simple on the

00:00:07.750 --> 00:00:10.010
surface, and actually the
solutions are pretty simple.

00:00:10.010 --> 00:00:13.300
But just the nature of the lack
of information they give, it

00:00:13.300 --> 00:00:15.760
becomes kind of hard to even
get started with

00:00:15.760 --> 00:00:16.310
these problems.

00:00:16.310 --> 00:00:18.820
And I'll be frank, this
problem, I kind of stumbled

00:00:18.820 --> 00:00:21.440
around with it for a couple of
minutes until I finally

00:00:21.440 --> 00:00:25.210
realized what they were asking
for or how to solve for it.

00:00:25.210 --> 00:00:28.920
So in this problem, we
have Bev, and she takes a

00:00:28.920 --> 00:00:31.000
train home at 4 o'clock.

00:00:31.000 --> 00:00:33.760
She arrives at the
station at 6 o'clock.

00:00:33.760 --> 00:00:36.800
Every day, driving at the
same rate, her husband

00:00:36.800 --> 00:00:38.630
meets her at the station.

00:00:38.630 --> 00:00:39.450
Fair enough.

00:00:39.450 --> 00:00:43.840
One day, she takes the train an
hour early and arrives at 5:00.

00:00:43.840 --> 00:00:47.165
Her husband leaves home to meet
her at the same usual time.

00:00:50.410 --> 00:00:53.860
Husband leaves home to meet
her at the usual time.

00:00:53.860 --> 00:00:56.180
So maybe an interesting thing
is to think about when does

00:00:56.180 --> 00:00:58.810
the husband normally leave?

00:00:58.810 --> 00:01:01.300
So let's say that this
is a normal scenario.

00:01:01.300 --> 00:01:02.700
I'll do that in green.

00:01:02.700 --> 00:01:06.000
At 6 o'clock, she arrives.

00:01:06.000 --> 00:01:06.530
We got there.

00:01:06.530 --> 00:01:08.130
She arrives at the
station at 6 o'clock.

00:01:08.130 --> 00:01:10.730
I don't know if this 4
o'clock is useful yet.

00:01:10.730 --> 00:01:14.530
And the husband, he gets there
right when she gets there.

00:01:14.530 --> 00:01:19.220
So he traveled-- he gets there,
and I don't know if I'm-- maybe

00:01:19.220 --> 00:01:23.380
I should draw a more-- well,
let me just draw it this way.

00:01:23.380 --> 00:01:27.150
I think this is how my
brain kind of handles it.

00:01:27.150 --> 00:01:30.960
So he gets there.

00:01:30.960 --> 00:01:32.760
And when did he have to leave?

00:01:32.760 --> 00:01:33.610
What was his time?

00:01:33.610 --> 00:01:36.960
So let's say it takes him
t minutes to get there,

00:01:36.960 --> 00:01:38.750
whatever t is given in.

00:01:38.750 --> 00:01:41.150
So if he gets there at 6
o'clock, that means he left

00:01:41.150 --> 00:01:43.260
at 6 o'clock minus t.

00:01:43.260 --> 00:01:45.400
If t is 30 minutes,
it would be 5:30.

00:01:45.400 --> 00:01:49.000
If t is one hour, it
would be 5 o'clock.

00:01:49.000 --> 00:01:58.600
So leaves at 6 o'clock minus t.

00:01:58.600 --> 00:02:00.970
And t, whatever units
t happens to be.

00:02:00.970 --> 00:02:02.450
Then he picks her up,
and what does he do?

00:02:02.450 --> 00:02:05.150
He goes back home, right?

00:02:05.150 --> 00:02:06.910
He goes back home.

00:02:06.910 --> 00:02:10.250
Let me do that with a-- I'll
do that with a skinnier line.

00:02:10.250 --> 00:02:14.850
He goes back home, same
distance, and we assume that

00:02:14.850 --> 00:02:16.890
the car just turns around
immediately, and that there's

00:02:16.890 --> 00:02:18.530
no time devoted to
picking her up.

00:02:18.530 --> 00:02:21.960
She just kind of jumps on
the car as he spins around.

00:02:21.960 --> 00:02:24.520
So it should take him the same
amount of time, right, t.

00:02:24.520 --> 00:02:25.890
So when does he
arrive back at home?

00:02:25.890 --> 00:02:28.940
So if it took him-- he left t
minutes before 6 o'clock and

00:02:28.940 --> 00:02:31.020
got there, so when
does he get back?

