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The goal of this video is to
explore some of the concepts,

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or formulas you might see in
a traditional physics class.

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But even more importantly to see
that they're really just common

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sense ideas.

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So let's just start
with a simple example.

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Let's say that-- and for
the sake of this video

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just so I don't
have to keep saying

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this is the magnitude
of the velocity,

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this is the direction of the
velocity, et cetera, et cetera.

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Let's just assume that if I
have a positive number that it

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means, for example, if I
have a positive velocity,

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it means that I am
going to the right.

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And let's say if I have a
negative number, which we won't

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see in this video, let's assume
that I'm going to the left.

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And that way I can just
write a number down,

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are we're only operating
in one dimension.

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You know that it
is specifying what

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the magnitude and
the direction is.

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If I said the velocity is
five meters per second,

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that means five meters
per second to the right.

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If I say it's negative
five meters per second,

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that means it's five meters
per second to the left.

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Now let's just say
just for simplicity,

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let's just say we start
with an initial velocity,

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of five meters per second.

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And once again, I am
specifying both the magnitude

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and the direction.

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Because of this convention here,
we know it is to the right.

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And let's say that we have
a constant acceleration.

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We have a constant
acceleration at play

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of two meters per
second, per second,

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or two meters per
second squared.

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And once again, since this is
positive, it is to the right.

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And let's say that we
do this for a duration.

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So my change in
time is- let's say

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we do this for a duration
of four-- I'll just use s.

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So I don't use seconds
and then sec, and then

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s in different places.

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So s for this video is seconds.

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So I want to do is think
about how far do we travel.

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Well there's two things.

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How fast are we going
after four seconds,

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and how far have we traveled
over the course of those four

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seconds.

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So let's draw ourselves a
little bit of a diagram here.

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So this is my velocity axis and
this over here is my time axis.

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I can draw a straighter
line than that.

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So that is my time axis.

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This is velocity.

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That is my velocity
right over there.

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And I'm starting off at
five meters per second.

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So this is five meters per
second right over here.

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So v i is equal to
five meters per second.

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And then every second
that goes by, I'm

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going two meters
per second faster.

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Because it's two meters
per second, per second.

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So every second that goes
by-- so after one second,

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I'm going to go two
meters per second faster.

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So I'll be at seven.

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Or another way to
think about it,

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is the slope of
this velocity line

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is my constant acceleration.

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Or I have a constant slope here.

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So it might look something
look something like that.

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So what has happened
after four seconds?

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So one, two, three, four.

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So this is my delta t.

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So my final velocity is
going to be right over there.

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I'll write it down
here just because it's

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getting in the way of
this word velocity.

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And so this is v. This
is my final velocity.

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And what would it be?

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Well I'm starting at
five meters per second.

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So I'm going to do it
both, using the variables,

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and using the concrete numbers.

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So starting at some
initial velocity.

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The subscript i
says i for initial.

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And then each second
that goes by, I'm

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getting this much faster.

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So if I want to see how
much faster have I gone,

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I multiply the
number of seconds--

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I'll just multiply the
number of seconds that go by,

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times my acceleration.

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And once again, this
right here, I just

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wrote the subscript
c saying that it's

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a constant acceleration.

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And so that will tell
me how fast I've gone.

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If I start at this
point, and I multiply

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the duration times my
slope, I will get this high.

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I will get to my final velocity.

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And just to make it
clear with the numbers,

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these numbers really
could be anything,

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I'm just picking these to
make it concrete in your mind.

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You have five meters per
second, plus four seconds,

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plus-- I want to
do that in yellow--

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plus four seconds times
our acceleration of two

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meters per second squared.

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And what is this
going to be equal to?

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You have the seconds canceling
out with one of the seconds

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down here.

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So let me write it.

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So we have five
meters per second,

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plus four times two is eight.

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This second's gone.

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We're just left with
meters per second.

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Eight meters per second.

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Or this is the same thing as
13 meters per second, which

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is going to be our
final velocity.

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And I want to take a pause here.

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And you can pause and
think about yourself.

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This hopefully
should be intuitive.

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We were starting at going
five meters per second.

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Every second that
goes by we're going

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to go two meters
per second faster.

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So after one second we will
be at seven meters per second.

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After two seconds we will be
at nine meters per second.

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After three seconds we will
be 11 meters per second.

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And then after four
seconds we will

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be at 13 meters per second.

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So you multiply how much time
passed times acceleration.

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This is how much faster,
we're going to be going.

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If we're already going
five meters per second.

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Five plus how much faster?

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13 meters per second.

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So this right up here
is 13 meters per second.

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So, I'll take a
little pause here.

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Hopefully that's intuitive.

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And the whole point
of that is to show you

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that this formula that you'll
often see in many physics

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books-- It's not just something
that randomly popped out

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of the air-- it just makes
complete common sense.

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Now the next thing
I want to talk about

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is what is the total distance
that we would have traveled.

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Now we know from the last
video that distance is just

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the area under this
curve right over here.

