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Welcome back.

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When I left off I was rushing
at the end of this problem

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because I tend to rush at the
end of problems when I am

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getting close to the YouTube
10 minute limit.

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But I just wanted to review the
end of it because I feel

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like I rushed it.

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And then, actually continue with
it and actually solve for

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the angle and then, introduce
a little bit of-- a little

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more trigonometry.

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So just to review what we did,
we said momentum is conserved

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and in two dimensions that means
momentum is conserved in

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each of the dimensions.

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So we figured out what the
initial momentum of the entire

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system was and we said, well,
in the x direction, the

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initial momentum-- and all the
momentum was coming from the

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ball A right.

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Because ball B wasn't moving,
so its velocity was 0.

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So its momentum was 0.

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So ball A in the x direction and
it was only moving in the

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x direction.

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So it's momentum in the x
direction was 3 meters per

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second times 10 kilogram
meters per second.

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And we got 30 kilogram
meters per second.

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And then there was no momentum
in the y direction.

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And then we knew that well after
they hit each other,

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ball A kind of ricochets off
at a 30 degree angle at 2

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meters per second.

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We used that information to
figure out the x and y

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components of A's velocity.

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So A's velocity in the y
direction was 1 meter per

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second and A's velocity
in the x direction was

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square root of 3.

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And we used that information to
figure out A's momentum in

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each direction.

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We said well, the momentum in
the y direction must be 1

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meter per second times A's mass,
which is 10 kilogram

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meters per second.

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Which I wrote-- what
I wrote here.

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And then we figured out A's
momentum in the B direction

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and we said well, that's
just going to be square

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root of 3 times 10.

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And that's 10 square
root of 3.

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And then we used that
information to

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solve for B's momentum.

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Because we said well, B's
momentum plus A's momentum in

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the x direction has
to add up to 30.

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This was the x direction
before.

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And we knew that B's momentum
plus A's momentum in the y

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direction had to add
up to 0, right?

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And so, since y's momentum going
upwards was 10 kilogram

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meters per second, we knew
that B's momentum going

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downwards would also
have to be 10

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kilogram meters per second.

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Or you could even say
it's negative 10.

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And we figure that out based
on the fact that B

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had half the mass.

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That its velocity going down
was 2 meters per second.

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And similarly, we knew that
A's momentum in the x

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direction, which was 10 square
root of 3 kilogram meters per

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second, plus B's momentum
in the x

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direction is equal to 30.

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And then we just subtracted out
and we got B's momentum in

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the x direction.

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And then we divided by B's
mass to get its velocity.

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Which we got as 2.54.

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So that's where I left off
and we were rushing.

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And already, this gives you a
sense of what B is doing.

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Although it's broken up into
the x and y direction.

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Now if we wanted to simplify
this, if we wanted to kind of

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write B's new velocity the same
way that the problem gave

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us A's velocity, right?

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They told us A's velocity was
2 meters per second at an

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angle of 30 degrees.

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We now have to use this
information to figure out B's

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velocity and the angle of it.

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And how do we do that?

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Well this is just straight up
trigonometry at this point, or

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really just straight
up geometry.

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Let me clear all of this.

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And let's remember these two
numbers, 2.54 and minus 2.

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So B, we learned that in the x
direction its velocity-- this

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is all for B-- is equal to 2.54
meters per second and

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then y direction, it
was moving down.

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We could write this
as minus 2.

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But I'll just write this as 2
meters per second downwards.

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Right?

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Same thing.

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Minus 2 up is the same thing as
2 meters per second down.

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So the resulting vector's
going to look

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something like this.

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When you add two vectors you
just put them-- put the one's

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end at the beginning of the
other-- put them front to end,

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like we did here.

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And then you add them
together and this is

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the resulting vector.

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And I think you're used
to that at this point.

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And now we have to figure out
this angle and this side.

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Well this side is easy because
this is a right angle, so we

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use Pythagorean theorem.

