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Now that we know a little bit
about vectors and scalars,

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let's try to apply what we
know about them for some pretty

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common problems you'd, one,
see in a physics class,

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but they're also common problems
you'd see in everyday life,

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because you're trying to
figure out how far you've gone,

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or how fast you're
going, or how long it

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might take you to
get some place.

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So first I have, if
Shantanu was able to travel

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5 kilometers north in
1 hour in his car, what

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was his average velocity?

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So one, let's just
review a little bit

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about what we know about
vectors and scalars.

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So they're giving us that
he was able to travel

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5 kilometers to the north.

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So they gave us a magnitude,
that's the 5 kilometers.

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That's the size of
how far he moved.

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And they also give a direction.

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So he moved a distance
of 5 kilometers.

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Distance is the scalar.

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But if you give the direction
too, you get the displacement.

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So this right here
is a vector quantity.

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He was displaced 5
kilometers to the north.

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And he did it in
1 hour in his car.

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What was his average velocity?

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So velocity, and
there's many ways

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that you might see it defined,
but velocity, once again,

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is a vector quantity.

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And the way that we
differentiate between vector

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and scalar quantities
is we put little arrows

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on top of vector quantities.

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Normally they are bolded,
if you can have a typeface,

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and they have an
arrow on top of them.

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But this tells you
that not only do I

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care about the
value of this thing,

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or I care about the
size of this thing,

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I also care about its direction.

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That's what the arrow.

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The arrow isn't
necessarily its direction,

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it just tells you that
it is a vector quantity.

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So the velocity of something
is its change in position,

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including the direction
of its change in position.

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So you could say
its displacement,

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and the letter for
displacement is

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S. And that is a
vector quantity,

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so that is displacement.

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And you might be
wondering, why don't they

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use D for displacement?

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That seems like a much
more natural first letter.

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And my best sense of that is,
once you start doing calculus,

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you start using D for
something very different.

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You use it for the
derivative operator,

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and that's so that the
D's don't get confused.

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And that's why we use
S for displacement.

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If someone has a better
explanation of that,

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feel free to comment
on this video,

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and then I'll add another
video explaining that better

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explanation.

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So velocity is your
displacement over time.

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If I wanted to write an
analogous thing for the scalar

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quantities, I could
write that speed,

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and I'll write
out the word so we

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don't get confused
with displacement.

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Or maybe I'll write "rate."

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Rate is another way that
sometimes people write speed.

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So this is the vector version,
if you care about direction.

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If you don't care
about direction,

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you would have your rate.

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So this is rate, or speed,
is equal to the distance

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that you travel over some time.

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So these two, you could
call them formulas, or you

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could call them
definitions, although I

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would think that they're
pretty intuitive for you.

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How fast something is
going, you say, how far

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did it go over some
period of time.

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These are essentially
saying the same thing.

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This is when you
care about direction,

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so you're dealing with
vector quantities.

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This is where you're not so
conscientious about direction.

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And so you use distance,
which is scalar,

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and you use rate or
speed, which is scalar.

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Here you use displacement,
and you use velocity.

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Now with that out of the
way, let's figure out

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what his average velocity was.

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And this key word,
average, is interesting.

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Because it's possible that
his velocity was changing

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over that whole time period.

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But for the sake of
simplicity, we're

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going to assume that it was
kind of a constant velocity.

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What we are calculating is going
to be his average velocity.

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But don't worry about
it, you can just

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assume that it wasn't changing
over that time period.

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So his velocity is,
his displacement

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was 5 kilometers to the north--
I'll write just a big capital.

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Well, let me just write it
out, 5 kilometers north--

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over the amount of
time it took him.

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And let me make it clear.

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This is change in time.

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This is also a change in time.

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Sometimes you'll just
see a t written there.

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Sometimes you'll see
someone actually put

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this little triangle,
the character delta,

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in front of it, which
explicitly means "change in."

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It looks like a very fancy
mathematics when you see that,

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but a triangle in
front of something

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literally means "change in."

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So this is change in time.

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So he goes 5 kilometers
north, and it took him 1 hour.

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So the change in
time was 1 hour.

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So let me write that over here.

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So over 1 hour.

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So this is equal
to, if you just look

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at the numerical
part of it, it is

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5/1-- let me just write
it out, 5/1-- kilometers,

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and you can treat the
units the same way

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you would treat the
quantities in a fraction.

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5/1 kilometers per hour,
and then to the north.

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Or you could say this
is the same thing

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as 5 kilometers per hour north.

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So this is 5 kilometers
per hour to the north.

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So that's his average velocity,
5 kilometers per hour.

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And you have to be careful,
you have to say "to the north"

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if you want velocity.

