The graphs show the change in
frequency with time of the Sound of the horn of a fast-moving car that is heard by a
person walking slowly by the side of the road as the car approaches and passes
them. Which of the graphs A, B, C, and D
most correctly shows the changes in frequency that the pedestrian hears?
Okay, so in this question, we’ve
got four graphs that show the change in frequency with time of the sound of the horn
of a fast-moving car which is heard by a person walking slowly by the side of the
road as the car approaches and then passes them. We need to work out which graph
most correctly shows the changes in frequency that the pedestrian hears. So let’s draw a quick little
Let’s say that we’re looking from
above, so this is our car. And let’s say that it’s moving
towards the right at a speed 𝑉, capital 𝑉. And it’s also honking its horn as
it goes. There we go, beep beep! And as it drives along, it goes
past a passenger here on the side of the road, walking also towards the right but
slowly. Let’s say at a speed lowercase
𝑣. Now truthfully, it doesn’t matter
what the values of capital 𝑉 and lowercase 𝑣 are.
The Doppler effect is going to
occur regardless because there’s some relative speed between them. And the relative speed between the
car and the pedestrian is capital 𝑉 minus lowercase 𝑣. Now what exactly does this
Well, let’s say that capital 𝑉 is
50 miles per hour. Now this is just a random number
that we’ve pulled from nowhere, but it’s a good example to try to explain what we’re
considering. Let’s also say that lowercase 𝑣 is
two miles per hour. So the pedestrian is walking along
towards the right at two miles per hour.
Well, the relative speed between
these two, the car and the pedestrian, is capital 𝑉 minus lowercase 𝑣, which
happens to be 50 miles per hour minus two miles per hour, which happens to be 48
miles per hour. Now it’s this relative speed that
affects the amount of Doppler shift that occurs to the sound emitted by the car,
which is the horn. It’s honking its horn.
In other words, the Doppler effect
which is what we’re considering here, the change in frequency of the sound, would be
exactly the same in this situation as if the car was travelling at 48 miles per hour
to the right and the pedestrian was stationary, because still the relative speed
between them is 48 miles per hour.
Relative to the pedestrian, in
other words, the car is moving to the right at 48 miles per hour. And once again, this speed affects
the extent to which the Doppler shift occurs on the sound emitted by the car. Anyway, that’s not super
It’s just an example to show how
the Doppler shift would occur when the car moves past the pedestrian. So we know that the car is moving
to the right at a much faster speed than the pedestrian is. And we’re trying to consider the
frequency change of the horn that the car is honking, as heard by the pedestrian of
And as we’ve already mentioned,
we’re looking at the Doppler effect. Now the Doppler effect tells us
that when the source of the sound, in this case, is moving towards the observer, the
frequency of the sound increases compared to when there’s no relative motion between
the source and the observer.
In other words, when the source and
the observer are moving either at the same speed or both are stationary, so there’s
no relative speed between them, the observer hears the horn at a certain
But when the source is moving
towards the observer, so in this case when the car is moving towards the pedestrian,
the observer hears the sound as a higher frequency. And vice versa as well, when the
source is moving away from the observer, the observer hears the sound as a lower
And of course the speed with which
the source is moving towards the observer or away from the observer affects how much
this frequency changes by.
The faster the source is moving,
the larger the shift in frequency. So we know that this car is
initially moving towards our observer here, our pedestrian. And it continues to move and passes
the pedestrian, at which point it starts moving away from the observer.
So in this region here, the
frequency of the sound emitted by the car is going to be high and then it decreases
as it starts moving away from the observer. In other words, what the observer
hears is [imitates a honk], because of course a high-frequency sound corresponds to
a high-pitched sound and a low-frequency sound corresponds to low-pitched sound.
So what we can do is to draw our
own graph of how the frequency changes as the car passes the observer. On the vertical axis, we can put
the frequency of the horn. And of course we’re going to be
drawing it arbitrarily.
So it doesn’t matter if it’s in
hertz. But it’s best to keep it in hertz
because that’s the standard unit of frequency. However, we don’t need to label any
numbers because we don’t have any information about the numbers.
We just need to draw a certain
shape. Now on the horizontal axis, we can
put the time, once again in seconds because that’s the standard unit. And the time is basically the time
elapsed since the car was at this point here. In other words, we can say that at
zero seconds the car is here, and then it moves to the right as time passes.
So to recap, the sound that the
observer would have heard as the car started approaching them, then pass them, and
continue to move away from them. What they would have heard is
[imitates a honk], so the sound starts as a high pitch or a high frequency and then
ends up as a low pitch or low frequency.
In other words, it’s something like
this [imitates a honk]. Now this might seem a little bit
intuitive to you if you’ve ever stood by the side of a road and heard cars passing
by you. In fact, however, this doesn’t
necessarily even have to apply to cars. It can apply to anything that’s
moving past you.
And the Doppler shift doesn’t even
have to apply to sound. It also applies to the light
emitted by a source as it moves past the observer. The frequency of the light emitted
changes depending on whether the source is moving towards the observer or away from
And the faster the source is
moving, the larger the shift in frequency. So anyway, coming back to our
question, we’ve drawn our own graph now. And it looks something like graph
B. So hooray! That’s our final answer. But just for fun, let’s imagine
what the other graphs would sound like.
Let’s look at graph A. Graph A starts at a low-frequency
then increases to a higher frequency and then comes back down again. That sounds something like [makes
sound]. And that’s not quite the sound a
car makes when it goes past you, is it?
Anyway, looking at graph A. It starts out with a low frequency
and then increases. So it goes [makes sound]. And that’s not what a car sounds
like either. And then the very last one, graph
D, starts off at a high frequency, then dips, and then increases again. In other words, it goes [makes
And once again, that’s not the
sound a car makes when it goes past you. And hence our final answer is that
graph B most correctly shows the changes in frequency that the pedestrian hears.