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So it's just the area
under this curve.

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You say wait, this is kind of
a strange shape right here.

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How do I calculate its area?

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And we can just use a
little simple geometry

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to break it down into
two different areas, that

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are very easy to
calculate their areas.

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Or two simple shapes.

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You can break it down into
this blue part, this rectangle

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right over here.

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Easy to figure out the
area of a rectangle.

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And we could break it down
into this purple part.

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This triangle right here.

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Easy to figure out the
area of a triangle.

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And that will be the
total distance we travel.

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And even this will hopefully
make some intuition.

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Because this blue
area is how far would

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we have traveled if we
were not accelerating.

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If we just went five meters
per second, for four seconds.

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So if you go five meters
per second times--

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so this is one second, two
second, three second, four

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second.

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So you're going from
times zero to time four,

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your change in time
is four seconds.

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So if you go five meters per
second, for four seconds,

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you're going to go 20 meters.

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This right here is 20 meters.

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That is the area
of this right here.

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Five times four.

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This purple or
magenta area tells you

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how further than
this, are you going,

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because you are
accelerating, because you

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kept going faster and
faster and faster.

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And it's pretty easy
to calculate this area.

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The base here is still
five, because that's

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five seconds have gone by.

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And what's the height here?

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The height here is
my final velocity

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minus my initial velocity.

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Or it's the change in velocity
due to the acceleration.

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And the change in velocity
due to the acceleration.

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13 minus five is eight.

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Or it's this 8 right over here.

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It is eight meters per second.

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So this height right over here
is eight meters per second.

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The base right over
here is four seconds.

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That's the time that past.

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So what's the area
of this triangle?

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Area of a triangle is
one half, times the base,

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which is four seconds,
times the height, which

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is eight meters per second.

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Seconds cancel out.

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One half times four
is two, times eight

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is equal to 16 meters.

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So the total distance we
traveled is 20 plus 16,

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is 36 meters.

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That is the total, or I could
say the total displacement,

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and once again, it's to the
right, since it is positive.

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So this is our displacement.

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Now what I want to do is do
this exact same calculation,

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but keep it in variable
form and that'll

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give another formula that many
people, many often memorize.

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But I want you to
understand it it's

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a completely intuitive
formula, and it just

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comes out of the logical
flow of reasoning

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that we went through
in this video.

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What is the area?

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Once again, if we just
think about the variables.

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Well, the area of
this rectangle right

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here is our initial velocity,
times our change in time.

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So this is the blue
rectangle right over here.

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And then plus-- what
do we have to do--

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we have the change
in time, once again,

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we have the change in
time, times this height.

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Which is our final velocity.

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Which is times
our final velocity

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minus our initial velocity.

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These are all vectors.

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They're just positive, telling
us we're going to the right.

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And we don't just multiply
the base times the height.

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That would give us the area
of this entire rectangle.

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We have to take it by half
because a triangle is only half

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of that rectangle.

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So times one half.

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And so this is the area.

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This is the purple area.

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That's not purple.

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This is the purple
area, right over here.

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This is the area of this.

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This is the area of that.

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And let's simplify
this a little bit.

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Let's factor out a delta t.

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So if you factor
out a delta t, you

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get delta t times, a
bunch of stuff, v sub i.

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So your initial velocity.

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We factored this out.

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Plus this stuff.

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Plus this thing right over here.

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And we can distribute
the one half.

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We factored the delta t's out.

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And let's multiply the one half
times each of these things.

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So it's going to be
plus, one half times v f,

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times our final velocity.

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That's not the right
color, let me actually

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do it in the right color so
you understand what I'm doing.

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So this is the one half.

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So plus one half times
our final velocity.

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Minus one half times
our initial velocity.

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I want to do that in blue.

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00:10:55,271 --> 00:10:57,270
Sorry, I'm having trouble
changing colors today.

256
00:10:57,270 --> 00:10:59,635
Minus one half times
our initial velocity.

257
00:10:59,635 --> 00:11:03,572


258
00:11:03,572 --> 00:11:05,030
And now what does
this simplify to?

259
00:11:05,030 --> 00:11:07,450
We have something plus one
half times something else,

260
00:11:07,450 --> 00:11:10,230
minus one half times
that original something.

261
00:11:10,230 --> 00:11:14,490
So what is v i minus
one half times v i.

262
00:11:14,490 --> 00:11:16,230
So anything minus
half of it, you're

263
00:11:16,230 --> 00:11:19,070
just going to have a
positive half left.

264
00:11:19,070 --> 00:11:21,490
So these two terms,
this term and this term

265
00:11:21,490 --> 00:11:27,300
will simplify to one half v i.

266
00:11:27,300 --> 00:11:30,980
One half the initial
velocity plus one half times

267
00:11:30,980 --> 00:11:31,780
the final velocity.

268
00:11:31,780 --> 00:11:35,480


269
00:11:35,480 --> 00:11:37,730
And all of that is
being multiplied

270
00:11:37,730 --> 00:11:42,740
by our change in time.

