Introduction to l'Hôpital's rule

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.560
Most of what we do early on
when we first learn about

00:00:03.560 --> 00:00:06.720
calculus is to use limits.

00:00:06.720 --> 00:00:10.440
We use limits to figure out
derivatives of functions.

00:00:13.720 --> 00:00:16.800
In fact, the definition
of a derivative uses

00:00:16.800 --> 00:00:18.120
the notion of a limit.

00:00:18.120 --> 00:00:22.030
It's a slope around the point
as we take the limit of

00:00:22.030 --> 00:00:24.390
points closer and closer
to the point in question.

00:00:24.390 --> 00:00:27.110
And you've seen that many,
many, many times over.

00:00:27.110 --> 00:00:29.790
In this video I guess we're
going to do it in the

00:00:29.790 --> 00:00:30.630
opposite direction.

00:00:30.630 --> 00:00:39.130
We're going to use derivatives
to figure out limits.

00:00:39.130 --> 00:00:43.360
And in particular, limits that
end up in indeterminate form.

00:00:43.360 --> 00:00:46.550
And when I say by indeterminate
form I mean that when we just

00:00:46.550 --> 00:00:51.550
take the limit as it is, we end
up with something like 0/0, or

00:00:51.550 --> 00:00:55.450
infinity over infinity, or
negative infinity over

00:00:55.450 --> 00:00:57.870
infinity, or maybe negative
infinity over negative

00:00:57.870 --> 00:01:00.330
infinity, or positive infinity
over negative infinity.

00:01:00.330 --> 00:01:05.190
All of these are indeterminate,
undefined forms.

00:01:05.190 --> 00:01:08.290
And to do that we're going
to use l'Hopital's rule.

00:01:17.730 --> 00:01:19.490
And in this video I'm just
going to show you what

00:01:19.490 --> 00:01:23.010
l'Hoptial's rule says and how
to apply it because it's fairly

00:01:23.010 --> 00:01:25.480
straightforward, and it's
actually a very useful tool

00:01:25.480 --> 00:01:27.510
sometimes if you're in some
type of a math competition and

00:01:27.510 --> 00:01:31.270
they ask you to find a
difficult limit that when you

00:01:31.270 --> 00:01:33.310
just plug the numbers in you
get something like this.

00:01:33.310 --> 00:01:37.270
L'Hopital's rule is normally
what they are testing you for.

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And in a future video I might
prove it, but that gets a

00:01:40.290 --> 00:01:40.975
little bit more involved.

00:01:40.975 --> 00:01:44.870
The application is actually
reasonably straightforward.

00:01:44.870 --> 00:01:50.780
So what l'Hopital's rule tells
us that if we have-- and I'll

00:01:50.780 --> 00:01:53.940
do it in abstract form first,
but I think when I show you

00:01:53.940 --> 00:01:56.080
the example it will
all be made clear.

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That if the limit as x approaches
c of f of x is equal to 0, and

00:02:11.050 --> 00:02:20.100
the limit as x approaches c of
g of x is equal to 0, and-- and

00:02:20.100 --> 00:02:30.620
this is another and-- and the
limit as x approaches c of f

00:02:30.620 --> 00:02:38.160
prime of x over g prime of
x exists and it equals L.

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then-- so all of these
conditions have to be met.

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This is the indeterminate
form of 0/0, so this

00:02:44.050 --> 00:02:45.560
is the first case.

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Then we can say that the
limit as x approaches c of

00:02:55.380 --> 00:03:03.070
f of x over g of x is also
going to be equal to L.

00:03:03.070 --> 00:03:05.810
So this might seem a little bit
bizarre to you right now, and

00:03:05.810 --> 00:03:07.220
I'm actually going to write the
other case, and then

00:03:07.220 --> 00:03:07.750
I'll do an example.

00:03:07.750 --> 00:03:09.710
We'll do multiple examples
and the examples are going

00:03:09.710 --> 00:03:10.580
to make it all clear.

00:03:10.580 --> 00:03:13.040
So this is the first case and
the example we're going to

00:03:13.040 --> 00:03:16.650
do is actually going to be
an example of this case.

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Now the other case is if the
limit as x approaches c of f of

00:03:24.390 --> 00:03:32.960
x is equal to positive or
negative infinity, and the

00:03:32.960 --> 00:03:39.470
limit as x approaches c of g of
x is equal to positive or

00:03:39.470 --> 00:03:46.280
negative infinity, and the
limit of I guess you could say

00:03:46.280 --> 00:03:51.220
the quotient of the derivatives
exists, and the limit as x

00:03:51.220 --> 00:03:56.200
approaches c of f prime of x
over g prime of x

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is equal to L.

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Then we can make this
same statement again.

00:04:01.860 --> 00:04:06.070
Let me just copy that out.

00:04:06.070 --> 00:04:10.220
Edit, copy, and then
let me paste it.

00:04:10.220 --> 00:04:13.450
So in either of these two
situations just to kind of make

00:04:13.450 --> 00:04:15.520
sure you understand what you're
looking at, this is the

00:04:15.520 --> 00:04:18.300
situation where if you just
tried to evaluate this limit

00:04:18.300 --> 00:04:21.880
right here you're going to
get f of c, which is 0.

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Or the limit as x approaches c
of f of x over the limit as

00:04:25.990 --> 00:04:27.480
x approaches c of g of x.

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That's going to give you 0/0.

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And so you say, hey, I don't
know what that limit is?

00:04:32.440 --> 00:04:33.600
But this says, well, look.

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If this limit exists, I could
take the derivative of each

00:04:37.440 --> 00:04:41.020
of these functions and then
try to evaluate that limit.

00:04:41.020 --> 00:04:44.220
And if I get a number, if that
exists, then they're going

00:04:44.220 --> 00:04:45.980
to be the same limit.

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This is a situation where when
we take the limit we get

00:04:48.710 --> 00:04:52.060
infinity over infinity, or
negative infinity or positive

00:04:52.060 --> 00:04:54.070
infinity over positive
or negative infinity.

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So these are the two
indeterminate forms.

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And to make it all clear let
me just show you an example

00:04:59.410 --> 00:05:03.610
because I think this will make
things a lot more clear.

00:05:03.610 --> 00:05:08.910
So let's say we are trying
to find the limit-- I'll

00:05:08.910 --> 00:05:10.610
do this in a new color.

00:05:10.610 --> 00:05:13.830
Let me do it in this
purplish color.

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Let's say we wanted to find
the limit as x approaches

00:05:17.420 --> 00:05:23.160
0 of sine of x over x.

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Now if we just view this, if we
just try to evaluate it at 0 or

00:05:27.430 --> 00:05:30.410
take the limit as we approach 0
in each of these functions,

00:05:30.410 --> 00:05:33.180
we're going to get something
that looks like 0/0.

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Sine of 0 is 0.

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Or the limit as x approaches
0 of sine of x is 0.

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And obviously, as x approaches
0 of x, that's also

00:05:42.120 --> 00:05:43.140
going to be 0.

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So this is our
indeterminate form.

00:05:45.100 --> 00:05:48.180
And if you want to think about
it, this is our f of x, that

00:05:48.180 --> 00:05:51.240
f of x right there
is the sine of x.

00:05:51.240 --> 00:05:55.510
And our g of x, this g of
x right there for this

00:05:55.510 --> 00:06:00.070
first case, is the x.

00:06:00.070 --> 00:06:07.170
g of x is equal to x and f
of x is equal to sine of x.

00:06:07.170 --> 00:06:10.130
And notice, well, we definitely
know that this meets the

00:06:10.130 --> 00:06:11.520
first two constraints.

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The limit as x, and in
this case, c is 0.

00:06:14.970 --> 00:06:19.540
The limit as x approaches 0 of
sine of sine of x is 0, and

00:06:19.540 --> 00:06:24.030
the limit as x approaches
0 of x is also equal to 0.

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So we get our
indeterminate form.

00:06:26.180 --> 00:06:29.340
So let's see, at least, whether
this limit even exists.

00:06:29.340 --> 00:06:32.100
If we take the derivative of f
of x and we put that over the

00:06:32.100 --> 00:06:35.900
derivative of g of x, and take
the limit as x approaches 0

00:06:35.900 --> 00:06:38.400
in this case, that's our c.

