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- [Instructor] All right,
I wanna talk to you about

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acceleration versus time graphs

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because as far as motion graphs go,

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these are probably the hardest.

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One reason is because
acceleration just naturally

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is an abstract concept for
a lot of people to deal with

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and now it's a graph and
people don't like graphs either

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particularly often times.

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Another reason is, if you wanted to know

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the motion of the object,

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let's say it was this doggie.

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This is my doggie Daisy.

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Let's say Daisy was accelerating.

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If you wanted to know the
velocity that daisy had,

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you can't figure it out
directly from this graph

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unless you have some extra information.

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You have to know information
about the velocity

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Daisy had at some moment
in order to figure out

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from this graph the velocity
Daisy had at some other moment.

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So, what can this graph tell
you about the motion of Daisy?

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Well, let's say this graph
described Daisy's acceleration.

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So Daisy can be accelerating.

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Maybe we're playing catch.

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We'll give her a ball.

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We'll throw the ball.

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Hopefully she actually lets
go and she brings it back.

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This graph is gonna
represent her acceleration.

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So this graph, we just read it,

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it says that Daisy had two
meters per second squared

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of acceleration for the first four seconds

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and then her acceleration dropped to zero

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at six seconds and then her
acceleration came negative

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until it was negative
three at nine seconds.

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But, from this we can't tell

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if she's speeding up or slowing down.

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what can we figure out?

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Well, we can figure out some stuff

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because acceleration
is related to velocity

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and we can figure out how
it's related to velocity

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by remembering that it is defined

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to be the change in velocity
over the change in time.

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So this is how we make
our link to velocity.

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So if we solve this for delta v,

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we get that the delta v,

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the change in velocity
over some time interval,

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will be the acceleration
during that time interval

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times interval itself,
how long did that take.

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This is the key to relating
this graph to velocity.

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In other words, let's consider
this first four seconds.

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Let's go between zero and four seconds.

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Daisy had an acceleration of
two meters per second squared.

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So that means, well,
two was the acceleration

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meters per second squared,

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times the accel, times
the time, excuse me,

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the time was four seconds.

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So there was four seconds
worth of acceleration.

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You get positive eight.

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What are the units?

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This second cancels with that second.

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You get positive eight meters per second.

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So the change in velocity
for the first four seconds

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was positive eight.

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This isn't the velocity.

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It's the change in velocity.

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How would you ever find that
for this diagonal region.

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This is as problem.

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Look at this.

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If I wanted to find, let's say
the velocity at six seconds,

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well the acceleration at this point is two

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but then the acceleration
at this point is one.

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The acceleration at this point is zero.

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That acceleration we keep changing.

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How would I ever figure this out?

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What acceleration would I
plug in during this portion?

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But we're in luck.

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This formula allows us to say
something really important.

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A geometric aspect of these graphs

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that are gonna make our life easier

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and the way it makes
our life easier is that,

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look at what this is.

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This is saying acceleration
times delta t, but look it.

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The acceleration we plug in was this, two.

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So for the first four seconds,
the acceleration was two.

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The time, delta t, was four.

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We took this two multiplied by that four

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and got a number, positive eight,

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but this is a height times the width.

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If you take height times width,

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that just represents
the area of a rectangle.

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So all we found was the
area of this rectangle.

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The area is giving us our delta v

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because area, right, of a rectangle is

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height times width.

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We know that the height is

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gonna represent the acceleration here

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and the width is gonna represent delta t.

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Just by the definition of
acceleration we arranged,

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we know that a times delta t

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has to just be the change in velocity.

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So area and change in
velocity are representing

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the exact same thing on this graph.

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Area is the change in velocity.

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That's gonna be really useful

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because when you come over to here

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the area is still gonna
be the change in velocity.

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That's useful because I know how to easily

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find the area of a triangle.

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The area of a triangle is
just 1/2 base times height.

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I don't easily know how to
deal with an acceleration

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that's varying within this formula

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but I do know how to find the area.

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For instance the area
here, though I have 1/2,

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the base is two seconds,

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the height is gonna be positive two

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meters per second squared.

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What are we gonna get?

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One of the halves, cancel.

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Well, the half cancels one of the twos

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and I'm gonna get that this
is gonna be equal to two

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meters per second.

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That's gonna be the area that represents

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the change in velocity.

