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- [Instructor] There's
a big fat, juicy apple

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hanging from a tree branch,
and you want this apple,

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but you can't climb the tree?

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Luckily, you've got an
orange in your pocket.

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So you take this orange and
you chuck it at the apple,

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and it strikes it at the
apex of its trajectory,

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causing the apple to fly off,

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and now you've got an orange and an apple.

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Now this is technically fruit vandalism

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if it's not your apple tree.

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So make sure you're only
picking your own apples

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or you've paid someone to do
this and everything's legit.

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But this is also a collision problem.

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And in physics, you could solve
for the velocities involved

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and the mass and the momentums involved

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by using conservation momentum,
if we had some numbers.

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So let's give ourselves some numbers.

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Let's see if we can solve
for some quantities here.

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So let's say this apple, I
told you it was big and fat,

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let's say this apple was 0.7 kilograms.

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Let's say the orange
is probably not as big,

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so 0.4 kilograms.

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And let's give some numbers,

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let's say the speed of the
orange right before the collision

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was five meters per second.

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So since this orange
was at its apex, right?

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It was just heading in this way

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and it was going
horizontally at that moment,

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five meters per second, right
before it struck the apple.

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And let's say the apple was
moving three meters per second

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right after the collision,

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so right after the orange hit the apple,

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the apple starts flying at
three meters per second.

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One question we could ask,
one obvious question, is,

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if this is the speed of the
apple after the collision,

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what was the speed of the
orange after the collision?

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What was the velocity of the orange?

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And which way was it going?

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So we'll call it VO for V orange,

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and was that orange going left or right

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immediately after the
collision took place?

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Sometimes this isn't so obvious,

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so let's see if we can solve for this now.

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We've got enough to solve.

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We can do this using
conservation momentum,

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and conservation momentum says

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that if there's no external
impulse on a system,

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and our system here is
the orange and apple,

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if there's no external
impulse on these fruit,

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that means the total momentum

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before the collision took place,

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so right before the collision took place,

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has got to equal the total momentum

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right after the collision took place,

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and it's important that we denote

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right before and right after.

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We're not talking, like, right
as someone threw this fruit

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before it got up here, and
we're not talkin' finally,

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like after the apple gets
back down to the ground.

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You can't do that.

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For most collision problems,
you're gonna want to consider

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right before the
collision and right after,

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and the reason is,
remember, this formula here

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is only true if there's
no external impulse.

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So only if external impulse is zero.

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And you might be like,
well, isn't it always zero?

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Shouldn't it be zero in this case?

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It's not so obvious.

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If you're clever, you might
be like, wait a minute,

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there's a force of gravity on this apple.

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There's a force of gravity on the orange.

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So doesn't that mean
there's an external force?

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And if there's an external force,

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doesn't that mean there's
an external impulse?

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And doesn't that mean that the momentum

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shouldn't be conserved?

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Well, not really.

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And the reason is,

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for one, this force of
gravity is directed downward,

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so it's only gonna effect
the vertical momentum.

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And we're just talking about
the horizontal momentum here.

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I wanna know what happens
to the horizontal momentum

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of this orange, but secondly,
the definition of impulse

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is that it's the force that acts

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multiplied by the time duration.

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We're gonna say that if
we consider right before

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the collision and right
after the collision

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is our initial and final points,

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this time interval's gonna be so small

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that the force of gravity is gonna have

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almost no time to act.

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And because it has almost no time to act,

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it has almost no external impulse.

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So we're gonna ignore the
impulse due to gravity,

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because it acts over such
a small period of time

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and it's such a modest force,
which means we get to use

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conservation momentum for our system.

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So what is this gonna look like?

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Well, the momentum formula
is mass times velocity,

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so the initial momentum of the system,

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let's see, I'd have to
add up initial momentum

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of the orange is 0.4
kilograms, that's the mass,

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times the initial velocity,
that's five meters per second,

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plus, I'm gonna add to
that the mass of the apple,

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0.7 kilograms, multiplied by

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the initial velocity of the apple.

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What was the initial
velocity of the apple?

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It wasn't three, people
try to plug in three,

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that wasn't the final
velocity of the apple.

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The initial velocity of
the apple was just zero,

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'cause it was hangin' on a tree branch

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and was just sittin' there.