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Let's see if this limit exists.

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So I'll do that in the blue.

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So let me write the derivatives
of the two functions.

00:06:48.160 --> 00:06:50.720
So f prime of x.

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If f of x is sine of x,
what's f prime of x?

00:06:53.590 --> 00:06:55.480
Well, it's just cosine of x.

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You've learned that many times.

00:06:57.390 --> 00:07:01.215
And if g of x is x,
what is g prime of x?

00:07:01.215 --> 00:07:02.150
That's super easy.

00:07:02.150 --> 00:07:06.280
The derivative of x is just 1.

00:07:06.280 --> 00:07:13.690
Let's try to take the limit as
x approaches 0 of f prime of x

00:07:13.690 --> 00:07:17.000
over g prime of x-- over
their derivatives.

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So that's going to be the
limit as x approaches 0

00:07:19.390 --> 00:07:25.620
of cosine of x over 1.

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I wrote that 1 a
little strange.

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And this is pretty
straightforward.

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What is this going to be?

00:07:30.990 --> 00:07:33.610
Well, as x approaches 0
of cosine of x, that's

00:07:33.610 --> 00:07:37.290
going to be equal to 1.

00:07:37.290 --> 00:07:39.320
And obviously, the limit as
x approaches 0 of 1, that's

00:07:39.320 --> 00:07:40.900
also going to be equal to 1.

00:07:40.900 --> 00:07:46.470
So in this situation we just
saw that the limit as x

00:07:46.470 --> 00:07:49.820
approaches-- our c
in this case is 0.

00:07:49.820 --> 00:07:54.590
As x approaches 0 of f
prime of x over g prime

00:07:54.590 --> 00:07:56.240
of x is equal to 1.

00:07:56.240 --> 00:07:59.060
This limit exists and it
equals 1, so we've met

00:07:59.060 --> 00:08:00.580
all of the conditions.

00:08:00.580 --> 00:08:02.500
This is the case
we're dealing with.

00:08:02.500 --> 00:08:06.700
Limit as x approaches 0 of
sine of x is equal to 0.

00:08:06.700 --> 00:08:10.650
Limit as z approaches 0
of x is also equal to 0.

00:08:10.650 --> 00:08:14.500
The limit of the derivative of
sine of x over the derivative

00:08:14.500 --> 00:08:17.490
of x, which is cosine of x over
1-- we found this

00:08:17.490 --> 00:08:21.000
to be equal to 1.

00:08:21.000 --> 00:08:24.610
All of these top conditions
are met, so then we know

00:08:24.610 --> 00:08:26.480
this must be the case.

00:08:26.480 --> 00:08:34.500
That the limit as x approaches
0 of sine of x over x

00:08:34.500 --> 00:08:36.690
must be equal to 1.

00:08:36.690 --> 00:08:42.510
It must be the same limit as
this value right here where we

00:08:42.510 --> 00:08:46.120
take the derivative of the
f of x and of the g of x.

00:08:46.120 --> 00:08:48.430
I'll do more examples in the
next few videos and I think

00:08:48.430 --> 00:08:50.760
it'll make it a lot
more concrete.


Mach numbers

WEBVTT
Kind: captions
Language: en

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You've probably heard that the
top speed of some type of

00:00:04.750 --> 00:00:09.060
aircraft is Mach 2.

00:00:09.060 --> 00:00:13.070
Or maybe you've watched some
type of science fiction movie,

00:00:13.070 --> 00:00:15.465
or some type of fighter movie,
where they say, hey we are

00:00:15.465 --> 00:00:21.050
going to travel at Mach 1.8.

00:00:21.050 --> 00:00:23.480
And you had a sense, because
they're using it in the same

00:00:23.480 --> 00:00:25.680
context that you would use
as speed, that this is

00:00:25.680 --> 00:00:27.660
some type of speed.

00:00:27.660 --> 00:00:31.350
That these numbers represent
some type of speed.

00:00:31.350 --> 00:00:33.750
And if you assume that,
you were right.

00:00:33.750 --> 00:00:38.340
And the speeds they represent
are actually a multiple, or

00:00:38.340 --> 00:00:40.340
depending on how you want
to view it, multiples or

00:00:40.340 --> 00:00:44.880
fractions of the speed of
sound in that medium.

00:00:44.880 --> 00:00:49.050
So if someone tells you that
the Mach number is 2, what

00:00:49.050 --> 00:00:56.570
that tells you is that they are
traveling two times the

00:00:56.570 --> 00:01:07.340
speed of sound, in that medium,
for those conditions.

00:01:07.340 --> 00:01:11.460
And I make those last two
qualifiers on it because the

00:01:11.460 --> 00:01:13.840
speed of sound is not
always the same.

00:01:13.840 --> 00:01:16.100
It has a different speed
depending on whether it's

00:01:16.100 --> 00:01:20.430
traveling in air, or water, or
even depending on what makes

00:01:20.430 --> 00:01:21.910
up the air.

00:01:21.910 --> 00:01:24.600
And even if it is only traveling
in a certain type of

00:01:24.600 --> 00:01:29.050
air, or a certain makeup, a
mix of gases, it'll also

00:01:29.050 --> 00:01:30.960
change depending on
the temperature.

00:01:30.960 --> 00:01:36.050
So if you're at sea level, at
roughly about 20 degrees

00:01:36.050 --> 00:01:40.270
Celsius, the speed of sound is
300, so sea level-- do a

00:01:40.270 --> 00:01:43.230
different color, maybe blue for
sea level-- so if you're

00:01:43.230 --> 00:01:48.960
at sea level, the speed
of sound is about

00:01:48.960 --> 00:01:52.070
340 meters per second.

00:01:52.070 --> 00:01:55.080
This is at 20 degrees Celsius,
so it's a nice comfortable

00:01:55.080 --> 00:02:03.820
temperature at sea level, which
is roughly about 760

00:02:03.820 --> 00:02:05.800
miles per hour.

00:02:05.800 --> 00:02:10.240
Now if the temperature were to
drop, so if temperature goes

00:02:10.240 --> 00:02:16.220
down, so does the
speed of sound.

00:02:21.170 --> 00:02:23.520
And likewise, if the temperature
goes up, the speed

00:02:23.520 --> 00:02:24.870
of sound goes up.

00:02:24.870 --> 00:02:28.580
So when someone tells you that
something is traveling at Mach

00:02:28.580 --> 00:02:31.320
2, they're saying its traveling
at two times the

00:02:31.320 --> 00:02:33.660
speed of sound for
that medium.

00:02:33.660 --> 00:02:35.250
Usually they're talking
about something

00:02:35.250 --> 00:02:36.380
traveling through air.

00:02:36.380 --> 00:02:39.300
And for the speed of sound at
the temperature that they're

00:02:39.300 --> 00:02:39.720
going through.

00:02:39.720 --> 00:02:42.250
So if they're at some super-high
altitude, and

00:02:42.250 --> 00:02:45.230
normally to travel at these
types of speeds, you have to

00:02:45.230 --> 00:02:49.300
be at a fairly high altitude
where the air is less dense.

00:02:49.300 --> 00:02:52.270
You're also traveling in an
environment where the

00:02:52.270 --> 00:02:57.240
temperature is significantly
lower than 20 degrees Celsius.

00:02:57.240 --> 00:03:00.100
So you might say, hey, does
that mean if someone's

00:03:00.100 --> 00:03:02.090
traveling at Mach 2, does that
mean that they're definitely

00:03:02.090 --> 00:03:05.560
traveling at 680 meters
per second, or at

00:03:05.560 --> 00:03:08.440
1520 miles per hour?

00:03:08.440 --> 00:03:10.160
And of course, whenever someone
gives an air speed,

00:03:10.160 --> 00:03:11.270
it's relative to the air.

00:03:11.270 --> 00:03:12.310
And the answer is no.

00:03:12.310 --> 00:03:14.180
They're probably going to be
traveling a little bit slower

00:03:14.180 --> 00:03:16.270
than that, because at the
altitude that they're

00:03:16.270 --> 00:03:18.580
traveling the speed
of sound is lower.