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So this is going to be the
square root of 2.54 squared

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plus 2 squared.

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And what's 2.54 squared?

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2.54 times-- whoops.

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2.54 times 2.54 is
equal to 6.45.

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So that's the square root of
6.45 plus 4, which equals the

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square root of 10.45.

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And take the square
root of that.

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So that's 3.2, roughly.

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So the resulting velocity in
this direction, whatever angle

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this is, is 3.2 meters
per second.

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And I just used Pythagorean
theorem.

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So now all we have to do is
figure out the angle.

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We could use really any of the
trig ratios because we know

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all of the sides.

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So I don't know, let's
use one that you

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feel comfortable with.

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Well let's use sine.

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So sine of theta is
equal to what?

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SOH CAH TOA.

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Sine is opposite over
hypotenuse.

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So the opposite side is the y
direction, so that's 2, over

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the hypotenuse, 3.2.

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So 2 divided by 2 divided by 3.2
is equal to 0.625, which

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equals 0.625.

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So sine of theta equals 0.625.

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And maybe you're not familiar
with arcsine yet because I

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don't think I actually have
covered yet in the trig

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modules, although I
will eventually.

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So we know it's just the inverse
function of sine.

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So sine of theta is
equal to 0.625.

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Then we know that theta is equal
to the arcsine of 0.625.

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This is essentially saying,
when you say arcsine, this

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says, tell me the angle whose
sine is this number?

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That's what arcsine is.

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And we can take out Google
because it actually happens

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that Google has a-- let's see.

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Google actually-- it's an
automatic calculator.

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So you could type in arcsine
on Google of 0.625.

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Although I think the
answer they give

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you will be in radians.

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So I'll take that answer that
will be in radians and I want

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to convert to degrees, so I
multiply it times 180 over pi.

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That's just how I convert
from radians to degrees.

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And let's see what I get.

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So Google, you see, Google
says 38.68 degrees.

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They multiplied the whole
thing times 180 and then

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divided by pi, but that should
be the same thing.

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So roughly 38.7 degrees
is theta.

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Hope you understand that.

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You could pause it here if
you don't, but let me

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just write that down.

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So it's 38 degrees.

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So theta is equal
to 38.7 degrees.

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So then we're done.

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We figured out that
ball B gets hit.

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This is ball B and it
got hit by ball A.

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Ball A went off in that
direction at a 30 degree

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angle, at a 30 degree angle
at 2 meters per second.

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And now ball B goes at 38.--
or we could say roughly 39

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degrees below the horizontal
at a velocity of 3.2 meters

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per second.

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And does this intuitively
make sense to you?

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Well if you remember the problem
from before-- and I

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know I erased everything.

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Ball A had a mass of 10
kilograms while ball B had a

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mass of 5 kilograms.
So it makes sense.

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So let's think about just
the y direction.

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Ball A, we figured out, the y
component of its velocity was

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1 meter per second.

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And ball B's y component is 2
meters per second downwards.

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And does that makes sense?

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Well sure.

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Because their momentums
have to add up to 0.

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There was no y component of the
momentum before they hit

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each other.

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And in order for B to have the
same momentum going downwards

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in the y direction as A going
upwards, its velocity has to

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be essentially double, because
its mass is half.

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And a similar logic, although
the cosine-- it doesn't work

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out exactly like that.

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But a similar logic would mean
that its overall velocity is

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going to be faster than the-
than A's velocity.

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And so what was I
just-- oh yeah.

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My phone was ringing and
I got caught up.

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My brain starts to
malfunction.

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But anyway, as I was
saying, so just

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intuitively it makes sense.

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B has a smaller mass than A, so
it makes sense that-- one,

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B will be going faster and
that it gets deflected a

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little bit more as well.

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The reason why it seems like
it gets deflected more is

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because its y component
is more.

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But anyway, that last piece is
just to kind of hopefully give

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you a sense of what's happening
and I will see you

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in the next video.

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