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So this is gonna be zero.

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What that means is, this
entire term is gonna be zero,

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'cause zero times 7.7 is still zero.

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So this term just goes away.

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It's gonna be zero

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equals the final momentum.

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All right, so we added
up the total momentum

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of our system initially,

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now we're gonna add up all the momentum

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of our system finally.

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So 0.4 kilograms is
the mass of the orange,

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multiplied by, we don't
know the final velocity

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of the orange, that's
the thing we wanna find.

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So I'm gonna write that as
VO, for V of the orange.

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This final velocity of the
orange is what we wanna find.

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This term here represents the
final momentum of the orange,

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but I have to, I can't stop yet.

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I have to add to that the
final momentum of the apple.

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So remember, when you're writing
down conservation momentum

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for a system, the statement
isn't that the initial momentum

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of one object equals the final momentum

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of some other object.

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It says that the total initial
momentum of the entire system

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equals the total final
momentum of the entire system.

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So I'll take my .7, multiplied
by my final velocity

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is three meters per second for the apple,

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and now I can solve.

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So we can solve this,
I've only got one unknown.

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So .4 times five is two
kilogram meters per second,

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plus zero, I'm not gonna write that,

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'cause it would just take up space.

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Equals .4 times VO is the unknown,

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so .4 kilograms times the unknown VO,

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and then plus .7 times three

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is gonna be 2.1 kilogram
meters per second.

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So our system started off with

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two kilogram meters per second
of momentum to the right.

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That's what the orange brought in.

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And our system ends with 2.1
kilogram meters per second

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to the right, which is what the apple has,

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plus whatever momentum the orange has

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right after the collision,

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and you might look at this and be like,

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wait a minute, hold on,

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we screwed somethin' up.

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Two kilogram meters per second

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equals 2.1 kilogram meters
per second plus something?

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How is this right hand
side ever gonna equal two

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if it's got 2.1 to start with,

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but remember it can, momentum is a vector.

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And vectors can be positive or negative,

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depending on whether
they point right or left.

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So this just tells us, okay,

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the orange is going to
have to have momentum

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leftward after the collision,

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so that this whole right hand
side can add up to two again,

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and we know we're gonna have
a final velocity of the orange

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that's negative.

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But you don't have to be clever.

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If you just wanted to solve this equation,

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it'll tell you whether
it's going right or left.

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I'll show you why.

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If we just do this, two minus 2.1.

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So if we subtract 2.1 from both sides

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we'll get negative 0.1
kilogram meters per second

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and that's gonna equal this
final momentum of the orange,

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so equals 0.4 kilograms for the orange,

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times VO, the final
velocity of the orange,

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and now if we just
divide both sides by .4,

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we'll get negative 0.25 meters per second.

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That's the final velocity of the orange,

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and you realize, oh, I
didn't have to figure out

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the sine beforehand, I could just solve

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and in the conservation of
momentum formula will tell me

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whether it's going right or left,

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'cause if I get a negative
sine here, it just says,

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oh that velocity had to be directed

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in the negative direction in
order to conserve momentum

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in this case, so this orange,
right after the collision

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was heading leftward, that's
what the negative sign means,

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and this .25 means it was
heading leftward at a rate

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of .25 meters per second.

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So recapping, we could use
conservation of momentum

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to solve for an unknown velocity

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by setting the total initial
momentum of the system

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equal to the total final
momentum of the system.

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We gotta be careful with negative signs.

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If there was an initial
velocity that was negative,

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we would've had to plug in that velocity

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with a negative number, and
if we find a negative velocity

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to end with, that means that
quantity of that velocity

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was directed in the negative direction.

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Also, we can only use
conservation of momentum

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whenever the external impulse is zero,

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which is why we consider
points immediately before

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the collision and immediately after,

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so that this time interval is so small,

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gravity can't apply much
of an impulse at all,

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and I should say, we should
assume that this stem

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was barely hanging on by a string,

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'cause if this stem was
secured to the tree,

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then there would've
been and external force

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that coulda caused an external impulse.

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So let's assume this apple was

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already just about to fall off,

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and the slightest of
forces could knock it off.

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That way, there's no external impulse

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and we get to use
conservation of momentum.