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So Daisy's velocity changed

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by two meters per second during this time.

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Now you might object.

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You might say, "Wait a minute.

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"I'll buy this over here
because height times width

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"is just a times delta t,

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"but triangle, that has an
extra factor of a half in it,

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"and there's no half up here.

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"How does this, I mean, how
can we still make this claim?"

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We can make this claim

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because we'll do the
same thing we always do.

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We can imagine, all right,
imagine a rectangle here.

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We're gonna estimate the area
with a bunch of rectangles.

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Then this rectangle,

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and this rectangle in your
line like that looks horrible.

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That doesn't look like the
area of a triangle at all.

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It's got all these extra
pieces right here, right?

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You don't want all of that.

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And okay, I agree.

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That didn't work so well.

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Let's make them even smaller, right?

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Smaller width.

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So we'll do a rectangle like that.

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We'll do this one.

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You see we're getting better.

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This is definitely closer.

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This is not as bad as the other one

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but it's still not exact.

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And I agree, that is not exact

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so we'll make it even smaller rectangle

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and an even smaller rectangle here

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all of these at the same width

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but they're even smaller
than the ones before.

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Now we're getting really close.

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This area is really gonna get close

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to the area of the triangle.

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The point is if you make
them infinite testable small,

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they'll exactly represent
the area of a triangle.

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Each one of them can be
found with this formula.

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The delta v for each one will be the area,

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or sorry, the acceleration of
the height of that rectangle

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times the small infinite testable width

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and you'll get the total delta v

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which is so gonna be the total area.

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Long story short, area on a,

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acceleration versus time graphs

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represents the change in velocity.

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This is one you got to remember.

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this is the most important
aspect of an acceleration graph,

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oftentimes the most useful aspect of it,

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the way you analyze it.

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So why do we care about
change in velocity?

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Because it will allow
us to find the velocity.

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We just need to know the
velocity at one point

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then we can find the
velocity at any other point.

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For instance, let's say I gave
you the velocity Daisy had.

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For some reason I'm gonna stopwatch.

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I start my stopwatch at
right at that moment.

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At t equals zero, Daisy had a velocity of,

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let's say positive one meter per second.

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So Daisy was traveling
that fast at t equals zero.

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That was her velocity at
t equals zero seconds.

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Now I can get the
velocity wherever I want.

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If I want the velocity at four,

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let's figure this out.

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To get the velocity at four,
I can say that the delta v

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during this time period right here,

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this four seconds.

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I know what that delta v was.

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That delta v was positive eight.

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We found that area, height times width.

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So positive eight is what
the delta v is gotta equal.

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What's delta v?

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That's v at four seconds

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minus v at zero seconds.

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That's gotta be positive eight.

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I know what v at zero second was.

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That was one.

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So we can get that v at four
minus one meter per second

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is equal to positive
eight meters per second.

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So I get the velocity at four was

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positive nine meters per second.

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And you're like, phew, that was hard.

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I don't wanna do that every time.

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Yeah, I wouldn't wanna
do that every time either

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so there's a quick way to do it.

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We can just do this.

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What's the velocity we had to start with?

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That was one.

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What was our change in velocity?

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That was positive eight.

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So what's our final velocity?

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Well, one plus eight gives
us our final velocity.

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It's positive nine.

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Well it's just gonna take
this change in velocity

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of this area which represents
the change in velocity

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which is gonna add our
initial velocity to it

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when we solve for this final velocity.

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for instance, if I didn't
make sense, for instance,

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if we want to find the velocity at six,

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well, we can just say we
started at t equals four seconds

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with a velocity of positive nine.

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We start here with positive nine.

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Our change was positive two

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so we're gonna end with
positive 11 meters per second.

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You might object.

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You might say, "Wait
a minute, hold on now.

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"If we want delta v,

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"right, and that's positive two,

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"shouldn't delta v be the whole thing

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"from like zero to six seconds?

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"Shouldn't I say v at six
seconds minus v at zero

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"is positive two meters per second?"

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I can't do that.

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The reason I can't do that is because

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look at what I did on the left hand side,

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my time interval goes from zero to six

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but on the right hand side,

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I only included the area from four to six.

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That's the area, there's a
yellow triangle right here.

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If I wanted to put six and
zero on this left hand side,

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I could do that

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but from my total area,

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I wouldn't use that.

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I have to use the total area.