271
00:11:42,740 --> 00:11:45,410
Or the time that has gone by.

272
00:11:45,410 --> 00:11:49,359
And this tells us the distance.

273
00:11:49,359 --> 00:11:50,650
The distance that we travelled.

274
00:11:50,650 --> 00:11:53,050
Or another way to
think about it-- let's

275
00:11:53,050 --> 00:11:56,090
factor out this one half.

276
00:11:56,090 --> 00:12:00,200
You get distance is equal
to change in time, times--

277
00:12:00,200 --> 00:12:06,930
factoring out the one
half-- v i plus v f.

278
00:12:06,930 --> 00:12:09,260
No, that's not the right color.

279
00:12:09,260 --> 00:12:13,150
v i plus v f.

280
00:12:13,150 --> 00:12:16,160


281
00:12:16,160 --> 00:12:17,220
So this is interesting.

282
00:12:17,220 --> 00:12:21,020
The distance we traveled
is equal to one half

283
00:12:21,020 --> 00:12:24,340
of the initial velocity
plus the final velocity.

284
00:12:24,340 --> 00:12:26,350
So this is really--
if you just took,

285
00:12:26,350 --> 00:12:28,330
this quantity right
over here is just

286
00:12:28,330 --> 00:12:32,140
the arithmetic-- I always
have trouble saying that word.

287
00:12:32,140 --> 00:12:35,829
It's the arithmetic mean
of these two numbers.

288
00:12:35,829 --> 00:12:37,870
And so I'm going to define
this as something new.

289
00:12:37,870 --> 00:12:39,920
I'm going to call this
the average velocity.

290
00:12:39,920 --> 00:12:42,000
But we have to be very
careful with this.

291
00:12:42,000 --> 00:12:47,130
This right here is
the average velocity.

292
00:12:47,130 --> 00:12:48,830
But the only reason
why I can just

293
00:12:48,830 --> 00:12:52,137
take the starting velocity and
the ending velocity and, adding

294
00:12:52,137 --> 00:12:53,720
them together, and
then divide by two.

295
00:12:53,720 --> 00:12:55,803
Essentially taking the
average of these two things

296
00:12:55,803 --> 00:12:57,370
which would be
someplace over here.

297
00:12:57,370 --> 00:12:59,220
And I take that as
the average velocity,

298
00:12:59,220 --> 00:13:02,280
is because my
acceleration is constant.

299
00:13:02,280 --> 00:13:05,771
Which is usually an assumption
in most introductory physics

300
00:13:05,771 --> 00:13:06,270
classes.

301
00:13:06,270 --> 00:13:07,770
But it's not always
the assumption.

302
00:13:07,770 --> 00:13:11,090
But if you do have a constant
acceleration like this,

303
00:13:11,090 --> 00:13:13,900
you can assume that
the average velocity

304
00:13:13,900 --> 00:13:17,890
is going to be the average
of the initial velocity

305
00:13:17,890 --> 00:13:19,647
and the final velocity.

306
00:13:19,647 --> 00:13:21,980
If this was a curve or if the
acceleration was changing,

307
00:13:21,980 --> 00:13:23,720
you could not do that.

308
00:13:23,720 --> 00:13:25,430
But what's useful
about this-- is

309
00:13:25,430 --> 00:13:28,270
if you want to figure out the
distance that was traveled

310
00:13:28,270 --> 00:13:30,530
you just need to know
the initial velocity

311
00:13:30,530 --> 00:13:32,350
and the final velocity.

312
00:13:32,350 --> 00:13:34,980
Average the two, and
then multiply that

313
00:13:34,980 --> 00:13:37,310
times the time that goes by.

314
00:13:37,310 --> 00:13:39,210
So in this situation,
our final velocity

315
00:13:39,210 --> 00:13:41,150
is 13 meters per second.

316
00:13:41,150 --> 00:13:44,180
Our initial velocity was
five meters per second.

317
00:13:44,180 --> 00:13:47,600
So you have 13 plus
5 is equal to 18.

318
00:13:47,600 --> 00:13:49,230
You divide that by 2.

319
00:13:49,230 --> 00:13:53,350
Your average velocity
is 9 meters per second,

320
00:13:53,350 --> 00:13:55,480
if you take the
average of 13 and 5.

321
00:13:55,480 --> 00:13:58,440
And then 9 meters per
second times 4 seconds

322
00:13:58,440 --> 00:13:59,977
gives you 36 meters.

323
00:13:59,977 --> 00:14:01,560
So hopefully that
doesn't confuse you.

324
00:14:01,560 --> 00:14:03,200
I just want to show you
where some of these things

325
00:14:03,200 --> 00:14:04,360
that you'll see in
your physics class,

326
00:14:04,360 --> 00:14:06,693
some of these formulas, why
you shouldn't memorize them,

327
00:14:06,693 --> 00:14:08,607
and they can all be deduced.

328
00:14:08,607 --> 00:00:00,000