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What I want to do in this video
is answer an age-old question,

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or at least an interesting
question to me.

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And the question is, let's
say I have a ledge here--

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I have a ledge or
cliff, or maybe

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this is a building of some kind.

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And let's say it has height, h.

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So let's say it has a height
of h, right over here.

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And what I'm curious about is
if I were to either-- Let's

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say that this is me over
here, so this is me.

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If I were to either
jump, myself--

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that's not recommended
for very large h's.

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Or If I were to throw something,
maybe a rock off of this

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ledge, how fast would
either myself or that rock

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be going right before
it hits the ground?

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And like all of the
other videos we're

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doing on projectile
motion right now,

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we're going to ignore
air resistance.

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And for small h's and
for small velocities,

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that's actually reasonable.

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Or if the object
is very aerodynamic

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and is kind of dense,
then the air resistance

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will matter less.

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If it's me kind of belly
flopping from a high altitude,

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then the air resistance
will start to matter a lot.

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But for the sake of simplicity,
we're going to assume no air.

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Or we're not going
to take into effect

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the effects of air resistance.

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Or we could assume
that we're doing

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this on an Earth-like planet
that has no atmosphere.

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However you want to do it.

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So let's just think
about the problem.

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And just so you know,
some of you might say,

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that's not realistic.

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But this actually would be
realistic for a small h.

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If you were to jump off of the
roof of a one-story building,

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air resistance will not
be a major component

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in determining your speed if.

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It was to be a much larger
building, then all of a sudden

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it matters.

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And I don't recommend you
do any of these things.

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Those are all very
dangerous things.

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Much better to do
it with a rock.

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So that's actually the example
we're going to be considering.

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So let's just think
about this a little bit.

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We want to figure
out-- So at the top,

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right when the
thing gets dropped,

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right when the
rock gets dropped,

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you have an initial
velocity of 0.

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And once again, we're going
to use the convention here

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that positive velocity
means upwards,

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or a positive vector means up,
a negative vector means down.

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So we're going to have an
initial velocity over here

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of 0.

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And then at the
bottom we're going

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to have some final
velocity here that

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is going to be a
negative number.

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So it's going to have some
negative value over here.

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So this is going to be negative.

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This is going to be a negative
number right over there.

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And we know that the
acceleration of gravity

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for an object on free
fall, an object in free

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fall near the
surface of the earth.

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We know it, and we're going
to assume that it's constant.

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So our constant
acceleration is going

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to be negative 9.8 meters
per second squared.

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So what we're going
to do is given an h,

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and given that their
initial velocity is 0

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and that our acceleration
is negative 9.8 meters per

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squared, we want to figure
out what our final velocity is

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going to be right before
we hit the ground.

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We're going to
assume that this h is

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given in meters,
right over here.

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And we'll get an
answer in meters

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per second for that
final velocity.

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So let's see how we
can figure it out.

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So we know some basic things.

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And the whole point of
these is to really show you

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that you can always derive
these more interesting questions

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from very basic
things that we know.

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So we know that displacement is
equal to average velocity times

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change in time.

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And we know that
average velocity--

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if we assume
acceleration is constant,

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which we are doing--
average velocity

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is the final velocity plus
the initial velocity over 2.

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And then our change in time,
the amount of elapsed time

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that goes by-- this is
our change in velocity.

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So elapsed time
is the same thing.

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I write it over here--
is our change in velocity

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divided by our acceleration.

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And just to make sure
you understand this,

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it just comes
straight from the idea

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that acceleration-- or
let me write it this way--

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that change in velocity is
just acceleration times time.

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Or I should say, acceleration
times change in time.

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So if you divide both sides of
this equation by acceleration,

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you get this right over here.

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So that is what our
displacement-- Remember,

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I want an expression
for displacement

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in terms of the things
we know and the one

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thing that we want to find out.

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Well, for this example
right over here,

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we know a couple of things.

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Well actually, let me
take it step by step.

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We know that our
initial velocity is 0.

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So this first expression
for the example we're doing,

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the average velocity is going
to be our final velocity divided

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by 2, since our
initial velocity is 0.

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Our change in velocity
is the same thing

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as final velocity
minus initial velocity.

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And once again, we know that
the initial velocity is 0 here.

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So our change in velocity
is the same thing

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as our final velocity.

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So once again,
this will be times.

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Instead of writing
change in velocity here,

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we could just write
our final velocity

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because we're starting at 0.

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Initial velocity is 0.

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So times our final velocity
divided by our acceleration.

