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What I want to do in this video,
now that we have displacement

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as a function of time,
given constant acceleration

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and an initial velocity.

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I want to plot displacement,
final velocity,

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and acceleration, all of
those as functions of time.

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Just so we really
understand what's

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happening as the ball is
going up and then down.

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So we know this is
our displacement

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as a function of time.

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We know what our final
velocity is going to be,

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as a function of time.

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We talked about it
in the last video.

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Our final velocity is going
to be our initial velocity

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plus our acceleration
times change in time.

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Right?

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If we start at some initial
velocity, and then you

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multiply the
acceleration times time.

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This part tells you how
much faster or slower

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you're going to go than
your initial velocity.

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And that will be, I
guess you could say,

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your current velocity,
or the final velocity

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at that point in time.

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And of course, our
acceleration, we know.

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Our acceleration is
pretty straightforward.

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The acceleration due
to gravity is just

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going to be negative 9.8
meters per second squared.

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Once again, negative
being the convention

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that it is in the
downward direction.

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Our initial velocity
is going to be

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in the upward direction,
19.6 meters per second.

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So let's plot these
out a little bit.

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Let's plot these out.

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So the first graph I want
to do, right over here,

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will be my displacement
versus time.

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So this axis right over
here is going to be time,

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or maybe I could call this
the change in time axis.

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Actually, lets
just call it time.

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And then this axis, right over
here, I will call displacement.

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And let me put
some markers here.

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So let's say that this is
5 meters, 10 meters, 15

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meters, and 20 meters.

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And then in the time, this is 0,
this is 1 this is 2, this is 3,

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and this is 4 seconds.

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So this is in
seconds right here.

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This is meters.

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5, 10, 5, 10, 15, 20.

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So this is displacement.

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Displacement graph.

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And I want to, at the
same time as that,

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I want to do a velocity graph.

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So let me draw my
velocity graph like this.

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I'll do it a little
bit different.

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So, this is because the velocity
will be going up and down.

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So we need to have positive
and negative values here.

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Time will only be positive.

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So once again, I care about 1
second, 2 seconds, 3 seconds,

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and 4 seconds in time.

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And velocity, I'm
going to call this.

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This is going to be
10 meters per second.

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This is 20 meters per second.

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This will be negative
10 meters per second.

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And this will be negative
20 meters per second.

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And so all of this is
in meters per second.

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This right here is velocity.

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This axis right here is time.

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So this is my velocity graph.

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And why don't we just throw an
acceleration graph over here,

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although that's, to some
degree, the easiest of them all.

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So the acceleration
graph, and I'll

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just do this right
from the get go,

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because we're going to assume
that acceleration is constant.

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So this is 1 second,
2 seconds, 3 seconds,

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and 4 seconds into it.

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And then let's call
this negative 10.

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And all of this is in
meters per second squared.

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And so we know our
acceleration is negative

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9.8 meters per second squared.

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So the acceleration
the entire time

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over the four seconds, the
acceleration over the four

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seconds is going to be
that's about negative 9.8.

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It's going to be that.

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It's going to be a constant
acceleration the entire time.

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But let's figure out
displacement and velocity.

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So let me draw a
little table here.

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So in one column, I
will do change in time.

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Or sometimes you
could do that as time.

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Let's figure out what
our final velocity is,

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or I should really say our
current velocity, or velocity

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at that time.

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And then in this
column, I'll figure out

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what our displacement is.

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And I will do it for
times 0, 1, 2, 3, 4.

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Or change in time.

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So when 0 seconds have gone
by, when 1 second has gone by,

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when 2 seconds, 3 seconds,
and 4 seconds have gone by.

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Actually let, me call this
the change in time axis,

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because this is essentially
how many seconds have gone by.

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So this is my
change in time axis.

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And let me make it clear that
this graph-- I didn't label

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it here-- this is my
acceleration graph.

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And I'm going off of the screen.

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All right.

