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In the last video, we figured
out the formulas for the

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observed period and frequency
for an observer sitting in the

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path of the source.

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So the source is moving
towards the observer.

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So this is the example where
the train is moving towards

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you, and you perceive the
train's horn as having a

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higher pitch or a higher
frequency.

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And we were able to do that by
doing a thought experiment.

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Saying, OK, my object
starts here.

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After one period-- a period is
just a measure of time, but

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it's the measure of time over
which the source emits a

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cycle, so it emits a
cycle every period.

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But after one period, we said,
OK, where is that first wave

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front, or that first pulse,
or that first crest?

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And where is the source?

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Because exactly one period has
passed by, and the source will

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be ready to emit another
crest or another cycle.

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So the distance between where
the source is and that front

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of the crest, or that first
crest, that is going to be the

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wavelength.

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Because this next thing that
emits is going to be traveling

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at the exact same velocity, and
it's going to be separated

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by that distance, which we
saw is this expression.

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We said how long will it take
it to travel that distance?

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Well, it's traveling at
a speed of v sub w.

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That'll tell you what the
observed period would be for

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this dude over here.

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We calculated it right here,
and then the observed

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frequency is just the
inverse of that.

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Now let's think about the
situation where the observer

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is over here.

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So these equations, or these
formulas that we came up with

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right here, this is observer--
or let me say source traveling

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in direction of observer.

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Now let's think about the
opposite case, where the

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source is traveling away from
the observer, and in this

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case, the observer is
that guy over there.

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Maybe I'll do it in
a different color.

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He'll be blue.

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So this is the observer.

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So when we started off, our
source was right here.

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After exactly one period from
the source's point of view,

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that first crest emitted
has traveled radially

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outward that far.

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This is the distance.

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The velocity of the wave times
the amount of time that

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passed-- velocity times time
is going to give you

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distance-- and where the source
of the wave will have

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traveled to the right exactly
this distance.

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It's velocity times the amount
of time that's gone by.

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Now, in the last video, we said,
OK, that wave is just

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passing this guy.

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How long will it take for that
pulse that's being emitted

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right then to also reach him?

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And then that tells us the
period between two pulses or

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between two crests.

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Now let's think about that exact
same situation here.

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That first crest is just
passing this guy.

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And a period, or the t sub s,
which is a period of the

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emitted wave has
just passed by.

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So this guy is just about to
emit another wave, So?

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That other wave is going
to be right here.

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So what is the distance between
the crest, or the

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cycle, or however you want to
think about it, the pulse that

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is passing him by right now,
what is the distance between

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that and the pulse, the front
edge of that pulse that is

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being emitted right
at that moment?

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Right at that moment.

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Well, it's going to be this
radius, which is this value.

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It'll be v sub w times
the period.

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That is that distance plus the
amount of distance that our

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source has traveled away
from this guy.

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So plus v sub s times
the period.

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So that's what this distance is,
and this is how far apart

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this wave pulse is going to be
from that one or this crest is

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going to be from that one.

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So if he's seeing this first
crest right now, right at this

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moment, how long will it take
him to see the crest that's

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being emitted right now, that's
this far away from him?

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Well, it's that far
away from him.

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So let me write this down.

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So the amount of time it takes
for him to see the next crest

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or the same point in the next
cycle, that's the period.

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That's the observed period.

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That's going to be equal
to this distance.

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The velocity of the wave times
the period from the

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perspective of the source plus
the velocity of the source,

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because the source has gotten
that much further away from

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him, velocity of the source
times the period of the

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source, so that's how far
the next crest is.

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And then you divide it by the
speed of the wave, by the

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speed of each of the crests,
which is just the velocity of

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the wave. And we can just factor
out the t sub s's here

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and say this is t sub s times
v sub w, the velocity of the

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wave plus the velocity of the
source divided by the velocity

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the wave. So this will be a
larger observed period than if

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this guy was stationary and
especially if the observer was

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in the path of the guy.

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That make sense, because every
time this guy issues a cycle,

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he is moving a little
bit further away.

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So every crest the same point
in the cycle is going to be

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further and further apart, so
you're going to have longer

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wavelengths, longer periods.

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And if you want the observed
frequency for that guy over

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there-- I'll do it in
the same color.

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The observed frequency for
a guy where the source is

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traveling away from him is
just the inverse of that.

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So one over the period, one over
the period, same argument

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we did there, one over the
period from the point of view

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of the source is the frequency
of the source.

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Let me Color code it.

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So 1 over t sub s is equal to
the frequency of the source.

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This is the inverse of that.

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So I'm just taking one
over everything here.

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So 1 over t sub s is the
frequency of the source, and

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then we take the inverse of this
over here: the velocity

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of the wave divided by the
velocity of the wave plus the

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velocity of the source.

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So we're done, at least
for the simple cases.

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Obviously, it becomes a little
more interesting when someone

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isn't exactly in the direction
of the source or exactly being

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moved away from, but these are
kind of the two extreme cases.

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So this is the situation when
it is moving away from you.

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Now, just to check our math and
maybe make it a little bit

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concrete in relation to the
video we did where we

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introduced to the idea of the
Doppler effect, let's actually

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apply those numbers.

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So on that video, two videos
ago, we had a situation where

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the velocity of our source was
5 meters per second to the

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right, and the velocity of the
wave was 10 meters per second

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radially outward, and the period
of our wave-- I'll call

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it the period of our-- let me do
it in another color-- from

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the point of view of the source
was 1 second per cycle,

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and the frequency was just
the inverse of that.

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So 1 cycle per second,
or 1 Hertz, which

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is a cycle per second.

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So using those numbers, let's
see if we get to the exact

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same answer we got in that first
video where we first

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learned about the
Doppler effect.

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So let's look at the frequency
from the point of view of this

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gentleman right here.

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So the frequency of the source
is going to be 1 cycle per

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second, 1 Hertz.

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The velocity of the wave is
10 meters per second.

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Let me write this.

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1 cycle per second.

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The velocity of the wave is
10 meters per second.

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The velocity of the wave is 10
meters per second minus the

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velocity of the source is
5 meters per second.

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So what's this going
to be equal to?

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The observed frequency for this
guy right there, is going

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to be 1 cycle per second times
10 over 10 minus 5, and the

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meters per second cancel out.

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Meters per second in the
numerator, meters per second

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in the denominator.

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So 10 divided by 10 minus
5, or 10 divided by 5,

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is going to be 2.

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So it's going to be 2
cycles per second.

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And if you want the observed
period for this guy, its going

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to be the inverse of that,
or it's going to be

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1/2 second per cycle.

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And this is exactly what we got
in the previous video, or

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actually, it was
two videos ago.

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Now what about the guy who this
guy's running away from?

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Well, we'll do the
exact same thing.

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You have 1 cycle per
second, or 1 Hertz.

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That's the emitted frequency
from the point of view of the

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source times the velocity of
the wave divided by the

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velocity of the wave plus the
velocity of the source,

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because it's moving
away from him.

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So it's 10 over 10 plus 5.

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That's 10 over 15.

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That is 2/3, so this
is equal to 2/3.

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The units over here
all cancel out.

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This was cycles per second.

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So 2/3 cycle per second, which
confirms the numbers we got in

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that first video, so that should
make us feel good and

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it also makes a lot of sense.

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This guy is going to see the
wave crest more frequently.

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He's going to observe
a higher frequency.

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If this is a sound,
a higher pitch.

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This guy, since each crest or
the cycles are getting spread

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out, he's going to see them less
frequently, and if this

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is sound, he's going to
observe a lower pitch.

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