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Final velocity is the same
thing as change in velocity

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because initial velocity was 0.

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And all of this is going
to be our displacement.

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And now it looks like
we have everything

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written in things we know.

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So if we multiply both
sides of this expression

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or both sides of this equation
by 2 times our acceleration

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on that side.

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And we multiply
the left-hand side

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by-- I'll do the same colors--
2 times our acceleration.

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On the left hand side, we
get 2 times our acceleration

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times our displacement
is going to be equal to,

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on the right hand side, the
2 cancels out with the 2,

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the acceleration cancels
out with the acceleration--

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it will be equal
to the velocity,

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our final velocity squared.

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Final velocity times
final velocity.

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And so we can just solve
for final velocity here.

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So we know our acceleration
is negative 9.8 meters

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per second squared.

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So let me write this over here.

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So this is negative 9.8.

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So we have 2 times
negative 9.8--

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let me just multiply that out.

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So that's negative 19.6
meters per second squared.

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And then what's our
displacement going to be?

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What's the displacement
over the course

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of dropping this rock off of
this ledge or off of this roof?

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So you might be tempted to say
that our displacement is h.

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But remember, these
are vector quantities,

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so you want to make sure
you get the direction right.

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From where the rock started to
where it ends, what's it doing?

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It's going to go
a distance of h,

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but it's going to go a
distance of h downwards.

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And our convention
is down is negative.

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So in this example,
our displacement

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from when it leaves your hand
to when it hits the ground,

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the displacement is going
to be equal to negative h.

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It's going to travel
a distance of h,

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but it's going to travel
that distance downwards.

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And that's why this vector
notion is very important here.

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Our convention is
very important here.

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So our displacement over here is
going to be negative h meters.

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So this is the
variable, and this

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is the shorthand for meters.

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So when you multiply
these two things out,

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lucky for us these
negatives cancel out,

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and you get 19.6h meters squared
per second squared is equal

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to our final velocity squared.

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And notice, when
you square something

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you lose the sign information.

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If our final velocity was
positive, you square it,

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you still get a positive value.

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If it was negative
and you square it,

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you still get a positive value.

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But remember, in
this example, we're

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going to be moving downward.

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So we want the negative
version of this.

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So to really figure
out our final velocity,

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we take, essentially,
the negative square root

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of both sides of this equation.

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So if we were to take the square
root of both sides of this,

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you take the square
root of that side,

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you take the square root of
that side, you will get--

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and I'll flip them around-- your
final velocity, we could say,

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is equal to the
square root of 19.6h.

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And you can even take the
square root of the meter squared

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per second squared, treat
them almost like variables,

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even though they're units.

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And then outside of
the radical sign,

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you will get a
meters per second.

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And the thing I want
to be careful here

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is if we just take the
principal root here,

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the principal root here is
the positive square root.

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But we know that our velocity
is going to be downwards here,

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because that is our convention.

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So we want to make sure we
get the negative square root.

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So let's try it out
with some numbers.

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We've essentially
solved what we set out

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to solve at the
beginning of this video,

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how fast would we be falling,
as a function of the height.

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Well, let's try it
out with something.

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So let's say that the
height is-- I don't know,

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let's say the height
is 5 meters, which

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would be probably
jumping off of a

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or throwing a rock off
of a one-story, maybe

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a commercial one-story building.

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That's about 5 meters,
would be about 15 feet.

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So yeah, about the roof
of a commercial building,

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give or take.

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So let's turn it on.

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And so what do we get?

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If we put 5 meters in here, we
get 19.6 times 5 gives us 98.

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So almost 100.

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And then, we want to take
the square root of that,

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so it's going to be almost 10.

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So the square root of
98 gives us roughly 9.9.

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And we want the negative
square root of that.

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in that situation, when
the height is 5 meters-- So

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if you jump off of a
one-story commercial building,

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right at the bottom, or if you
throw a rock off that, right

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at the bottom, right
before it hits the ground,

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it will have a velocity of
negative 9.9 meters per second.

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So negative 9.9
meters per second.

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I'll leave it up to
you, as an exercise,

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to figure out how fast
this is either kilometers

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per hour or miles per hour
because it's pretty fast.

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It's not something
you's want to do.

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And this is just off of
a one-story building.

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But you can usually
figure this out.

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You could use this for,
really, any height as

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long as we're reasonably close
to the surface of the earth

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and you ignore the
effects of air resistance.

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At really high heights,
especially if the object is not

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that aerodynamic,
then air resistance

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will start to matter a lot.