00:02:31.020 --> 00:02:32.720
Well, it's going to take him t
minutes to get back, so he's

00:02:32.720 --> 00:02:39.020
going to arrive at
6 o'clock plus t.

00:02:39.020 --> 00:02:40.670
And so what was the
total amount of time

00:02:40.670 --> 00:02:41.280
that he traveled?

00:02:41.280 --> 00:02:43.650
And this is almost
superficially easy.

00:02:43.650 --> 00:02:47.420
Well, if you-- 6 o'clock plus t
minus 6 o'clock minus t, the

00:02:47.420 --> 00:02:49.180
difference in time is just 2t.

00:02:49.180 --> 00:02:52.830
He traveled a distance of-- or
a total time of 2t minutes,

00:02:52.830 --> 00:02:54.120
and that's almost obvious.

00:02:54.120 --> 00:02:56.310
If it takes him t minutes to go
and t minutes to come back,

00:02:56.310 --> 00:02:58.130
he traveled t minutes.

00:02:58.130 --> 00:02:59.240
Fair enough.

00:02:59.240 --> 00:03:01.370
Now, what happened on this day?

00:03:01.370 --> 00:03:04.420
It says that he leaves-- her
husband leaves home to meet

00:03:04.420 --> 00:03:05.790
her at the usual time.

00:03:05.790 --> 00:03:09.810
So once again, he is going
to leave-- he leaves--

00:03:09.810 --> 00:03:11.520
I'll do this in red.

00:03:11.520 --> 00:03:22.780
He leaves at 6 o'clock
minus t, right?

00:03:22.780 --> 00:03:26.060
And then, she arrived early,
so she's going to be walking

00:03:26.060 --> 00:03:28.510
back, so there's going to be
some smaller time that it

00:03:28.510 --> 00:03:30.250
takes him to reach her.

00:03:30.250 --> 00:03:33.530
So let's just leave that
unlabeled right now.

00:03:33.530 --> 00:03:36.840
So if I have a line here-- so
he's going to travel a shorter

00:03:36.840 --> 00:03:39.350
distance, and, of course, it's
going to take him less time

00:03:39.350 --> 00:03:41.150
because she started walking.

00:03:41.150 --> 00:03:44.440
That's what the problem
tells us, right?

00:03:44.440 --> 00:03:47.420
Today she arrives at 5 o'clock,
and she begins to walk home.

00:03:47.420 --> 00:03:49.610
So she's going to make some
distance up so he's not going

00:03:49.610 --> 00:03:51.650
to have to travel quite as far.

00:03:51.650 --> 00:03:54.810
And then whatever time
that was, he goes

00:03:54.810 --> 00:03:57.710
back the other way.

00:03:57.710 --> 00:04:01.360
It's going to be the
same amount of time.

00:04:01.360 --> 00:04:03.670
It's the same amount of time.

00:04:03.670 --> 00:04:06.130
And what does it tell us?

00:04:06.130 --> 00:04:08.330
He meets her on the way,
and then they arrive

00:04:08.330 --> 00:04:09.870
home 20 minutes early.

00:04:09.870 --> 00:04:13.910
So normally, they arrive at
6 o'clock plus t, right?

00:04:13.910 --> 00:04:14.820
That's normal.

00:04:14.820 --> 00:04:17.800
So today, they're going to
arrive 20 minutes early.

00:04:17.800 --> 00:04:23.930
So they're going to arrive
at, you could say, 6 o'clock

00:04:23.930 --> 00:04:27.830
plus t-- that's just that--
minus 20 minutes, right?

00:04:27.830 --> 00:04:29.690
They're going to arrive
20 minutes early.

00:04:29.690 --> 00:04:31.660
So what's the total amount
of time that he would have

00:04:31.660 --> 00:04:33.220
traveled on that day?

00:04:33.220 --> 00:04:35.110
The total amount of time?

00:04:35.110 --> 00:04:39.480
Well, he essentially travels
20 minutes less than he does

00:04:39.480 --> 00:04:40.480
on a normal day, right?

00:04:40.480 --> 00:04:42.630
He leaves at the same time and
he gets there 20 minutes early.

00:04:42.630 --> 00:04:46.080
And if you take the difference
between 6 plus t minus 20 and 6

00:04:46.080 --> 00:04:50.490
minus t, you're going
to get 2t minus 20.

00:04:50.490 --> 00:04:52.730
And that's almost-- I
probably didn't even have

00:04:52.730 --> 00:04:53.750
to draw all of this.