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So let's fill these things out.

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So at time 0, what is
our what is our velocity?

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Well, if we use this expression
right here, time 0, or delta t

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is equal to 0.

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This expression right
here, is going to be 0.

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And it's just going to
be our initial velocity.

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And in the last video, we
gave our initial velocity,

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is going to be as 19.6
meters per second.

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So it is going to be
19.6 meters per second.

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And let me plot that over here.

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At time 0, it is going to
be 19.6 meters per second.

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What is our initial displacement
at time zero, or change

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in time 0?

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So you look at this up here.

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Well, our delta t is 0, so this
expression is going to be 0,

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and this expression
is going to be 0.

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So we haven't done
any displacement yet,

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when no time has gone by.

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So we have done no displacement.

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We are right over there.

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Now, what happens after
1 second has gone by?

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What is now our velocity?

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Well, our initial velocity,
right over here, is 19.6.

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19.6 meters per second,
that was a given.

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And our acceleration is negative
9.8 meters per second squared.

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So it's a negative,
right over there.

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And then you multiply that times
delta t in every situation.

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So in this situation
we're going to multiply it

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by 1, because delta t is 1.

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So we get 19.6 minus
9.8, that gives us

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exactly 9.8 meters per second.

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And the units work out,
because you multiply this times

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seconds, this gives
you meters per second.

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So 19.6 meters per second minus
9.8 meters per second-- one

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of these seconds goes away when
you multiply it by second--

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gives you 9.8 meters per second.

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So after 1 second,
our velocity is now

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half of what it was before.

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So we're now going
9.8 meters per second.

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Let me draw a line here.

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9.8 meters per second.

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Now what is our displacement?

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So you look up here.

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And let me rewrite this
displacement formula,

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with all of the
information that we know.

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So we know that
displacement is going

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to be equal to our initial
velocity, which is 19.6--

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and I won't write
the units here, just

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for the sake of space--
times our change in time.

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I'll do it in that same
color, so you what's what.

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Times our change
in time plus 1/2.

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Now let me be clear, 1/2
times negative 9.8 meters

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per second squared.

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So 1/2 times a is
going to be-- actually

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I can rewrite this,
right over here--

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because this is going to
be negative 9.8 meters

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per second times 1/2, so this
is going to be negative 4.9.

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All I did, is I took 1/2
times negative 9.8 over here.

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1/2 times negative 9.8.

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And this is important.

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And this is why the vector
quantities start to matter.

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Because if you didn't, if
you put a positive here,

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you wouldn't have
the object actually

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slowing down as it went up,
because you would have gravity

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somehow accelerating
it as it went up.

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But it's actually
slowing it down.

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It's accelerating it in
the downwards direction.

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So that's why you have to have
that negative right over there.

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That was our convention at the
beginning of the last video.

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Up is positive.

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Down is negative.

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So let's focus.

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So this part right over
here, negative 4.9 meters

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per second squared
times delta t squared.

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And this will make it
a little bit easier,

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although we'll still-- let
me get the calculator out.

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So when one second has passed--
I'll get my trusty TI-85 out.

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When one second has passed, the
displacement is 19.6 times 1.

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Well, that's just 19.6.

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Minus 4.9 times 1 squared.

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So that's just minus 4.9.

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Gives us 14.7 meters.

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So after 1 second, the ball
has traveled 14.7 meters

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in the air.

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So that's roughly over there.

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Now, what happens
after 2 seconds?

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I'll do this in magenta.

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So after 2 seconds, our velocity
is 19.6 minus 9.8 times 2.

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00:09:21,740 --> 00:09:23,110
2 seconds have gone by.

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00:09:23,110 --> 00:09:28,950
Well, 9.8 meters per second
squared times 2 seconds

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gives us 19.6 meters per second.

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00:09:32,350 --> 00:09:33,770
So these just cancel out.

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So we get, our
velocity is now 0.