00:03:18.580 --> 00:03:21.980
Now there are ground speed
records where they are

00:03:21.980 --> 00:03:24.860
traveling in kind of the Mach
range where their velocities

00:03:24.860 --> 00:03:27.070
are high enough to say
it in terms of Mach.

00:03:27.070 --> 00:03:30.400
And for those types of
situations, they are traveling

00:03:30.400 --> 00:03:36.400
at multiples of 760 miles per
hour, 340 meters per second.

00:03:36.400 --> 00:03:39.280
Now you're probably saying, hey,
who is this person right

00:03:39.280 --> 00:03:41.780
here that I've cut and pasted
from Wikipedia?

00:03:41.780 --> 00:03:43.240
And this Ernst Mach.

00:03:43.240 --> 00:03:49.330
And this is who the Mach
number is named after.

00:03:49.330 --> 00:03:53.440
Ernst Mach, and he was this
Austrian physicist and

00:03:53.440 --> 00:03:54.214
philosopher.

00:03:54.214 --> 00:03:57.780
He did a ton of research
on shockwaves and

00:03:57.780 --> 00:03:59.060
soundwaves, and whatnot.

00:03:59.060 --> 00:04:03.360
So the Mach number is
named after him.


When the source and the wave move at the same velocity

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.380
In the last several videos, we
assumed that the velocity of

00:00:04.380 --> 00:00:08.430
the source, of the object that's
emitting the wave,

00:00:08.430 --> 00:00:12.730
we've assumed that that's less
than the velocity of the wave.

00:00:12.730 --> 00:00:14.420
And we saw what happens
with the Doppler

00:00:14.420 --> 00:00:15.670
effect and all of that.

00:00:15.670 --> 00:00:18.240
But what I want to do in this
video is not make this

00:00:18.240 --> 00:00:18.760
assumption.

00:00:18.760 --> 00:00:21.570
In particular, let's see what
happens, at least, at first

00:00:21.570 --> 00:00:24.320
for our formulas, and then get
a conceptual understanding.

00:00:24.320 --> 00:00:27.900
Let's see what happens when the
velocity of the source is

00:00:27.900 --> 00:00:32.040
equal to the velocity
of the wave.

00:00:32.040 --> 00:00:35.790
The first thing we might try to
do is just apply this new

00:00:35.790 --> 00:00:39.030
assumption into the formulas
that we had in the last video.

00:00:39.030 --> 00:00:41.210
And those formulas were here.

00:00:41.210 --> 00:00:44.650
These are the observed period
and frequency for an observer

00:00:44.650 --> 00:00:47.980
that's in the direction
of the object.

00:00:47.980 --> 00:00:50.100
And if we make this assumption,
that the velocity

00:00:50.100 --> 00:00:52.650
of the sound-- and the velocity
of the source-- we're

00:00:52.650 --> 00:00:55.110
not necessarily dealing with
soundwaves, although that

00:00:55.110 --> 00:00:56.700
might be an easy visualization
for you.

00:00:56.700 --> 00:00:59.210
That tends to be how
I visualize it.

00:00:59.210 --> 00:01:02.990
But what happens to these
formulas when the velocity of

00:01:02.990 --> 00:01:07.070
the source is equal to the
velocity of the wave?

00:01:07.070 --> 00:01:09.110
If these two quantities are
equal up here, you have

00:01:09.110 --> 00:01:11.300
something, subtract the
same thing from it.

00:01:11.300 --> 00:01:13.270
This numerator right
here becomes 0.

00:01:13.270 --> 00:01:15.880
So it'll turn this whole
thing into being 0.

00:01:15.880 --> 00:01:20.440
So the period, or the observed
period, will be 0, which means

00:01:20.440 --> 00:01:22.580
you don't have to wait
any time at all

00:01:22.580 --> 00:01:24.400
between successive crests.

00:01:24.400 --> 00:01:29.280
The entire waveform just gets
infinitely bunched together.

00:01:29.280 --> 00:01:32.290
So it's kind of like
one impulse.

00:01:32.290 --> 00:01:34.600
And if we look at the frequency,
we can either look

00:01:34.600 --> 00:01:37.000
directly at the formula, and
you'll see, you have something

00:01:37.000 --> 00:01:39.180
divided by 0 right here.

00:01:39.180 --> 00:01:41.275
So you can say this is 1 over 0,
or you could just say that

00:01:41.275 --> 00:01:44.300
the frequency is 1 over the
period, and you get this thing

00:01:44.300 --> 00:01:45.550
that's undefined.

00:01:49.080 --> 00:01:50.980
But if you want to think about,
what does the frequency

00:01:50.980 --> 00:01:55.120
approach as the velocity of
the source approaches the

00:01:55.120 --> 00:01:57.850
velocity of the wave, if this
thing is only a little bit

00:01:57.850 --> 00:02:00.340
less that thing, this is going
to be a very, very, very

00:02:00.340 --> 00:02:03.770
small, very, very small
positive number.

00:02:03.770 --> 00:02:06.300
So when you divide these
quantities by that very, very,

00:02:06.300 --> 00:02:08.300
very small positive
number, you're

00:02:08.300 --> 00:02:09.350
going to approach infinity.

00:02:09.350 --> 00:02:14.070
So the frequency is undefined at
the speed, at the velocity

00:02:14.070 --> 00:02:16.120
of the wave, but it's going
to approach infinity.

00:02:20.870 --> 00:02:28.100
It will approach infinity as
the source approaches the

00:02:28.100 --> 00:02:30.560
velocity of the wave. Not
necessarily a soundwave.

00:02:30.560 --> 00:02:33.150
I keep using soundwaves, because
that's how I tend to

00:02:33.150 --> 00:02:33.930
visualize things.

00:02:33.930 --> 00:02:36.450
And we'll talk in future videos
specifically about

00:02:36.450 --> 00:02:36.563
soundwaves.

00:02:36.563 --> 00:02:39.510
And we'll touch on it a little
bit in this video.

00:02:39.510 --> 00:02:40.830
So what is this telling us?

00:02:40.830 --> 00:02:42.440
Does this make any sense?

00:02:42.440 --> 00:02:46.640
And if you think about it, at
least to me, it starts to make

00:02:46.640 --> 00:02:49.230
sense just what you saw in the
last couple of videos.

00:02:49.230 --> 00:02:51.850
The last couple of videos,
when something was moving

00:02:51.850 --> 00:02:54.470
slower than the speed
of sound, you had,

00:02:54.470 --> 00:02:56.190
OK, I'm here now.

00:02:56.190 --> 00:03:01.800
And I'm about to release the
next crest. If I go one period

00:03:01.800 --> 00:03:04.170
ago, maybe I was right there.

00:03:04.170 --> 00:03:07.720
And the crest that I had
released at that time period

00:03:07.720 --> 00:03:10.830
maybe has traveled this
far, just like that.

00:03:10.830 --> 00:03:12.570
If you go a period before
that, I would

00:03:12.570 --> 00:03:13.680
have been over there.

00:03:13.680 --> 00:03:17.400
And crest that I released then
would have traveled that far.

00:03:17.400 --> 00:03:19.850
We saw this in the
last two videos.

00:03:19.850 --> 00:03:21.400
And if you go the period
before that, I

00:03:21.400 --> 00:03:22.330
would have been there.

00:03:22.330 --> 00:03:24.120
And the crest that I would
have released would

00:03:24.120 --> 00:03:26.580
have been that far.

00:03:26.580 --> 00:03:28.790
This was the whole reason why
the Doppler effect happens.

00:03:28.790 --> 00:03:32.400
Because the observer sitting
right here-- let me do this in

00:03:32.400 --> 00:03:35.890
a separate color-- the observer
sitting right here is

00:03:35.890 --> 00:03:39.310
going to experience these crests
more frequently than an

00:03:39.310 --> 00:03:40.380
observer sitting out here.

00:03:40.380 --> 00:03:43.520
Because the wavelength gets
compressed, because every time

00:03:43.520 --> 00:03:46.980
this guy releases a new
crest, or a new cycle,

00:03:46.980 --> 00:03:48.510
he has moved forward.