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In other words, the total are
from zero all the way to six

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because that's what I define on this side.

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These sides have to agree with each other.

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So from zero to six,
my total area would be,

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this area here was eight, right?

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We found that rectangle was eight.

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This area here was two.

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So my total area would be 10.

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I can do that if I want.

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I could say v at six minus v at zero was,

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well v at zero we said was one

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because I just gave you that,

260
00:10:28,281 --> 00:10:31,044
equals 10 meters per second.

261
00:10:31,044 --> 00:10:34,368
I get that the v at six
would be 11 meters per second

262
00:10:34,368 --> 00:10:35,920
just like we got it before.

263
00:10:35,920 --> 00:10:38,008
So you can still do it
mathematically like this

264
00:10:38,008 --> 00:10:41,636
but make sure your time
intervals agree on those sides.

265
00:10:41,636 --> 00:10:44,094
Now let's do the last part here.

266
00:10:44,094 --> 00:10:45,062
So we can find this area.

267
00:10:45,062 --> 00:10:48,250
This area and the area always represents

268
00:10:48,250 --> 00:10:51,518
the area from the curve
to the horizontal axis.

269
00:10:51,518 --> 00:10:53,461
So in this case it's
below the horizontal axis.

270
00:10:53,461 --> 00:10:55,484
That means it can negative area.

271
00:10:55,484 --> 00:10:57,751
The reason is it's a triangle again.

272
00:10:57,751 --> 00:11:00,540
So 1/2 base times height.

273
00:11:00,540 --> 00:11:04,707
So 1/2, the base is
one, two, three seconds.

274
00:11:06,020 --> 00:11:09,075
The height is negative three,

275
00:11:09,075 --> 00:11:10,025
negative now,

276
00:11:10,025 --> 00:11:12,318
negative three meters per second squared.

277
00:11:12,318 --> 00:11:13,704
I get that the total area is gonna be

278
00:11:13,704 --> 00:11:16,287
negative 4.5 meters per second.

279
00:11:17,883 --> 00:11:20,398
All right, now Daisy's gonna
have a change in velocity

280
00:11:20,398 --> 00:11:22,507
of negative 4.5.

281
00:11:22,507 --> 00:11:24,917
If we want to get the velocity at nine,

282
00:11:24,917 --> 00:11:26,671
there's a few ways we can do it.

283
00:11:26,671 --> 00:11:29,842
Right, just conceptually, we
can say that Daisy started

284
00:11:29,842 --> 00:11:32,259
at six with a velocity of 11.

285
00:11:33,411 --> 00:11:37,785
Her change during this
period was negative 4.5.

286
00:11:37,785 --> 00:11:40,011
If you just add the
two, you add the change

287
00:11:40,011 --> 00:11:42,436
to the value she started with.

288
00:11:42,436 --> 00:11:45,185
Well you're gonna get positive 6.5

289
00:11:45,185 --> 00:11:49,337
if I add 11 and negative
4.5 meters per second

290
00:11:49,337 --> 00:11:52,994
or, if that sounded like
mathematical witchcraft,

291
00:11:52,994 --> 00:11:55,647
you can say that, all right, delta v

292
00:11:55,647 --> 00:11:57,814
equals, what, negative 4.5

293
00:11:59,722 --> 00:12:00,988
meters per second.

294
00:12:00,988 --> 00:12:03,295
Delta v would be, all
right, you gotta be careful,

295
00:12:03,295 --> 00:12:06,523
this negative 4.5 represents this triangle

296
00:12:06,523 --> 00:12:09,379
so it's gotta be the delta
v between six and nine.

297
00:12:09,379 --> 00:12:12,393
So v at nine minus v at six

298
00:12:12,393 --> 00:12:15,810
has to be negative 4.5 meters per second.

299
00:12:16,774 --> 00:12:19,121
V at nine minus the v at six we know,

300
00:12:19,121 --> 00:12:20,840
v at six was 11.

301
00:12:20,840 --> 00:12:24,280
So I've got minus 11 meters per second

302
00:12:24,280 --> 00:12:26,401
equals negative 4.5.

303
00:12:26,401 --> 00:12:28,022
Wow, we ran out of room.

304
00:12:28,022 --> 00:12:31,765
V at nine would be negative 4.5 plus 11.

305
00:12:31,765 --> 00:12:32,874
That's what we did up here.