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00:09:36,420 --> 00:09:40,475
So after 2 seconds,
our velocity is now 0.

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Actually, let me-- So let me
make it so it's-- this thing

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should look more like a line.

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I don't make you get a sense--
So this is-- So let me just

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draw the line like this.

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Our velocity is now
0 after 2 seconds.

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What is our displacement?

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00:09:57,620 --> 00:10:01,670
So we're literally at the point
where the ball has no velocity.

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00:10:01,670 --> 00:10:03,230
At exactly 2 seconds.

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00:10:03,230 --> 00:10:08,520
So it's kind of gone up, and
for that exact moment in time,

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00:10:08,520 --> 00:10:09,930
it is stationary.

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00:10:09,930 --> 00:10:13,490
And then what do we have
going on in our displacement?

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00:10:13,490 --> 00:10:17,452
We have 19.6-- let me get
the calculator out for this.

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00:10:17,452 --> 00:10:19,910
I could do it by
hand, but for the sake

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00:10:19,910 --> 00:10:25,280
of quickness-- 19.6
times 2-- 19.6 times

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00:10:25,280 --> 00:10:32,280
2 seconds minus 4.9
times 2 seconds squared.

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00:10:32,280 --> 00:10:33,950
This is 2 seconds squared.

220
00:10:33,950 --> 00:10:35,360
Oh, I lost the calculator.

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00:10:35,360 --> 00:10:36,620
Times 2 seconds squared.

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So that's times 4.

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So that gives us 19.6 meters.

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So we're at 19-- let me
do that in magenta-- we

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00:10:44,430 --> 00:10:48,140
are at 19.6 meters.

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So after 2 seconds, we are
19.6 meters in the air.

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00:10:54,130 --> 00:10:57,090
Now let's go to 3 seconds.

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So after 3 seconds,
our velocity is now--

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00:11:01,850 --> 00:11:04,980
I'll just get the-- it's
19.6 meters per second

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00:11:04,980 --> 00:11:07,465
minus 9.8 times 3.

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And we could do
that in our head,

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but just to verify it for us,
let me get the calculator out.

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It's 19.6 minus 9.8 times 3.

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00:11:16,880 --> 00:11:19,910
That gives us negative
9.8 meters per second.

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00:11:19,910 --> 00:11:24,110


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So after 3 seconds,
our velocity is now

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00:11:26,080 --> 00:11:28,560
negative 9.8 meters per second.

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00:11:28,560 --> 00:11:29,470
What does that mean?

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It's now going in the
downward direction

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at 9.8 meters per second.

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00:11:34,080 --> 00:11:36,180
So this is our velocity graph.

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And then what is our
displacement at this point?

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So once again, let's
get the calculator out.

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00:11:41,400 --> 00:11:43,204
If you're getting
the hang of this.

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00:11:43,204 --> 00:11:45,370
At any time, I encourage
you to pause it, and try it

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00:11:45,370 --> 00:11:46,770
for yourself.

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So now, what is-- now
this is a little, OK.

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00:11:49,420 --> 00:11:51,540
So I'm looking at
my displacement,

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00:11:51,540 --> 00:11:52,540
I wrote right over here.

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So our displacement
where delta t

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00:11:55,100 --> 00:12:04,340
is 3 seconds, 19.6 times 3
minus 4.9 times-- and this

252
00:12:04,340 --> 00:12:06,260
is delta t, so
this is 3 seconds.

253
00:12:06,260 --> 00:12:08,610
We're talking about when delta
t, or our change in time,

254
00:12:08,610 --> 00:12:09,530
is 3 seconds.

255
00:12:09,530 --> 00:12:10,920
So that squared.

256
00:12:10,920 --> 00:12:12,760
So times 9.

257
00:12:12,760 --> 00:12:18,500
And that gives us 14.7 meters.

258
00:12:18,500 --> 00:12:22,410
So after 3 seconds, we're
at 14.7 meters again.