00:03:48.510 --> 00:03:50.570
He's moved forward in the
direction of this

00:03:50.570 --> 00:03:52.160
motion right here.

00:03:52.160 --> 00:03:55.120
So let's think about what
happens when he is exactly

00:03:55.120 --> 00:03:59.030
moving at the speed of the wave.
So let's say that the

00:03:59.030 --> 00:04:00.930
source is here now.

00:04:00.930 --> 00:04:02.200
This is right where he is.

00:04:02.200 --> 00:04:06.530
And he's right about to release
a new crest. So where

00:04:06.530 --> 00:04:07.760
was he at one period ago?

00:04:07.760 --> 00:04:11.490
So let's say he was here
one period ago.

00:04:15.550 --> 00:04:18.990
So one period ago, if he's right
going to release crest

00:04:18.990 --> 00:04:22.390
or cycle right now,
one period ago, he

00:04:22.390 --> 00:04:24.130
released another cycle.

00:04:24.130 --> 00:04:25.540
And where has that cycle gone?

00:04:25.540 --> 00:04:28.560
Well, we're assuming that the
wave is traveling at the same

00:04:28.560 --> 00:04:30.540
velocity as this guy.

00:04:30.540 --> 00:04:32.020
But it's going radially
outward.

00:04:32.020 --> 00:04:35.260
So whatever he released then,
it will have traveled at the

00:04:35.260 --> 00:04:37.260
same velocity as himself.

00:04:37.260 --> 00:04:41.210
So it will have gotten
this far.

00:04:41.210 --> 00:04:44.280
He released it one period ago,
and that's where he was one

00:04:44.280 --> 00:04:45.210
period ago.

00:04:45.210 --> 00:04:48.210
Over the course of the next
period, he traveled there and

00:04:48.210 --> 00:04:50.990
so did the wave. The wave
also traveled there.

00:04:50.990 --> 00:04:54.630
Now, where was this character
two periods ago?

00:04:54.630 --> 00:04:56.910
When I talk about the period,
I'm talking about the actual

00:04:56.910 --> 00:05:01.490
period of the wave. Every
period, or how long does it

00:05:01.490 --> 00:05:03.750
take between similar points
in the cycle?

00:05:03.750 --> 00:05:06.210
And I like to think of them as
the crests in the cycle.

00:05:06.210 --> 00:05:08.270
So two periods ago,
he was here.

00:05:08.270 --> 00:05:11.700
And he released a-- you can
imagine, a pulse, or crest.

00:05:11.700 --> 00:05:13.190
And where will that be now?

00:05:13.190 --> 00:05:14.940
Well that will have traveled
as far as he did.

00:05:14.940 --> 00:05:17.300
He traveled that far,
and so will the

00:05:17.300 --> 00:05:19.400
pulse that he released.

00:05:19.400 --> 00:05:21.800
It will have traveled-- actually
I have to make it a

00:05:21.800 --> 00:05:23.360
little bit more symmetric.

00:05:23.360 --> 00:05:24.890
It will have traveled
that far.

00:05:31.310 --> 00:05:33.750
And if you go three periods ago,
I think you get the idea.

00:05:33.750 --> 00:05:35.750
If you go three periods
ago, he was here.

00:05:35.750 --> 00:05:41.650
And he released a pulse then,
or crest, or a cycle of the

00:05:41.650 --> 00:05:42.870
wave. And where will
that be now?

00:05:42.870 --> 00:05:44.820
Well, it will have traveled
as fast as he's gotten.

00:05:44.820 --> 00:05:46.620
So it will have gotten
this far.

00:05:51.680 --> 00:05:54.830
Of course, it's traveling that
velocity in every direction,

00:05:54.830 --> 00:05:56.340
radially outward.

00:05:56.340 --> 00:05:59.980
Now think about the situation
for the observer.

00:05:59.980 --> 00:06:02.770
Think, in particular, about
the observer who's sitting

00:06:02.770 --> 00:06:03.140
right here.

00:06:03.140 --> 00:06:05.330
Let's say he's just out of the
way so that this thing doesn't

00:06:05.330 --> 00:06:07.600
run into him and kill him,
and what whatever else.

00:06:07.600 --> 00:06:09.330
But he's just out of the way,
just enough to experience the

00:06:09.330 --> 00:06:12.420
sound, but not directly collide
with this object

00:06:12.420 --> 00:06:15.340
that's emitting the-- well, I
shouldn't say sound-- emitting

00:06:15.340 --> 00:06:17.410
the wave. I want to be
general right here.

00:06:17.410 --> 00:06:20.180
We're not assuming that
this is a soundwave.

00:06:20.180 --> 00:06:21.870
What's he going to experience?

00:06:21.870 --> 00:06:25.020
Well, he's not going to-- well,
if we assume this is a

00:06:25.020 --> 00:06:28.050
soundwave, he's not going to
hear anything until the thing

00:06:28.050 --> 00:06:29.200
passes right there.

00:06:29.200 --> 00:06:31.770
And right when the thing passes,
it has all of the

00:06:31.770 --> 00:06:34.620
sound that it generated coming
with it at exactly

00:06:34.620 --> 00:06:36.140
that moment in time.

00:06:36.140 --> 00:06:40.150
Instead of hearing things
periodically, all of the wave

00:06:40.150 --> 00:06:43.630
fronts smack this
guy all at once.

00:06:43.630 --> 00:06:46.910
And perceptually, instead of
hearing a pitch, because

00:06:46.910 --> 00:06:49.570
you're hearing something
periodic, you're just going to

00:06:49.570 --> 00:06:52.540
hear a big thump, because all
of that sound energy is

00:06:52.540 --> 00:06:54.020
getting to you at
the same time.

00:06:54.020 --> 00:06:58.150
You're just going
to hear a thump.

00:06:58.150 --> 00:06:59.900
Because it's no longer
really a frequency.

00:06:59.900 --> 00:07:02.010
All of the energy is coming
to you at the same time.

00:07:02.010 --> 00:07:04.400
And when you are talking about
sound, specifically, and

00:07:04.400 --> 00:07:06.800
especially, when you are
trans-sonic, which means

00:07:06.800 --> 00:07:09.410
you're around the speed of
sound, or parts of you are

00:07:09.410 --> 00:07:11.570
above or below the speed of
sound, and you move into

00:07:11.570 --> 00:07:14.410
supersonic speeds, that's
what people relate

00:07:14.410 --> 00:07:15.850
to the sonic boom.

00:07:15.850 --> 00:07:17.880
And we're going to talk a little
bit more about that in

00:07:17.880 --> 00:07:19.690
the next video and mach numbers,
all that, because I

00:07:19.690 --> 00:07:22.110
just find all of that
fascinating.

00:07:22.110 --> 00:07:24.510
But I think this is intuitive.

00:07:24.510 --> 00:07:28.110
Because when you look at this,
everything is just reaching

00:07:28.110 --> 00:07:29.660
you at exactly the same time.

00:07:29.660 --> 00:07:31.520
And this was the case of
soundwaves, but it would be

00:07:31.520 --> 00:07:33.730
true of any type of waves.


Doppler effect formula when source is moving away

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.110
In the last video, we figured
out the formulas for the

00:00:04.110 --> 00:00:08.660
observed period and frequency
for an observer sitting in the

00:00:08.660 --> 00:00:10.260
path of the source.

00:00:10.260 --> 00:00:12.470
So the source is moving
towards the observer.

00:00:12.470 --> 00:00:15.910
So this is the example where
the train is moving towards

00:00:15.910 --> 00:00:19.170
you, and you perceive the
train's horn as having a

00:00:19.170 --> 00:00:21.170
higher pitch or a higher
frequency.

00:00:21.170 --> 00:00:24.490
And we were able to do that by
doing a thought experiment.

00:00:24.490 --> 00:00:27.620
Saying, OK, my object
starts here.

00:00:27.620 --> 00:00:31.000
After one period-- a period is
just a measure of time, but

00:00:31.000 --> 00:00:34.900
it's the measure of time over
which the source emits a

00:00:34.900 --> 00:00:38.320
cycle, so it emits a
cycle every period.

00:00:38.320 --> 00:00:41.130
But after one period, we said,
OK, where is that first wave

00:00:41.130 --> 00:00:44.300
front, or that first pulse,
or that first crest?