306
00:12:32,874 --> 00:12:36,624
We got that it was just
6.5 meters per second

307
00:12:37,567 --> 00:12:39,733
and that agrees with what we said earlier.

308
00:12:39,733 --> 00:12:43,688
So finding the area can get
you the change in velocity

309
00:12:43,688 --> 00:12:46,013
and then knowing the velocity
at one unknown at a time

310
00:12:46,013 --> 00:12:48,209
can get you the velocity at
any other moment in time.

311
00:12:48,209 --> 00:12:49,279
Just be careful.

312
00:12:49,279 --> 00:12:51,275
Make sure you're associating
the right time interval

313
00:12:51,275 --> 00:12:53,619
on both the length and the right side.

314
00:12:53,619 --> 00:12:54,880
They have to agree.

315
00:12:54,880 --> 00:12:56,927
One more thing before you go.

316
00:12:56,927 --> 00:12:58,415
The slope on these graphs

317
00:12:58,415 --> 00:13:00,511
often represents something meaningful.

318
00:13:00,511 --> 00:13:02,081
That's the same in this graph.

319
00:13:02,081 --> 00:13:03,656
So the slope of this graph,

320
00:13:03,656 --> 00:13:05,939
let's try to interpret what this means.

321
00:13:05,939 --> 00:13:09,854
The slope on an acceleration
versus time graph.

322
00:13:09,854 --> 00:13:11,202
Well the slope is always represented

323
00:13:11,202 --> 00:13:13,401
as the rise over the run

324
00:13:13,401 --> 00:13:16,620
and the rise is y two minus y one

325
00:13:16,620 --> 00:13:18,080
over x two minus x one

326
00:13:18,080 --> 00:13:21,471
except instead of y
and x, we have a and t.

327
00:13:21,471 --> 00:13:25,322
So we're gonna have a two minus a one

328
00:13:25,322 --> 00:13:27,973
over t two minus t one.

329
00:13:27,973 --> 00:13:30,202
This is gonna be delta a,

330
00:13:30,202 --> 00:13:32,207
the change in a over the change in time.

331
00:13:32,207 --> 00:13:33,417
What is that?

332
00:13:33,417 --> 00:13:37,228
It's the rate of change
of the acceleration.

333
00:13:37,228 --> 00:13:39,326
That is even one more layer removed

334
00:13:39,326 --> 00:13:41,455
from what we're used dealing with, right?

335
00:13:41,455 --> 00:13:45,122
Velocity, velocity is
the change in position

336
00:13:45,996 --> 00:13:47,949
with respect to time.

337
00:13:47,949 --> 00:13:52,366
Acceleration is the change in
velocity with respect to time.

338
00:13:52,366 --> 00:13:54,972
Now we're saying that the
something is the change

339
00:13:54,972 --> 00:13:56,651
in acceleration with respect to time.

340
00:13:56,651 --> 00:13:57,484
What is it?

341
00:13:57,484 --> 00:13:58,552
It's the jerk.

342
00:13:58,552 --> 00:14:00,848
So this is often called the jerk.

343
00:14:00,848 --> 00:14:01,682
That's the name of it.

344
00:14:01,682 --> 00:14:04,015
It's not used all that often.

345
00:14:04,015 --> 00:14:06,291
It's quite honestly not the most useful

346
00:14:06,291 --> 00:14:08,494
motion variable you'll ever meet

347
00:14:08,494 --> 00:14:10,574
and you won't get asked
for that often most likely

348
00:14:10,574 --> 00:14:11,705
on test and whatnot

349
00:14:11,705 --> 00:14:14,788
but it has its application sometimes

350
00:14:15,924 --> 00:14:19,123
that exist and it has a
name that's called the jerk.

351
00:14:19,123 --> 00:14:22,997
So recapping, the area,
the important fact here

352
00:14:22,997 --> 00:14:25,569
is that the area under
acceleration versus time graphs

353
00:14:25,569 --> 00:14:28,382
gives you the change in velocity.

354
00:14:28,382 --> 00:14:30,251
Once you know the velocity at one point,

355
00:14:30,251 --> 00:14:32,614
you could find the velocity
at any other point.

356
00:14:32,614 --> 00:14:35,369
The slope of an acceleration
versus time graph

357
00:14:35,369 --> 00:00:00,000
gives you the jerk.