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Let's say you and I are
playing a game where we're

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trying to figure out how high a
ball is being thrown in the air

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or how fast that we're
throwing that ball in the air.

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And what we do is
one of us has a ball

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and the other one has
a stopwatch over here.

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So this is my best attempt.

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It looks more like a
cat than a stopwatch,

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but I think you get the idea.

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And what we do is one
of us throws the ball,

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and the other one times how
long the ball is in the air.

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And then what we
do is we're going

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to use that time in
the air to figure out

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how fast the ball was
thrown straight up,

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and how long it was in the
air and how high it got.

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And there's going to be one
assumption I make here--

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and frankly, this
is an assumption

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that we're going to make
in all of these projectile

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motion-type problems-- is that
air resistance is negligible.

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And for something like-- if
this is a baseball, or something

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like that, that's a
pretty good approximation.

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So we're not going to
get the exact answer.

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And I encourage you to
experiment on your own

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to see what air resistance does
relative to your calculations.

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But we're going to assume
for this projectile motion--

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and really all of
the future ones

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or at least in the
basic physics playlist--

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we're going to assume that
air resistance is negligible.

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And what that does for
us is that we can assume

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that the time for the ball
to go up to its peak height

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is the same thing as the time
that it takes to go down.

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If you look at
this previous video

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where we plotted
displacement versus time,

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you see after 2 seconds the ball
went from being on the ground--

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or I guess the thrower's hand--
all the way to its peak height.

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And in the next 2 seconds, it
took that same amount of time

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to go back down to the
ground, which makes sense.

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Whatever the
initial velocity is,

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it takes half the
time to go to 0.

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And it takes that
same amount of time

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to now be accelerating
in the downward direction

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back to that same
magnitude of velocity

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but now in the
downward direction.

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So let's play around
with some numbers

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here just so we get a little
bit more of a concrete sense.

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So let's say I throw
a ball in the air.

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And you measure,
using the stopwatch,

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that the ball is in
the air for 5 seconds.

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So how do we figure out
how fast I threw the ball?

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Well, the first
thing we can do is

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we could say, look, if
the total time in the air

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was 5 seconds, that means that
the time-- let me write it.

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That means that
the change in time

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to go up during the
first half of, I guess,

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the ball's time in the air
is going to be 2.5 seconds.

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Which tells us that
over this 2.5 seconds,

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we went from our initial
velocity, whatever it was,

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we went from our
initial velocity

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to our final velocity,
which is a velocity

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of 0 meters per second
in the 2 and 1/2 seconds.

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And this isn't the
graph for that example.

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This is the graph for the
previous example where

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we knew the initial velocity.

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But in whatever
that time is, you're

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going from your initial velocity
to being stationary at the top,

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right?

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Right when the ball
is stationary then

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it starts getting
increasing velocity

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in the downward direction.

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So it takes 2.5 seconds to
go from some initial velocity

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to 0 seconds.

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So we do know what the
acceleration of gravity is.

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We know that the acceleration
of gravity here--

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we're assuming it's constant,
although it's slightly not

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constant.

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But we're going to
assume it's constant,

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if we're just dealing close
to the surface of the earth--

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is negative 9.8 meters
per second squared.

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So let's think about it.

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Our change in velocity
is the final velocity

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minus the initial
velocity, which

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is the same thing as 0 minus
the initial velocity, which

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is the negative of
the initial velocity.

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And what's another way to
think about change in velocity?

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Well, just from the
definition of acceleration,

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change in velocity is
equal to acceleration--

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negative 9.8 meters per
second squared-- times time,

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or times change in time.

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We're just talking
about the first half

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of the ball's time in the air.

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So our change in time is 2.5
seconds-- times 2.5 seconds.

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So what is our
change in velocity,

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which is also the same
thing as the negative

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of our initial velocity?

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Let me get my calculator,
bring it onto the screen.

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So it is negative 9.8 meters
per second times 2.5 seconds.

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It gives us negative 24.5.

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So this gives us-- let me
write it in a new color.

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This gives us negative
24.5 meters per second.

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This second cancels out
with one of these seconds

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in the denominator, so we only
have one in the denominator

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now, so it's meters per second.

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And this is the same thing as
the negative initial velocity.

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That's the same thing as
our change in velocity.

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And so you multiply both
sides by a negative.

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We get our initial velocity.

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So that simply we were
able to figure out

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what our velocity was.

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So literally you take the
total time in the air,

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take it and divide
it by 2, and then

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multiply that by the
acceleration of gravity.

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And I guess you could take
the absolute value of that

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or you take the positive
version of that.

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And then that gives you
your initial velocity.

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So your initial velocity
here is literally

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24.5 meters per second.

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And since it's a
positive quantity,

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it is upwards in this example.

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So that's my initial velocity.

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So we already figured
out part of this game,

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the initial velocity
that I threw it upwards.

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And that's also
going to be-- we're

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going to have the same
magnitude of velocity

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when the ball's about
to hit the ground,

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although it's going to be
in the other direction.

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So what is the distance--
or let me make it clear.

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What is the displacement of the
ball from its lowest point--

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right when it leaves your
hand-- all the way to the peak?

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Well, we just have to remember--
and once again, all this

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comes from very straightforward
ideas, change in velocity

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is equal to acceleration
times change in time.

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And then the other
simple idea is

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that displacement is equal
to average velocity times

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change in time.

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Now, what is our
average velocity?

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Our average velocity is
your initial velocity

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plus your final
velocity divided by 2,

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if we assume
acceleration is constant.

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So it's literally just
the arithmetic mean

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of your initial and
final velocities.

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So what is that?

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That's going to be 24.5
meters per second plus--

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what's our final velocity?

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In this situation,
remember, we're

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just going over the
first 2.5 seconds.

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So our final velocity is once
again 0 meters per second.

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We're just talking
about when we get

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to this point right over here.

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So our final velocity is
just 0 meters per second.

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And we're just going
to divide that by 2.

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This will give us
the average velocity.

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And then we want to multiply
that times 2.5 seconds.

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So we get-- this part right
over here-- 24.5 divided by 2.

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We can ignore the 0.

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That still is 24.5 That
gives us 12.25 times 2.5.

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And remember, this right
over here is in seconds.

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Let me write the units down.

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So this is 12.25 meters per
second times 2.5 seconds.

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And just to remind
ourselves, we're

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calculating the displacement
over the first 2 and 1/2

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seconds.

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So this gives us-- I'll get
the calculator out once again.

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We have 12.25 times 2.5
seconds gives us 30.625.

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So this gives us--
so our displacement

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is 30.625 meters-- these
seconds cancel out-- meters.

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This is actually a ton.

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This is roughly, give or
take, about 90 feet thrown

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in the air.

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So this would be like
a nine-story building.

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And I, frankly, do not
have the arm for that.

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But if someone is
able to throw the ball

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for 5 seconds in
the air, they have

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thrown it 30 meters in the air.

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Well, hopefully you
found that entertaining.

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In the next video,
I'll generalize this.

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Maybe we can get a
little bit of a formula

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so maybe you can generalize it.

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So regardless of the
measurement of time,

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you can get the
displacement in the air.

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Or even better, try to
derive it yourself and we'll

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see how at least I tackle
it in the next video.