00:00:44.300 --> 00:00:46.090
And where is the source?

00:00:46.090 --> 00:00:49.950
Because exactly one period has
passed by, and the source will

00:00:49.950 --> 00:00:53.550
be ready to emit another
crest or another cycle.

00:00:53.550 --> 00:00:57.180
So the distance between where
the source is and that front

00:00:57.180 --> 00:01:00.200
of the crest, or that first
crest, that is going to be the

00:01:00.200 --> 00:01:00.800
wavelength.

00:01:00.800 --> 00:01:03.340
Because this next thing that
emits is going to be traveling

00:01:03.340 --> 00:01:05.720
at the exact same velocity, and
it's going to be separated

00:01:05.720 --> 00:01:08.530
by that distance, which we
saw is this expression.

00:01:08.530 --> 00:01:10.770
We said how long will it take
it to travel that distance?

00:01:10.770 --> 00:01:13.440
Well, it's traveling at
a speed of v sub w.

00:01:13.440 --> 00:01:16.500
That'll tell you what the
observed period would be for

00:01:16.500 --> 00:01:17.580
this dude over here.

00:01:17.580 --> 00:01:19.470
We calculated it right here,
and then the observed

00:01:19.470 --> 00:01:21.530
frequency is just the
inverse of that.

00:01:21.530 --> 00:01:23.940
Now let's think about the
situation where the observer

00:01:23.940 --> 00:01:27.360
is over here.

00:01:27.360 --> 00:01:31.310
So these equations, or these
formulas that we came up with

00:01:31.310 --> 00:01:36.020
right here, this is observer--
or let me say source traveling

00:01:36.020 --> 00:01:38.133
in direction of observer.

00:01:48.900 --> 00:01:50.830
Now let's think about the
opposite case, where the

00:01:50.830 --> 00:01:54.120
source is traveling away from
the observer, and in this

00:01:54.120 --> 00:01:55.830
case, the observer is
that guy over there.

00:01:55.830 --> 00:01:56.890
Maybe I'll do it in
a different color.

00:01:56.890 --> 00:01:58.400
He'll be blue.

00:01:58.400 --> 00:02:00.510
So this is the observer.

00:02:00.510 --> 00:02:03.350
So when we started off, our
source was right here.

00:02:03.350 --> 00:02:08.789
After exactly one period from
the source's point of view,

00:02:08.789 --> 00:02:11.520
that first crest emitted
has traveled radially

00:02:11.520 --> 00:02:12.800
outward that far.

00:02:12.800 --> 00:02:14.270
This is the distance.

00:02:14.270 --> 00:02:18.020
The velocity of the wave times
the amount of time that

00:02:18.020 --> 00:02:20.310
passed-- velocity times time
is going to give you

00:02:20.310 --> 00:02:24.370
distance-- and where the source
of the wave will have

00:02:24.370 --> 00:02:26.650
traveled to the right exactly
this distance.

00:02:26.650 --> 00:02:30.500
It's velocity times the amount
of time that's gone by.

00:02:30.500 --> 00:02:33.620
Now, in the last video, we said,
OK, that wave is just

00:02:33.620 --> 00:02:34.350
passing this guy.

00:02:34.350 --> 00:02:37.080
How long will it take for that
pulse that's being emitted

00:02:37.080 --> 00:02:39.030
right then to also reach him?

00:02:39.030 --> 00:02:41.900
And then that tells us the
period between two pulses or

00:02:41.900 --> 00:02:43.430
between two crests.

00:02:43.430 --> 00:02:45.990
Now let's think about that exact
same situation here.

00:02:45.990 --> 00:02:50.110
That first crest is just
passing this guy.

00:02:50.110 --> 00:02:53.780
And a period, or the t sub s,
which is a period of the

00:02:53.780 --> 00:02:55.430
emitted wave has
just passed by.

00:02:55.430 --> 00:02:58.150
So this guy is just about to
emit another wave, So?

00:02:58.150 --> 00:03:01.270
That other wave is going
to be right here.

00:03:01.270 --> 00:03:04.940
So what is the distance between
the crest, or the

00:03:04.940 --> 00:03:07.740
cycle, or however you want to
think about it, the pulse that

00:03:07.740 --> 00:03:10.610
is passing him by right now,
what is the distance between

00:03:10.610 --> 00:03:15.790
that and the pulse, the front
edge of that pulse that is

00:03:15.790 --> 00:03:18.460
being emitted right
at that moment?

00:03:18.460 --> 00:03:19.920
Right at that moment.

00:03:19.920 --> 00:03:23.770
Well, it's going to be this
radius, which is this value.

00:03:23.770 --> 00:03:27.960
It'll be v sub w times
the period.

00:03:27.960 --> 00:03:32.170
That is that distance plus the
amount of distance that our

00:03:32.170 --> 00:03:34.740
source has traveled away
from this guy.

00:03:34.740 --> 00:03:39.880
So plus v sub s times
the period.

00:03:39.880 --> 00:03:43.760
So that's what this distance is,
and this is how far apart

00:03:43.760 --> 00:03:46.940
this wave pulse is going to be
from that one or this crest is

00:03:46.940 --> 00:03:48.350
going to be from that one.

00:03:48.350 --> 00:03:52.030
So if he's seeing this first
crest right now, right at this

00:03:52.030 --> 00:03:55.100
moment, how long will it take
him to see the crest that's

00:03:55.100 --> 00:03:58.790
being emitted right now, that's
this far away from him?

00:03:58.790 --> 00:04:00.350
Well, it's that far
away from him.

00:04:00.350 --> 00:04:02.590
So let me write this down.

00:04:02.590 --> 00:04:05.525
So the amount of time it takes
for him to see the next crest

00:04:05.525 --> 00:04:09.150
or the same point in the next
cycle, that's the period.

00:04:09.150 --> 00:04:11.590
That's the observed period.

00:04:11.590 --> 00:04:14.130
That's going to be equal
to this distance.

00:04:14.130 --> 00:04:16.660
The velocity of the wave times
the period from the

00:04:16.660 --> 00:04:20.110
perspective of the source plus
the velocity of the source,

00:04:20.110 --> 00:04:22.220
because the source has gotten
that much further away from

00:04:22.220 --> 00:04:25.520
him, velocity of the source
times the period of the

00:04:25.520 --> 00:04:29.220
source, so that's how far
the next crest is.

00:04:29.220 --> 00:04:31.350
And then you divide it by the
speed of the wave, by the

00:04:31.350 --> 00:04:34.970
speed of each of the crests,
which is just the velocity of

00:04:34.970 --> 00:04:38.000
the wave. And we can just factor
out the t sub s's here

00:04:38.000 --> 00:04:43.090
and say this is t sub s times
v sub w, the velocity of the

00:04:43.090 --> 00:04:47.000
wave plus the velocity of the
source divided by the velocity

00:04:47.000 --> 00:04:52.090
the wave. So this will be a
larger observed period than if

00:04:52.090 --> 00:04:55.300
this guy was stationary and
especially if the observer was

00:04:55.300 --> 00:04:56.070
in the path of the guy.

00:04:56.070 --> 00:05:00.160
That make sense, because every
time this guy issues a cycle,

00:05:00.160 --> 00:05:02.340
he is moving a little
bit further away.

00:05:02.340 --> 00:05:05.970
So every crest the same point
in the cycle is going to be

00:05:05.970 --> 00:05:07.810
further and further apart, so
you're going to have longer

00:05:07.810 --> 00:05:10.080
wavelengths, longer periods.

00:05:10.080 --> 00:05:12.540
And if you want the observed
frequency for that guy over

00:05:12.540 --> 00:05:14.500
there-- I'll do it in
the same color.

00:05:14.500 --> 00:05:17.360
The observed frequency for
a guy where the source is

00:05:17.360 --> 00:05:19.700
traveling away from him is
just the inverse of that.

00:05:19.700 --> 00:05:23.050
So one over the period, one over
the period, same argument

00:05:23.050 --> 00:05:26.050
we did there, one over the
period from the point of view

00:05:26.050 --> 00:05:28.650
of the source is the frequency
of the source.

