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Let's say I have
something moving

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with a constant velocity
of five meters per second.

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

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

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

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And let me plot its
velocity against time.

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So this is my velocity.

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So I'm actually
going to only plot

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the magnitude of the
velocity, and you

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can specify that like this.

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So this is the magnitude
of the velocity.

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

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

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So its magnitude is
five meters per second.

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And it's constant.

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It's not changing.

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As the seconds tick away the
velocity does not change.

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So it's just moving
five meters per second.

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

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travel after five seconds?

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

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

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So how far did this thing
travel after five seconds?

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Well, we could think
about it two ways.

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One, we know that velocity
is equal to displacement over

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change in time.

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And displacement is
just change in position

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over change in time.

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Or another way to
think about it--

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If you multiply
both sides by change

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

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is equal to displacement.

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So what was of the
displacement over here?

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Well, I know what
the velocity is--

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it's five meters per second.

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

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That is the velocity.

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

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And so you get the seconds
cancel out the seconds,

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

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And that's pretty
straightforward.

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But the slightly more
interesting thing

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

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

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that is in general,
if you plot velocity,

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the magnitude of velocity.

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So you could say
speed to versus time.

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Or let me just stay
with the magnitude

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of the velocity versus time.

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The area under
that curve is going

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

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Because displacement is
just the velocity times

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the change in time.

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

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So let me draw a
slightly different one

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where the velocity is changing.

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So let me draw a situation
where you have a constant

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acceleration .

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The acceleration
over here is going

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to be one meter per
second, per second.

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So one meter per
second, squared.

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And let me draw the
same type of graph,

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although this is going to
look a little different now.

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So this is my velocity axis.

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I'll give myself a
little bit more space.

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So this is my velocity axis.

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

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and this right over
here is my time axis.

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So this is time.

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And let me mark
some stuff off here.

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

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

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And the magnitude
of velocity is going

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to be measured in
meters per second.

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And the time is going to
be measured in seconds.

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So my initial
velocity, or I could

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say the magnitude of
my initial velocity--

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so just my initial
speed, you could say,

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this is just a
fancy way of saying

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my initial speed is zero.

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So my initial speed is zero.

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So after one second
what's going to happen?

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

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So now I'm going one
meter per second.

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After two seconds,
whats happened?

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

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

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

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going a second faster than that.

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

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class, that's exactly
what the acceleration

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is in this diagram
right over here.

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The acceleration, we
know that acceleration

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is equal to change in
velocity over change in time.

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

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So this right over here
is a change in time.

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And this right over here
is a change in velocity.

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When we plot velocity or
the magnitude of velocity

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

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And since we're assuming the
acceleration is constant,

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we have a constant slope.

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So we have just a line here.

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We don't have a curve.

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

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

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

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is going to be five seconds.

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

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Which is a slightly more
interesting question

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than what we've
been asking so far.

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So we start off with an
initial velocity of zero.

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And then for five
seconds we accelerate

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it one meter per second squared.

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So one, two, three, four, five.

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So this is where we go.

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This is where we are.

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So after five seconds,
we know our velocity.

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Our velocity is now
five meters per second.

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But how far have we traveled?

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So we could think about
it a little bit visually.

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

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Maybe right over here,
we have the velocity

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of one meter per second.

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

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that'll give me a
little bit of distance.

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

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calculated the same way.

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I could keep drawing
these rectangles here,

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

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because I wasn't for
the whole second,

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I wasn't only going
one meter per second.

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I kept accelerating.

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

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I could split up the
rectangles even more.

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So maybe I go every half second.

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

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And I go that velocity
for a half-second.

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Velocity times the time would
give me the displacement.

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And I do it for the
next half second.

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Same exact idea here.

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Gives me the displacement.

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So on and so forth.

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

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accurate-- the smaller
the rectangles,

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

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under this curve.

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And just like the
situation here.

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This area under
the curve is going

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to be the distance traveled.

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

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

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So the area of a triangle
is equal to one half

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times base times height.

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Which hopefully
makes sense to you,

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because if you just
multiply base times height,

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you get the area for
the entire rectangle,

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and the triangle is
exactly half of that.

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So the distance traveled
in this situation,

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or I should say
the displacement,

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

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The displacement
here is going to be--

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or I should say the magnitude
of the displacement,

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maybe, which is the same
thing as the distance,

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is going to be one
half times the base,

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which is five seconds,
times the height,

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which is five meters per second.

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Times five meters.

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Let me do that in another color.

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Five meters per second.

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The seconds cancel
out with the seconds.

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

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

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

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there's a couple of
interesting things.

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

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versus time, the
area under the curve,

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given a certain amount
of time, tells you

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how far you have traveled.

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The other interesting thing
is that the slope of the curve

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tells you your acceleration.

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What's the slope over here?

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Well, It's completely flat.

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

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So in this situation, we
have a constant acceleration.

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The magnitude of that
acceleration is exactly zero.

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Our velocity is not changing.

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

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

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The other interesting
thing, is, if even

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if you have constant
acceleration,

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you could still figure
out the distance

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by just taking the area
under the curve like this.

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We were able to
figure out there we

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were able to get 12.5 meters.

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

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let me just do it
until next video,

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

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Now that we feel
comfortable with the idea,

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that the distance
you traveled is

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the area under the
velocity versus time curve.

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