259
00:12:22,410 --> 00:12:24,660
And so we're at the same
position we were at 1 second,

260
00:12:24,660 --> 00:12:26,743
but the difference is, now
we're moving downwards.

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00:12:26,743 --> 00:12:28,630
Over here we were
moving upwards.

262
00:12:28,630 --> 00:12:31,330
And then finally, what
happens after 4 seconds?

263
00:12:31,330 --> 00:12:32,590
Well, what's our velocity?

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00:12:32,590 --> 00:12:33,860
Well let me just get
the calculator out,

265
00:12:33,860 --> 00:12:36,480
although you might be able to
figure this out in your head.

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00:12:36,480 --> 00:12:49,530
Our velocity is going to be
19.6 minus 9.8 times 4 seconds.

267
00:12:49,530 --> 00:12:52,050
Which is minus 19.6
meters per second.

268
00:12:52,050 --> 00:12:55,430


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00:12:55,430 --> 00:12:58,160
So our magnitude of our
velocity is the same

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00:12:58,160 --> 00:13:01,120
as when we initially threw
the ball, except now it's

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00:13:01,120 --> 00:13:02,500
going in the opposite direction.

272
00:13:02,500 --> 00:13:03,720
It's now going downwards.

273
00:13:03,720 --> 00:13:07,380


274
00:13:07,380 --> 00:13:09,520
And what is our displacement?

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00:13:09,520 --> 00:13:11,660
Get the calculator out.

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00:13:11,660 --> 00:13:16,840
So we have, our displacement
is 19.6 times 4-- 4 seconds

277
00:13:16,840 --> 00:13:21,990
have gone by-- minus
4.9 times 4 squared--

278
00:13:21,990 --> 00:13:26,490
which is 16-- so times
16, which is equal to 0.

279
00:13:26,490 --> 00:13:28,450
Our displacement is 0.

280
00:13:28,450 --> 00:13:30,500
We are back on the ground.

281
00:13:30,500 --> 00:13:32,890
We are back on the ground.

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00:13:32,890 --> 00:13:35,520
So if you were to
plot its displacement

283
00:13:35,520 --> 00:13:38,480
you would actually get a
parabola, a downward opening

284
00:13:38,480 --> 00:13:40,760
parabola that looks
something like this.

285
00:13:40,760 --> 00:13:44,320
Doing my best to draw
it relatively neatly.

286
00:13:44,320 --> 00:13:46,160
Actually, I can do a
better job than that.

287
00:13:46,160 --> 00:13:47,076
I'll do a dotted line.

288
00:13:47,076 --> 00:13:49,215
Dotted lines are always
easier to adjust midstream.

289
00:13:49,215 --> 00:13:53,440


290
00:13:53,440 --> 00:13:55,082
If you plot displacement
versus time,

291
00:13:55,082 --> 00:13:56,290
it looks something like this.

292
00:13:56,290 --> 00:13:59,180
Its velocity is just this
downward sloping line,

293
00:13:59,180 --> 00:14:01,260
and then the
acceleration is constant.

294
00:14:01,260 --> 00:14:03,680
And the whole reason
why I wanted to do this,

295
00:14:03,680 --> 00:14:06,560
is I wanted to show you that
the velocity the whole time

296
00:14:06,560 --> 00:14:08,610
is decreasing at
a constant pace.

297
00:14:08,610 --> 00:14:12,120
And that makes sense, because
the rate at which the velocity

298
00:14:12,120 --> 00:14:14,790
increases or decreases
is the acceleration.

299
00:14:14,790 --> 00:14:19,350
And the acceleration, based on
our convention, is downwards.

300
00:14:19,350 --> 00:14:20,630
So that's why it's decreasing.

301
00:14:20,630 --> 00:14:22,710
We have a negative slope here.

302
00:14:22,710 --> 00:14:27,180
We have a negative slope
of negative 9.8 meters

303
00:14:27,180 --> 00:14:28,890
per second squared.