00:04:53.750 --> 00:04:55.545
You could just say, well, you
know, on a regular day, it

00:04:55.545 --> 00:04:57.990
takes them two-- if t is the
amount of time it normally

00:04:57.990 --> 00:05:00.380
takes him to go to the
station, on a normal day,

00:05:00.380 --> 00:05:03.080
he travels 2t minutes.

00:05:03.080 --> 00:05:06.140
Today, he leaves at the same
time, gets 20 minutes early, so

00:05:06.140 --> 00:05:09.870
he travels 2t minus 20 minutes.

00:05:09.870 --> 00:05:10.950
Fair enough.

00:05:10.950 --> 00:05:12.110
Now, what does that tell us?

00:05:12.110 --> 00:05:14.630
Well, how long did it
take him to get to her?

00:05:14.630 --> 00:05:17.200
How long did this take her?

00:05:17.200 --> 00:05:20.410
Well, to go to pick her up
takes the same amount of time

00:05:20.410 --> 00:05:23.620
as to come back, so he must
have taken half of this time

00:05:23.620 --> 00:05:25.940
to go and half of this
time to come back.

00:05:25.940 --> 00:05:29.650
So it must have been t
minus 10 to go, and then

00:05:29.650 --> 00:05:32.340
t minus 10 to come back.

00:05:32.340 --> 00:05:35.800
So it took him 10 minutes
less to reach her this time.

00:05:35.800 --> 00:05:37.630
So what time does
he pick her up?

00:05:37.630 --> 00:05:39.570
And this is the key.

00:05:39.570 --> 00:05:43.090
So normally, when it takes
him t minutes, he gets

00:05:43.090 --> 00:05:44.820
there at 6 o'clock.

00:05:44.820 --> 00:05:48.040
This time she started walking
because she got there early,

00:05:48.040 --> 00:05:51.080
and he gets there 10 minutes--
he reaches her 10 minutes

00:05:51.080 --> 00:05:53.190
earlier than he normally
would have reached her,

00:05:53.190 --> 00:05:54.910
right? t minus 10.

00:05:54.910 --> 00:05:57.020
Normally, he reaches
her at 6 o'clock.

00:05:57.020 --> 00:05:59.340
Today he reaches her
10 minutes earlier.

00:05:59.340 --> 00:06:05.740
So he reaches her at 5:50.

00:06:05.740 --> 00:06:10.750
Now, the question is, how
many minutes did Bev walk?

00:06:10.750 --> 00:06:13.500
How many minutes did Bev walk?

00:06:13.500 --> 00:06:17.340
Remember, she arrived an hour
early and arrives at 5 o'clock,

00:06:17.340 --> 00:06:19.550
and then she starts walking.

00:06:19.550 --> 00:06:24.250
She arrives at 5:00, 5 o'clock,
and just starts walking.

00:06:24.250 --> 00:06:25.330
And when does he pick her up?

00:06:25.330 --> 00:06:26.950
He picks her up at 5:50.

00:06:26.950 --> 00:06:30.540
So she walked for 50 minutes.

00:06:30.540 --> 00:06:31.900
That's a neat problem.

00:06:31.900 --> 00:06:35.010
Because on some level, it's
very easy, but on a lot of

00:06:35.010 --> 00:06:38.000
other levels, they give you
all of this other information

00:06:38.000 --> 00:06:39.110
that's not necessary.

00:06:39.110 --> 00:06:40.870
For example, you don't
have to know that she

00:06:40.870 --> 00:06:42.480
leaves at 4 o'clock.

00:06:42.480 --> 00:06:45.040
That's actually probably the
most unnecessary piece of

00:06:45.040 --> 00:06:47.480
information, but everything
else is kind of left abstract.

00:06:47.480 --> 00:06:49.750
But even though you can--
they're not giving a lot of

00:06:49.750 --> 00:06:52.780
details, you can actually
figure out how far she walked

00:06:52.780 --> 00:06:56.100
without knowing how fast she
walks, or how fast her husband

00:06:56.100 --> 00:06:58.640
drives, or how far they are
from the train station, or how

00:06:58.640 --> 00:07:00.670
far the house is from the train
station, or any of that.

00:07:00.670 --> 00:07:02.970
You're still able to figure
out how far she walked.

00:07:02.970 --> 00:07:04.620
Anyway, neat problem.

00:07:04.620 --> 00:07:05.680
Thanks again to Kortaggio.