Why Distance is Area under Velocity-Time Line

Let's say I have
something moving

with a constant velocity
of five meters per second.

And we're just assuming
it's moving to the right,

just to give us a direction,
because this is a vector

quantity, so it's moving in
that direction right over there.

And let me plot its
velocity against time.

So this is my velocity.

So I'm actually
going to only plot

the magnitude of the
velocity, and you

can specify that like this.

So this is the magnitude
of the velocity.

And then on this axis
I'm going to plot time.

So we have a constant velocity
of five meters per second.

So its magnitude is
five meters per second.

And it's constant.

It's not changing.

As the seconds tick away the
velocity does not change.

So it's just moving
five meters per second.

Now, my question to you
is how far does this thing

travel after five seconds?

So after five seconds-- so
this is one second, two second,

three seconds, four seconds,
five seconds, right over here.

So how far did this thing
travel after five seconds?

Well, we could think
about it two ways.

One, we know that velocity
is equal to displacement over

change in time.

And displacement is
just change in position

over change in time.

Or another way to
think about it--

If you multiply
both sides by change

in time-- you get velocity
times change in time,

is equal to displacement.

So what was of the
displacement over here?

Well, I know what
the velocity is--

it's five meters per second.

That's the velocity,
let me color-code this.

That is the velocity.

And we know what the change in
time is, it is five seconds.

And so you get the seconds
cancel out the seconds,

you get five times five-- 25
meters-- is equal to 25 meters.

And that's pretty
straightforward.

But the slightly more
interesting thing

is that's exactly the area under
this rectangle right over here.

What I'm going to show
you in this video,

that is in general,
if you plot velocity,

the magnitude of velocity.

So you could say
speed to versus time.

Or let me just stay
with the magnitude

of the velocity versus time.

The area under
that curve is going

to be the distance traveled,
because, or the displacement.

Because displacement is
just the velocity times

the change in time.

So if you just take out a
rectangle right over there.

So let me draw a
slightly different one

where the velocity is changing.

So let me draw a situation
where you have a constant

acceleration .

The acceleration
over here is going

to be one meter per
second, per second.

So one meter per
second, squared.

And let me draw the
same type of graph,

although this is going to
look a little different now.

So this is my velocity axis.

I'll give myself a
little bit more space.

So this is my velocity axis.

I'm just going to draw the
magnitude of the velocity,

and this right over
here is my time axis.

So this is time.

And let me mark
some stuff off here.

So one, two, three, four, five,
six, seven, eight, nine, ten.

And one, two, three, four, five,
six, seven, eight, nine, ten.

And the magnitude
of velocity is going

to be measured in
meters per second.

And the time is going to
be measured in seconds.

So my initial
velocity, or I could

say the magnitude of
my initial velocity--

so just my initial
speed, you could say,

this is just a
fancy way of saying

my initial speed is zero.

So my initial speed is zero.

So after one second
what's going to happen?

After one second I'm going
one meter per second faster.

So now I'm going one
meter per second.

After two seconds,
whats happened?

Well now I'm going another meter
per second faster than that.

After another second--
if I go forward in time,

if change in time is
one second, then I'm

going a second faster than that.

And if you remember the idea of
the slope from your algebra one

class, that's exactly
what the acceleration

is in this diagram
right over here.

The acceleration, we
know that acceleration

is equal to change in
velocity over change in time.

Over here change in time
is along the x-axis.

So this right over here
is a change in time.

And this right over here
is a change in velocity.

When we plot velocity or
the magnitude of velocity

relative to time, the slope of
that line is the acceleration.

And since we're assuming the
acceleration is constant,

we have a constant slope.

So we have just a line here.

We don't have a curve.

Now what I want to do is
think about a situation.

Let's say that we accelerate it
one meter per second squared.

And we do it for--
so the change in time

is going to be five seconds.

And my question to you is
how far have we traveled?

Which is a slightly more
interesting question

than what we've
been asking so far.

So we start off with an
initial velocity of zero.

And then for five
seconds we accelerate

it one meter per second squared.

So one, two, three, four, five.

So this is where we go.

This is where we are.

So after five seconds,
we know our velocity.

Our velocity is now
five meters per second.

But how far have we traveled?

So we could think about
it a little bit visually.

We could say, look, we could try
to draw rectangles over here.

Maybe right over here,
we have the velocity

of one meter per second.

So if I say one meter per
second times the second,

that'll give me a
little bit of distance.

And then the next one I have
a little bit more of distance,

calculated the same way.

I could keep drawing
these rectangles here,

but then you're like, wait,
those rectangles are missing,

because I wasn't for
the whole second,

I wasn't only going
one meter per second.

I kept accelerating.

So I actually, I should maybe
split up the rectangles.

I could split up the
rectangles even more.

So maybe I go every half second.

So on this half-second I
was going at this velocity.

And I go that velocity
for a half-second.

Velocity times the time would
give me the displacement.

And I do it for the
next half second.

Same exact idea here.

Gives me the displacement.

So on and so forth.

But I think what you see as
you're getting-- is the more

accurate-- the smaller
the rectangles,

you try to make here, the closer
you're going to get to the area

under this curve.

And just like the
situation here.

This area under
the curve is going

to be the distance traveled.

And lucky for us, this is
just going to be a triangle,

and we know how to figure
out the area for triangle.

So the area of a triangle
is equal to one half

times base times height.

Which hopefully
makes sense to you,

because if you just
multiply base times height,

you get the area for
the entire rectangle,

and the triangle is
exactly half of that.

So the distance traveled
in this situation,

or I should say
the displacement,

just because we want to make
sure we're focused on vectors.

The displacement
here is going to be--

or I should say the magnitude
of the displacement,

maybe, which is the same
thing as the distance,

is going to be one
half times the base,

which is five seconds,
times the height,

which is five meters per second.

Times five meters.

Let me do that in another color.

Five meters per second.

The seconds cancel
out with the seconds.

And we're left with one half
times five times five meters.

So it's one half times 25,
which is equal to 12.5 meters.

And so there's an interesting
thing here, well one,

there's a couple of
interesting things.

Hopefully you'll realize that
if you're plotting velocity

versus time, the
area under the curve,

given a certain amount
of time, tells you

how far you have traveled.

The other interesting thing
is that the slope of the curve

tells you your acceleration.

What's the slope over here?

Well, It's completely flat.

And that's because the
velocity isn't changing.

So in this situation, we
have a constant acceleration.

The magnitude of that
acceleration is exactly zero.

Our velocity is not changing.

Here we have an acceleration of
one meter per second squared,

and that's why the slope of this
line right over here is one.

The other interesting
thing, is, if even

if you have constant
acceleration,

you could still figure
out the distance

by just taking the area
under the curve like this.

We were able to
figure out there we

were able to get 12.5 meters.

The last thing I want to
introduce you to-- actually,

let me just do it
until next video,

and I'll introduce you to
the idea of average velocity.

Now that we feel
comfortable with the idea,

that the distance
you traveled is

the area under the
velocity versus time curve.