304
00:14:28,890 --> 00:14:30,992
And so just to think
about what's happening

305
00:14:30,992 --> 00:14:32,450
for this ball, or
this rock-- and I

306
00:14:32,450 --> 00:14:34,283
know this video is
getting long-- as it goes

307
00:14:34,283 --> 00:14:38,460
through the air, I'm going to
draw the vectors for velocity.

308
00:14:38,460 --> 00:14:40,750
And I'm going to
do that in orange,

309
00:14:40,750 --> 00:14:42,600
or no, maybe I'll
do that in blue.

310
00:14:42,600 --> 00:14:44,210
So velocity in blue.

311
00:14:44,210 --> 00:14:47,150
So right when we start,
it has a positive velocity

312
00:14:47,150 --> 00:14:49,340
of 19.6 meters per second.

313
00:14:49,340 --> 00:14:51,560
So I'll draw a big
vector like this,

314
00:14:51,560 --> 00:14:54,980
19.6 meters per second,
that's its velocity.

315
00:14:54,980 --> 00:14:57,790
Then after 1 second, it's
9.8 meters per second.

316
00:14:57,790 --> 00:14:59,484
So it's half of that.

317
00:14:59,484 --> 00:15:01,150
Maybe it would look
something like this,

318
00:15:01,150 --> 00:15:03,060
9.8 meters per second.

319
00:15:03,060 --> 00:15:06,710
Then at this peak right over
here, it has a velocity of 0.

320
00:15:06,710 --> 00:15:10,580
Then as you go to 3 seconds,
the magnitude of its velocity

321
00:15:10,580 --> 00:15:12,360
is 9.8 meters per second.

322
00:15:12,360 --> 00:15:14,150
But it is now downwards.

323
00:15:14,150 --> 00:15:16,770
It is now downwards,
so it looks like this.

324
00:15:16,770 --> 00:15:20,930
And then, finally, right when
it hits the ground, right

325
00:15:20,930 --> 00:15:23,920
before it hits the ground,
it has a negative velocity

326
00:15:23,920 --> 00:15:27,170
of 19.6 meters per second.

327
00:15:27,170 --> 00:15:32,850
So it would look like
this, roughly like this.

328
00:15:32,850 --> 00:15:35,070
If I use the same
scale over here.

329
00:15:35,070 --> 00:15:37,992
But what was the
acceleration the entire time?

330
00:15:37,992 --> 00:15:40,200
Well, the entire time, the
acceleration was negative.

331
00:15:40,200 --> 00:15:42,900
It was negative 9.8
meters per second squared.

332
00:15:42,900 --> 00:15:44,250
And I'll do that in orange.

333
00:15:44,250 --> 00:15:46,810
So the acceleration,
over here, negative-- no,

334
00:15:46,810 --> 00:15:49,750
I want to do that in
orange-- the acceleration was

335
00:15:49,750 --> 00:15:52,580
negative 9.8 meters
per second squared.

336
00:15:52,580 --> 00:15:55,990
Acceleration, negative 9.8
meters per second squared.

337
00:15:55,990 --> 00:15:57,960
Negative 9.8 meters
per second squared.

338
00:15:57,960 --> 00:16:01,610
The acceleration is
constant the entire time.

339
00:16:01,610 --> 00:16:05,520
This last one is negative 9.8
meters per second squared.

340
00:16:05,520 --> 00:16:07,350
It does not change,
depending where

341
00:16:07,350 --> 00:16:09,090
you are in the
curve, when you're

342
00:16:09,090 --> 00:16:10,560
near the surface of the Earth.

343
00:16:10,560 --> 00:16:12,190
So hopefully that clarifies
things a little bit

344
00:16:12,190 --> 00:16:13,606
and gives you a
good sense of what

345
00:16:13,606 --> 00:00:00,000
happens when you throw a
projectile into the air.