Calculating Average Velocity or Speed

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

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

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

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

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

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

might take you to
get some place.

So first I have, if
Shantanu was able to travel

5 kilometers north in
1 hour in his car, what

was his average velocity?

So one, let's just
review a little bit

about what we know about
vectors and scalars.

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

5 kilometers to the north.

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

That's the size of
how far he moved.

And they also give a direction.

So he moved a distance
of 5 kilometers.

Distance is the scalar.

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

So this right here
is a vector quantity.

He was displaced 5
kilometers to the north.

And he did it in
1 hour in his car.

What was his average velocity?

So velocity, and
there's many ways

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

is a vector quantity.

And the way that we
differentiate between vector

and scalar quantities
is we put little arrows

on top of vector quantities.

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

and they have an
arrow on top of them.

But this tells you
that not only do I

care about the
value of this thing,

or I care about the
size of this thing,

I also care about its direction.

That's what the arrow.

The arrow isn't
necessarily its direction,

it just tells you that
it is a vector quantity.

So the velocity of something
is its change in position,

including the direction
of its change in position.

So you could say
its displacement,

and the letter for
displacement is

S. And that is a
vector quantity,

so that is displacement.

And you might be
wondering, why don't they

use D for displacement?

That seems like a much
more natural first letter.

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

you start using D for
something very different.

You use it for the
derivative operator,

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

And that's why we use
S for displacement.

If someone has a better
explanation of that,

feel free to comment
on this video,

and then I'll add another
video explaining that better

explanation.

So velocity is your
displacement over time.

If I wanted to write an
analogous thing for the scalar

quantities, I could
write that speed,

and I'll write
out the word so we

don't get confused
with displacement.

Or maybe I'll write "rate."

Rate is another way that
sometimes people write speed.

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

If you don't care
about direction,

you would have your rate.

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

that you travel over some time.

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

could call them
definitions, although I

would think that they're
pretty intuitive for you.

How fast something is
going, you say, how far

did it go over some
period of time.

These are essentially
saying the same thing.

This is when you
care about direction,

so you're dealing with
vector quantities.

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

And so you use distance,
which is scalar,

and you use rate or
speed, which is scalar.

Here you use displacement,
and you use velocity.

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

what his average velocity was.

And this key word,
average, is interesting.

Because it's possible that
his velocity was changing

over that whole time period.

But for the sake of
simplicity, we're

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

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

But don't worry about
it, you can just

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

So his velocity is,
his displacement

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

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

over the amount of
time it took him.

And let me make it clear.

This is change in time.

This is also a change in time.

Sometimes you'll just
see a t written there.

Sometimes you'll see
someone actually put

this little triangle,
the character delta,

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

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

but a triangle in
front of something

literally means "change in."

So this is change in time.

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

So the change in
time was 1 hour.

So let me write that over here.

So over 1 hour.

So this is equal
to, if you just look

at the numerical
part of it, it is

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

and you can treat the
units the same way

you would treat the
quantities in a fraction.

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

Or you could say this
is the same thing

as 5 kilometers per hour north.

So this is 5 kilometers
per hour to the north.

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

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

if you want velocity.

If someone just said "5
kilometers per hour,"

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

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

You could do the same
thing if someone just said,

what was his average
speed over that time?

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

would be the
distance he travels.

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

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

His change in time is 1 hour.

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

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

This is a scalar quantity.

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

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

we talked about things in
terms of meters per second.

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

depending on how you
want to pronounce it,

kilometers per hour.

What if someone wanted
it in meters per second,

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

travels in a second?

And there, it just becomes
a unit conversion problem.

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

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

how would we do it?

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

are traveling in an hour.

So let's take that 5
kilometers per hour,

and we want to
convert it to meters.

So I put meters
in the numerator,

and I put kilometers
in the denominator.

And the reason why I do that
is because the kilometers

are going to cancel out
with the kilometers.

And how many meters are
there per kilometer?

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

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

So these two
characters cancel out.

And if you multiply,
you get 5,000.

So you have 5 times 1,000.

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

1,000.

So I just multiplied
the numbers.

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

Multiplication is
commutative-- I always

have trouble pronouncing
that-- and associative.

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

and in the denominator,
you have hours.

Meters per hour.

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

And you might say,
hey, Sal, I know

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

I could do that in my head.

And you probably could.

But this canceling out
dimensions, or what's

often called
dimensional analysis,

can get useful once you
start doing really, really

complicated things with less
intuitive units than something

like this.

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

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

that's a ton of meters.

So you should get
a larger number

if you're talking
about meters per hour.

And now when we want
to go to seconds,

let's do an intuitive gut check.

If something is traveling a
certain amount in an hour,

it should travel a much
smaller amount in a second,

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

are in an hour.

So that's your gut check.

We should get a smaller
number than this

when we want to say
meters per second.

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

So we want to cancel
out the hours,

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

So the best way to cancel
this hours in the denominator

is by having hours
in the numerator.

So you have hours per second.

So how many hours
are there per second?

Or another way to
think about it, 1 hour,

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

Well, you have 60 seconds
per minute times 60 minutes

per hour.

The minutes cancel out.

60 times 60 is 3,600
seconds per hour.

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

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

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

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

And so now this hour
cancels out with that hour,

and then you multiply,
or appropriately divide,

the numbers right here.

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

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

Meters per second.

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

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

already getting a
little bit long,

let me get my trusty
calculator out.

I get my trusty calculator
out just for the sake of time.

5,000 divided by 3,600, which
would be really the same thing

as 50 divided by
36, that is 1.3--

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

So this is equal to
1.39 meters per second.

So Shantanu was traveling
quite slow in his car.

Well, we knew that just
by looking at this.

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

the car roll pretty slowly.