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Just want to follow up
on the last video, where

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we threw balls in the air, and
saw how long they stayed up

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in the air.

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And we used that to figure
out how fast we initially

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threw the ball and how
high they went in the air.

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And in the last video, we
did it with specific numbers.

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In this video, I
just want to see

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if we can derive some
interesting formulas so that we

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can do the computations
really fast in our brains,

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while we're playing this game
out on some type of a field,

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and we don't necessarily
have any paper around.

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So let's say that the ball
is in the air for delta t.

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Delta t is equal
to time in the air.

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Then we know that
the time up is going

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to be half that, which is the
same thing as the time down.

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The time up is going to
be equal to delta t-- I'm

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going to do that
in the same color--

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is going to be equal to the
time in the air divided by 2.

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So what was our
initial velocity?

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Well, all we have to
do is remind ourselves

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that the change
in velocity, which

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is the same thing as
the final velocity

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minus the initial velocity.

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So the final
velocity-- remember,

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we're just talking about half
of the path of this ball.

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So the time that
it gets released,

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and it's going at kind of
its maximum upward velocity,

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and it goes slower, and slower,
slower, all the way until it's

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stationary for just
a moment, and then

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it starts going down again.

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Now remember, the acceleration
is constant downwards

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this entire time.

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So what is the final
velocity if we just

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consider half of the time?

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Well, it's the time.

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It's 0.

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So it's going to be 0
minus our initial velocity,

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when it was taking off.

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That's our change in velocity.

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This is our change
in velocity, is

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going to be equal to the
acceleration of gravity,

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negative 9.8 meters
per second squared--

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or the acceleration
due to gravity

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when an object is in free fall,
to be technically correct--

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times the time that
we are going up.

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So times delta t up,
which is the same thing.

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I won't even write it.

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Delta t up is the same
thing as our total time

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in the air divided by 2.

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And so we get negative
the initial velocity is

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equal to-- this thing,
when you divide it by 2,

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is going to be 4.9 meters
per second squared--

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we still have our negative
out front-- times our delta t.

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Remember, this is our
total time in the air,

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not just the time up.

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This is our total
time in the air.

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And then we multiply both
sides times a negative.

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We get our initial
velocity is just

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going to be equal to 4.9
meters per second squared times

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the total time that
we are in the air.

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Or you could say
it's going to be

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9.8 meters per second squared
times half of the time

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that we're in the air.

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Either of those would get
you the same calculation.

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So let's figure out our total
distance, or the distance

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that we travel in the time up.

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So that'll give us
our peak distance.

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Remember that distance-- or
I should say displacement,

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in this situation--
displacement is

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equal to average velocity
times change in time.

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The change in time that we
care about is the time up.

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So that is our delta t over 2.

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Our total time divided by 2.

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This is our time up.

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And what's our average velocity?

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Well, the average
velocity, if we

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assume constant acceleration,
is your initial velocity

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plus your final velocity over 2.

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It's really just the
mean of the two things.

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Well, we know what our
initial velocity is.

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Our initial velocity is
this thing over here.

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So this is this thing over here.

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Our final velocity--
remember, we're

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just talking about
the first half

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of the time the
ball is in the air--

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so it's final velocity is 0.

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We're talking when it gets
to this peak point, right

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over here-- that's from two
videos ago-- that peak point

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right over there.

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So our average
velocity is just going

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to be this stuff divided by 2.

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So it's going to be 4.9 meters
per second squared times delta

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t over 2.

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So this right here, this
is our average velocity.

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Velocity average.

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So let's stick that
back over here.

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So our maximum
displacement is going

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to be our average
velocity-- so that

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is 4.9 meters per second
squared-- times delta

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t, all of that over 2.

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And then we multiply it
again times the time up.

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So times delta t over 2 again.

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This is the same thing.

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These are the same thing.

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And then we can simplify it.

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Our maximum displacement is
equal to 4.9 meters per second

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squared times delta t
squared, all of that over 4.

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And then we can just
divide 4.9 divided by 4.

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4.9 divided by 4 is-- let me
just get the calculator out.

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I don't want to do that
in my head, get this far

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and make a careless mistake.

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4.9 divided by 4 is 1.225.

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So our maximum
displacement is going

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to be 1.225 times our total
time in the air squared,

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which is a pretty
straightforward calculation.

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So this is our max
displacement, kind

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of how high we get displaced.

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Right when that
ball is stationary,

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or has no net velocity, just
for a moment, and starts

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decelerating downwards.

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So we can use that.

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If a ball is in the air for
5 seconds-- we can verify

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our computation from
the last video--

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our maximum displacement, 1.225,
times 5 squared, which is 25,

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will give us 30.625.

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That's what we got
in the last video.

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If the ball's in the
air for, I don't know,

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2.3 seconds-- so
it's 1.225 times 2.3

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squared-- then that means it
went 6.48 meters in the air.

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So anyway, I just wanted to give
you a simple expression that

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gives you the maximum
displacement from the ground,

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assuming air resistance
is negligible,

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as a function of the
total time in the air.

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I don't know.

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I find that pretty fun.

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And it's a neat game to play.