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If someone just said "5
kilometers per hour,"

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they're giving you a speed,
or rate, or a scalar quantity.

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You have to give the direction
for it to be a vector quantity.

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You could do the same
thing if someone just said,

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what was his average
speed over that time?

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You could have said, well, his
average speed, or his rate,

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would be the
distance he travels.

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The distance, we don't care
about the direction now,

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is 5 kilometers, and
he does it in 1 hour.

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His change in time is 1 hour.

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So this is the same thing
as 5 kilometers per hour.

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So once again, we're only
giving the magnitude here.

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This is a scalar quantity.

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If you want the vector, you
have to do the north as well.

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Now, you might be saying,
hey, in the previous video,

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we talked about things in
terms of meters per second.

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Here, I give you kilometers,
or "kil-om-eters,"

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depending on how you
want to pronounce it,

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kilometers per hour.

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What if someone wanted
it in meters per second,

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or what if I just wanted to
understand how many meters he

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travels in a second?

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And there, it just becomes
a unit conversion problem.

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And I figure it doesn't hurt
to work on that right now.

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So if we wanted to do
this to meters per second,

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how would we do it?

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Well, the first step is to
think about how many meters we

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are traveling in an hour.

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So let's take that 5
kilometers per hour,

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and we want to
convert it to meters.

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So I put meters
in the numerator,

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and I put kilometers
in the denominator.

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And the reason why I do that
is because the kilometers

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are going to cancel out
with the kilometers.

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And how many meters are
there per kilometer?

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Well, there's 1,000 meters
for every 1 kilometer.

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And I set this up right here so
that the kilometers cancel out.

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So these two
characters cancel out.

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And if you multiply,
you get 5,000.

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So you have 5 times 1,000.

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So let me write this-- I'll do
it in the same color-- 5 times

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1,000.

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So I just multiplied
the numbers.

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When you multiply something,
you can switch around the order.

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Multiplication is
commutative-- I always

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have trouble pronouncing
that-- and associative.

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And then in the units, in the
numerator, you have meters,

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and in the denominator,
you have hours.

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Meters per hour.

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And so this is equal to
5,000 meters per hour.

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And you might say,
hey, Sal, I know

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that 5 kilometers is the
same thing as 5,000 meters.

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I could do that in my head.

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And you probably could.

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But this canceling out
dimensions, or what's

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often called
dimensional analysis,

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can get useful once you
start doing really, really

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complicated things with less
intuitive units than something

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like this.

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But you should always do an
intuitive gut check right here.

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You know that if you do
5 kilometers in an hour,

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that's a ton of meters.

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So you should get
a larger number

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if you're talking
about meters per hour.

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And now when we want
to go to seconds,

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let's do an intuitive gut check.

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If something is traveling a
certain amount in an hour,

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it should travel a much
smaller amount in a second,

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or 1/3,600 of an hour, because
that's how many seconds there

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are in an hour.

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So that's your gut check.

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We should get a smaller
number than this

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when we want to say
meters per second.

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But let's actually do it with
the dimensional analysis.

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So we want to cancel
out the hours,

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and we want to be left with
seconds in the denominator.

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So the best way to cancel
this hours in the denominator

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is by having hours
in the numerator.

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So you have hours per second.

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So how many hours
are there per second?

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Or another way to
think about it, 1 hour,

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think about the larger unit,
1 hour is how many seconds?

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Well, you have 60 seconds
per minute times 60 minutes

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per hour.

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The minutes cancel out.

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60 times 60 is 3,600
seconds per hour.

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So you could say this is 3,600
seconds for every 1 hour,

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or if you flip them, you would
get 1/3,600 hour per second,

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or hours per second, depending
on how you want to do it.

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So 1 hour is the same
thing as 3,600 seconds.

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And so now this hour
cancels out with that hour,

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and then you multiply,
or appropriately divide,

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the numbers right here.

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And you get this is equal
to 5,000 over 3,600 meters

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per-- all you have left in the
denominator here is second.

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Meters per second.

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And if we divide both the
numerator and the denominator--

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I could do this by hand, but
just because this video's

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already getting a
little bit long,

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let me get my trusty
calculator out.

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00:11:00,990 --> 00:11:05,470
I get my trusty calculator
out just for the sake of time.

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00:11:05,470 --> 00:11:09,280
5,000 divided by 3,600, which
would be really the same thing

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00:11:09,280 --> 00:11:13,410
as 50 divided by
36, that is 1.3--

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I'll just round it
over here-- 1.39.

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So this is equal to
1.39 meters per second.

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So Shantanu was traveling
quite slow in his car.

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Well, we knew that just
by looking at this.

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5 kilometers per hour, that's
pretty much just letting

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the car roll pretty slowly.

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