00:05:28.650 --> 00:05:31.810
Let me Color code it.

00:05:31.810 --> 00:05:35.240
So 1 over t sub s is equal to
the frequency of the source.

00:05:35.240 --> 00:05:36.440
This is the inverse of that.

00:05:36.440 --> 00:05:38.570
So I'm just taking one
over everything here.

00:05:38.570 --> 00:05:41.950
So 1 over t sub s is the
frequency of the source, and

00:05:41.950 --> 00:05:44.595
then we take the inverse of this
over here: the velocity

00:05:44.595 --> 00:05:48.110
of the wave divided by the
velocity of the wave plus the

00:05:48.110 --> 00:05:49.910
velocity of the source.

00:05:49.910 --> 00:05:51.710
So we're done, at least
for the simple cases.

00:05:51.710 --> 00:05:53.700
Obviously, it becomes a little
more interesting when someone

00:05:53.700 --> 00:05:57.930
isn't exactly in the direction
of the source or exactly being

00:05:57.930 --> 00:06:02.210
moved away from, but these are
kind of the two extreme cases.

00:06:02.210 --> 00:06:07.590
So this is the situation when
it is moving away from you.

00:06:07.590 --> 00:06:09.990
Now, just to check our math and
maybe make it a little bit

00:06:09.990 --> 00:06:13.370
concrete in relation to the
video we did where we

00:06:13.370 --> 00:06:16.220
introduced to the idea of the
Doppler effect, let's actually

00:06:16.220 --> 00:06:17.310
apply those numbers.

00:06:17.310 --> 00:06:21.550
So on that video, two videos
ago, we had a situation where

00:06:21.550 --> 00:06:27.520
the velocity of our source was
5 meters per second to the

00:06:27.520 --> 00:06:35.310
right, and the velocity of the
wave was 10 meters per second

00:06:35.310 --> 00:06:40.165
radially outward, and the period
of our wave-- I'll call

00:06:40.165 --> 00:06:45.580
it the period of our-- let me do
it in another color-- from

00:06:45.580 --> 00:06:51.080
the point of view of the source
was 1 second per cycle,

00:06:51.080 --> 00:06:53.580
and the frequency was just
the inverse of that.

00:06:53.580 --> 00:06:56.700
So 1 cycle per second,
or 1 Hertz, which

00:06:56.700 --> 00:06:58.550
is a cycle per second.

00:06:58.550 --> 00:07:00.450
So using those numbers, let's
see if we get to the exact

00:07:00.450 --> 00:07:02.930
same answer we got in that first
video where we first

00:07:02.930 --> 00:07:04.330
learned about the
Doppler effect.

00:07:04.330 --> 00:07:06.910
So let's look at the frequency
from the point of view of this

00:07:06.910 --> 00:07:08.770
gentleman right here.

00:07:08.770 --> 00:07:14.680
So the frequency of the source
is going to be 1 cycle per

00:07:14.680 --> 00:07:16.560
second, 1 Hertz.

00:07:16.560 --> 00:07:20.940
The velocity of the wave is
10 meters per second.

00:07:20.940 --> 00:07:21.430
Let me write this.

00:07:21.430 --> 00:07:24.820
1 cycle per second.

00:07:24.820 --> 00:07:27.660
The velocity of the wave is
10 meters per second.

00:07:27.660 --> 00:07:32.000
The velocity of the wave is 10
meters per second minus the

00:07:32.000 --> 00:07:35.150
velocity of the source is
5 meters per second.

00:07:35.150 --> 00:07:36.460
So what's this going
to be equal to?

00:07:36.460 --> 00:07:42.060
The observed frequency for this
guy right there, is going

00:07:42.060 --> 00:07:51.210
to be 1 cycle per second times
10 over 10 minus 5, and the

00:07:51.210 --> 00:07:52.620
meters per second cancel out.

00:07:52.620 --> 00:07:54.370
Meters per second in the
numerator, meters per second

00:07:54.370 --> 00:07:54.540
in the denominator.

00:07:54.540 --> 00:07:58.560
So 10 divided by 10 minus
5, or 10 divided by 5,

00:07:58.560 --> 00:08:00.270
is going to be 2.

00:08:00.270 --> 00:08:02.470
So it's going to be 2
cycles per second.

00:08:05.860 --> 00:08:09.450
And if you want the observed
period for this guy, its going

00:08:09.450 --> 00:08:10.650
to be the inverse of that,
or it's going to be

00:08:10.650 --> 00:08:12.540
1/2 second per cycle.

00:08:12.540 --> 00:08:15.470
And this is exactly what we got
in the previous video, or

00:08:15.470 --> 00:08:17.000
actually, it was
two videos ago.

00:08:17.000 --> 00:08:19.870
Now what about the guy who this
guy's running away from?

00:08:19.870 --> 00:08:22.480
Well, we'll do the
exact same thing.

00:08:22.480 --> 00:08:25.610
You have 1 cycle per
second, or 1 Hertz.

00:08:25.610 --> 00:08:27.890
That's the emitted frequency
from the point of view of the

00:08:27.890 --> 00:08:33.049
source times the velocity of
the wave divided by the

00:08:33.049 --> 00:08:37.150
velocity of the wave plus the
velocity of the source,

00:08:37.150 --> 00:08:38.850
because it's moving
away from him.

00:08:38.850 --> 00:08:40.659
So it's 10 over 10 plus 5.

00:08:40.659 --> 00:08:42.340
That's 10 over 15.

00:08:42.340 --> 00:08:45.920
That is 2/3, so this
is equal to 2/3.

00:08:45.920 --> 00:08:47.440
The units over here
all cancel out.

00:08:47.440 --> 00:08:48.700
This was cycles per second.

00:08:48.700 --> 00:08:54.820
So 2/3 cycle per second, which
confirms the numbers we got in

00:08:54.820 --> 00:08:56.515
that first video, so that should
make us feel good and

00:08:56.515 --> 00:08:58.230
it also makes a lot of sense.

00:08:58.230 --> 00:09:02.840
This guy is going to see the
wave crest more frequently.

00:09:02.840 --> 00:09:04.700
He's going to observe
a higher frequency.

00:09:04.700 --> 00:09:06.370
If this is a sound,
a higher pitch.

00:09:06.370 --> 00:09:10.300
This guy, since each crest or
the cycles are getting spread

00:09:10.300 --> 00:09:14.050
out, he's going to see them less
frequently, and if this

00:09:14.050 --> 00:09:17.450
is sound, he's going to
observe a lower pitch.


Doppler effect formula for observed frequency

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:04.170
I've got this source of a wave
right here that's moving to

00:00:04.170 --> 00:00:06.150
the right at some velocity.

00:00:06.150 --> 00:00:14.510
So let's just say that the
velocity of the source-- let's

00:00:14.510 --> 00:00:19.460
call it v sub s to the right--
so we're really going to do

00:00:19.460 --> 00:00:21.180
what we do in the last video,
but we're going to do it in

00:00:21.180 --> 00:00:23.630
more abstract terms so we can
come up with a generalized

00:00:23.630 --> 00:00:26.160
formula for the observed
frequency.

00:00:26.160 --> 00:00:30.190
So that's how fast he's moving
to the right, and he's

00:00:30.190 --> 00:00:35.400
emitting a wave. Let's say the
wave that he's emitting-- so

00:00:35.400 --> 00:00:40.930
the velocity of wave--
let's call that v

00:00:40.930 --> 00:00:43.270
sub w radially outward.

00:00:43.270 --> 00:00:45.510
We've got to give a magnitude
and a direction.

00:00:45.510 --> 00:00:46.760
So radially outward.

00:00:50.070 --> 00:00:53.660
That's the velocity of the wave,
and that wave is going

00:00:53.660 --> 00:00:55.950
to have a period and a
frequency, but it's going to

00:00:55.950 --> 00:00:58.550
have a period and a frequency
associated from the point of

00:00:58.550 --> 00:01:00.630
view of the source.

00:01:00.630 --> 00:01:01.515
And we're going to
do everything.

00:01:01.515 --> 00:01:02.840
This is all classical
mechanics.

00:01:02.840 --> 00:01:04.959
We're not going to be talking
about relativistic speed, so

00:01:04.959 --> 00:01:07.580
we don't have to worry about all
of the strange things that

00:01:07.580 --> 00:01:10.580
happen as things approach
the speed of light.

00:01:10.580 --> 00:01:15.670
So let's just say it has
some period of-- let me

00:01:15.670 --> 00:01:17.130
write it this way.

00:01:17.130 --> 00:01:22.220
The source period, which is the
period of the wave from

00:01:22.220 --> 00:01:26.890
the perspective of the source,
so the source period, we'll

00:01:26.890 --> 00:01:31.850
call it t sub source, And the
source frequency, which would

00:01:31.850 --> 00:01:34.400
just be-- we've learned,
hopefully it's intuitive now--

00:01:34.400 --> 00:01:35.900
would be the inverse of this.

00:01:35.900 --> 00:01:42.790
So the source frequency would
be-- we'll call it f sub s.

00:01:42.790 --> 00:01:44.960
And these two things are the
inverse of each other.

00:01:44.960 --> 00:01:47.300
The inverse of the period
of a wave is its

00:01:47.300 --> 00:01:49.060
frequency, vice versa.

00:01:49.060 --> 00:01:52.010
So let's think about what's
going to happen.

00:01:52.010 --> 00:01:56.930
Let's say at time equal zero,
he emits that first crest,

00:01:56.930 --> 00:01:59.430
that first pulse, so he's
just emitted it.

00:01:59.430 --> 00:02:01.540
You can't even see it because
it just got emitted.

00:02:01.540 --> 00:02:05.020
And now let's fast forward
t seconds.

00:02:05.020 --> 00:02:07.695
Let's say that this is in
seconds, so every t seconds,

00:02:07.695 --> 00:02:09.410
it emits a new pulse.

00:02:09.410 --> 00:02:11.970
First of all, where is
that first pulse

00:02:11.970 --> 00:02:14.800
after t sub s seconds?

00:02:14.800 --> 00:02:18.170
Well, you multiply the velocity
of that first pulse

00:02:18.170 --> 00:02:19.650
times the time.

00:02:19.650 --> 00:02:22.780
Velocity times time is going
to give you a distance.

00:02:22.780 --> 00:02:24.200
If you don't believe me, I'll
show you an example.

00:02:24.200 --> 00:02:27.980
If I tell you the velocity is 5
meters per second, and let's

00:02:27.980 --> 00:02:30.850
say that this period is 2
seconds, that's going to give

00:02:30.850 --> 00:02:32.340
you 10 meters.

00:02:32.340 --> 00:02:34.580
The seconds cancel out.

00:02:34.580 --> 00:02:38.230
So to figure out how far that
wave will have gone after t

00:02:38.230 --> 00:02:42.090
sub s seconds, you just multiply
t sub s times the

00:02:42.090 --> 00:02:43.830
velocity of the wave.

00:02:43.830 --> 00:02:46.160
And let's say it's
gotten over here.

00:02:46.160 --> 00:02:47.960
It's radially outward.

00:02:47.960 --> 00:02:50.080
So, I'll draw it radially
outward.

00:02:50.080 --> 00:02:53.690
That's my best attempt
at a circle.

00:02:53.690 --> 00:03:00.080
And this distance right here,
this radius right there, that

00:03:00.080 --> 00:03:03.140
is equal to velocity
times time.

00:03:03.140 --> 00:03:08.780
The velocity of that first
pulse, v sub w, that's

00:03:08.780 --> 00:03:09.510
actually the speed.

00:03:09.510 --> 00:03:11.630
I'm saying it's v sub
w radially outward.

00:03:11.630 --> 00:03:12.800
This isn't a vector quantity.

00:03:12.800 --> 00:03:14.750
This is just a number
you can imagine.

00:03:14.750 --> 00:03:19.450
v sub w times the period,
times t of s.

00:03:22.100 --> 00:03:23.780
I know it's abstract, but just
think, this is just the

00:03:23.780 --> 00:03:25.300
distance times the time.

00:03:25.300 --> 00:03:29.330
If this was moving at 10 meters
per second and if the

00:03:29.330 --> 00:03:31.190
period is 2 seconds,
this is how far.

00:03:31.190 --> 00:03:34.520
It will have gone 10 meters
after 2 seconds.

00:03:34.520 --> 00:03:36.635
Now, this thing we said
at the beginning of

00:03:36.635 --> 00:03:38.100
the video is moving.

00:03:38.100 --> 00:03:40.410
So although this is radially
outward from the point at

00:03:40.410 --> 00:03:43.340
which it was emitted, this thing
isn't standing still.

00:03:43.340 --> 00:03:44.720
We saw this in the last video.

00:03:44.720 --> 00:03:46.690
This thing has also moved.

00:03:46.690 --> 00:03:47.540
How far?

00:03:47.540 --> 00:03:48.700
Well, we do the same thing.

00:03:48.700 --> 00:03:52.330
We multiply its velocity times
the same number of time.

00:03:52.330 --> 00:03:55.770
Remember, we're saying what does
this look like after t

00:03:55.770 --> 00:03:59.230
sub s seconds, or some period
of time t sub s.

00:03:59.230 --> 00:04:01.130
Well, this thing is moving
to the right.

00:04:01.130 --> 00:04:02.660
Let's say it's here.

00:04:02.660 --> 00:04:05.570
Let's say it's moved
right over here.

00:04:05.570 --> 00:04:08.470
In this video, we're assuming
that the velocity of our

00:04:08.470 --> 00:04:12.440
source is strictly less than the
velocity of the wave. Some

00:04:12.440 --> 00:04:14.750
pretty interesting things happen
right when they're

00:04:14.750 --> 00:04:16.890
equal, and, obviously, when
it goes the other way.

00:04:16.890 --> 00:04:18.860
But we're going to assume that
it's strictly less than.

00:04:18.860 --> 00:04:23.150
The source is traveling slower
than the actual wave.

00:04:23.150 --> 00:04:24.290
But what is this distance?

00:04:24.290 --> 00:04:25.980
Remember, we're talking
about-- let me do it

00:04:25.980 --> 00:04:27.550
in orange as well.

00:04:27.550 --> 00:04:31.650
This orange reality is what's
happened after t sub s

00:04:31.650 --> 00:04:33.160
seconds, you can say.

00:04:33.160 --> 00:04:35.360
So this distance right here.

00:04:35.360 --> 00:04:38.390
That distance right there--
I'll do it in a different

00:04:38.390 --> 00:04:41.540
color-- is going to be the
velocity of the source.

00:04:41.540 --> 00:04:46.350
It's going to be v sub
s times the amount of

00:04:46.350 --> 00:04:47.190
time that's gone by.

00:04:47.190 --> 00:04:49.340
And I said at the beginning,
that amount of time is the

00:04:49.340 --> 00:04:51.480
period of the wave. That's
the time in question.

00:04:51.480 --> 00:04:54.340
So period of the wave t sub s.

00:04:54.340 --> 00:04:57.950
So after one period of the wave,
if that's 5 seconds,

00:04:57.950 --> 00:05:00.910
then we'll say, after 5 seconds,
the source has moved

00:05:00.910 --> 00:05:07.800
this far, v sub s times t sub s,
and that first crest of our

00:05:07.800 --> 00:05:12.300
wave has moved that far,
V sub w times t sub s.

00:05:12.300 --> 00:05:14.300
Now, the time that we're talking
about, that's the

00:05:14.300 --> 00:05:16.270
period of the wave
being emitted.

00:05:16.270 --> 00:05:19.810
So exactly after that amount of
time, this guy is ready to

00:05:19.810 --> 00:05:21.960
emit the next crest.
He has gone

00:05:21.960 --> 00:05:23.500
through exactly one cycle.

00:05:23.500 --> 00:05:27.390
So he is going to emit
something right now.

00:05:27.390 --> 00:05:30.680
So it's just getting emitted
right at that point.

00:05:30.680 --> 00:05:33.760
So what is the distance between
the crest that he

00:05:33.760 --> 00:05:37.990
emitted t sub s seconds ago or
hours ago or microseconds ago,

00:05:37.990 --> 00:05:38.730
we don't know.

00:05:38.730 --> 00:05:41.650
What's the distance between this
crest and the one that

00:05:41.650 --> 00:05:43.210
he's just emitting?

00:05:43.210 --> 00:05:45.600
Well, they're going to move at
the same velocity, but this

00:05:45.600 --> 00:05:48.810
guy is already out here, while
this guy is starting off from

00:05:48.810 --> 00:05:50.260
the source's position.

00:05:50.260 --> 00:05:52.530
So the difference in their
distance, at least when you

00:05:52.530 --> 00:05:54.630
look at it this way, is the
distance between the source

00:05:54.630 --> 00:05:56.670
here and this crest.

00:05:56.670 --> 00:05:59.740
So what is this distance
right here?

00:05:59.740 --> 00:06:02.790
What is that distance
right there?

00:06:02.790 --> 00:06:07.270
Well, this whole radial
distance, we already said,

00:06:07.270 --> 00:06:12.090
this whole radial distance is
v sub w, the velocity of the

00:06:12.090 --> 00:06:16.440
wave, times the period of the
wave from the perspective of

00:06:16.440 --> 00:06:19.170
the source, and we're going to
subtract out how far the

00:06:19.170 --> 00:06:20.540
source itself has moved.

00:06:20.540 --> 00:06:23.350
The source has moved in the
direction, in this case, if

00:06:23.350 --> 00:06:24.910
we're looking at it from
this point of view,

00:06:24.910 --> 00:06:27.050
of that wave front.

00:06:27.050 --> 00:06:33.140
So it's going to be minus v
sub s, the velocity of the

00:06:33.140 --> 00:06:38.990
source, times the period of the
wave from the perspective

00:06:38.990 --> 00:06:40.340
of the source.

00:06:40.340 --> 00:06:41.780
So let me ask you a question.

00:06:41.780 --> 00:06:44.900
If you're sitting right here, if
you're the observer, you're

00:06:44.900 --> 00:06:49.730
this guy right here, you're
sitting right over there, and

00:06:49.730 --> 00:06:52.680
you've just had that first
crest, at that exact moment

00:06:52.680 --> 00:06:56.190
that first crest has passed you
by, how long are you going

00:06:56.190 --> 00:06:58.720
to have to wait for
the next crest?

00:06:58.720 --> 00:07:01.520
How long until this one that
this guy's emitting right now

00:07:01.520 --> 00:07:03.010
is going to pass you by?

00:07:03.010 --> 00:07:04.920
Well, it's going to have
to cover this distance.

00:07:04.920 --> 00:07:06.750
It's going to have to
cover that distance.

00:07:06.750 --> 00:07:07.650
Let me write this down.

00:07:07.650 --> 00:07:10.580
So the question I'm asking is
what is the period from the

00:07:10.580 --> 00:07:14.060
point of view of this observer
that's right in the direction

00:07:14.060 --> 00:07:15.320
of the movement of the source?

00:07:15.320 --> 00:07:19.660
So the period from the point
of view of the observer is

00:07:19.660 --> 00:07:22.180
going to be equal to the
distance that the next pulse

00:07:22.180 --> 00:07:25.240
has to travel, which is that
business up there.

00:07:25.240 --> 00:07:26.900
So let me copy and paste that.

00:07:30.260 --> 00:07:32.390
So it's going to be that.

00:07:32.390 --> 00:07:33.150
Let me get rid of that.

00:07:33.150 --> 00:07:35.690
It shouldn't look like an equal
sign, so I can delete

00:07:35.690 --> 00:07:36.540
that right over there.

00:07:36.540 --> 00:07:39.070
Or a negative sign.

00:07:39.070 --> 00:07:41.335
So it's going to be this
distance that the next pulse

00:07:41.335 --> 00:07:42.840
is going to travel, that one
that's going to be emitted

00:07:42.840 --> 00:07:46.260
right at that moment, divided by
the speed of that pulse, or

00:07:46.260 --> 00:07:49.310
the speed of the wave, or the
velocity the wave, and we know

00:07:49.310 --> 00:07:50.400
what that is.

00:07:50.400 --> 00:07:52.565
That is v sub w.

00:07:58.680 --> 00:08:01.600
Now this gives us the period
of the observation.

00:08:01.600 --> 00:08:01.770
Now.

00:08:01.770 --> 00:08:03.850
If we wanted the frequency-- and
we can manipulate this a

00:08:03.850 --> 00:08:04.190
little bit.

00:08:04.190 --> 00:08:05.980
Let's do that a little bit.

00:08:05.980 --> 00:08:08.940
So we can also write this.

00:08:08.940 --> 00:08:12.390
We could factor out the
period of the source.

00:08:12.390 --> 00:08:14.630
So t sub s we could
factor out.

00:08:14.630 --> 00:08:20.440
So it becomes t sub s times the
velocity of the wave minus

00:08:20.440 --> 00:08:26.690
the velocity of the source, all
of that over the velocity

00:08:26.690 --> 00:08:30.230
of the wave. And so just like
that, we've gotten our formula

00:08:30.230 --> 00:08:33.309
for the observed period for this
observer who's sitting

00:08:33.309 --> 00:08:38.270
right in the path of this moving
object as a function of

00:08:38.270 --> 00:08:42.270
the actual period of this wave
source, the wave's velocity

00:08:42.270 --> 00:08:44.930
and the velocity
of the source.

00:08:44.930 --> 00:08:46.670
Now, if we wanted the frequency,
we just take the

00:08:46.670 --> 00:08:48.140
inverse of this.

00:08:48.140 --> 00:08:49.340
So let's do that.

00:08:49.340 --> 00:08:52.640
So the frequency of the
observer-- so this is how many

00:08:52.640 --> 00:08:54.630
seconds it takes for him
to see the next cycle.

00:08:54.630 --> 00:08:57.110
If you want cycles per second,
you take the inverse.

00:08:57.110 --> 00:08:58.880
So the frequency of the observer
is just going to be

00:08:58.880 --> 00:08:59.600
the inverse of this.

00:08:59.600 --> 00:09:02.060
So if we take the inverse of
this whole expression, we're

00:09:02.060 --> 00:09:07.800
going to get 1 over t sub
s times v sub w over the

00:09:07.800 --> 00:09:10.840
velocity of the wave minus
the velocity the source.

00:09:10.840 --> 00:09:13.740
And of course, 1 over the period
from the point of view

00:09:13.740 --> 00:09:16.580
of the source, this
is the same thing.

00:09:16.580 --> 00:09:18.840
This right here is the
same thing as the

00:09:18.840 --> 00:09:20.260
frequency of the source.

00:09:20.260 --> 00:09:21.070
So there you have it.

00:09:21.070 --> 00:09:22.160
We have our two relations.

00:09:22.160 --> 00:09:26.010
At least if you are in the path,
if the velocity of the

00:09:26.010 --> 00:09:28.690
source is going in
your direction,

00:09:28.690 --> 00:09:30.310
then we have our formulas.

00:09:30.310 --> 00:09:34.190
And I'll rewrite them, just
because the observed period of

00:09:34.190 --> 00:09:37.600
the observer is going to be the
period from the point of

00:09:37.600 --> 00:09:42.060
view of the source times the
velocity of the wave minus the

00:09:42.060 --> 00:09:44.080
velocity of the source-- that's
the velocity of the

00:09:44.080 --> 00:09:47.650
source-- divided by the velocity
of the wave itself.

00:09:47.650 --> 00:09:51.640
The frequency, from the point
of view of this observer, is

00:09:51.640 --> 00:09:53.560
just the inverse of that,
which is the frequency.

00:09:53.560 --> 00:09:56.590
The inverse of the period is the
frequency from the point

00:09:56.590 --> 00:10:00.840
of view of the source times
the velocity of the wave

00:10:00.840 --> 00:10:03.730
divided by the velocity
of the wave minus the

00:10:03.730 --> 00:10:05.450
velocity of the source.

00:10:05.450 --> 00:10:07.640
In the next video, I'll do the
exact same exercise, but I'll

00:10:07.640 --> 00:10:10.400
just think about what happens to
the observer that's sitting

00:10:10.400 --> 00:10:11.960
right there.