2023 A/L Physics Past Paper · Question 24 (English Medium)

Answer: (1) only (A) is true.

What the question asks

What we know:

The wheel turns clockwise. So $P$ is the top, $Q$ is on the right, $R$ is the bottom and $S$ is on the left.

The speaker $X$ is close to the wheel, on its right. In the figure it is level with $S$, the centre and $Q$.

The speaker keeps sending sound of the constant frequency $f_0$.

We want: which of the statements (A), (B) and (C) about the pitch heard by the man are true.

What each letter means:

$f_0$ ($\text{Hz}$): the frequency of the sound from the speaker.

$f$ ($\text{Hz}$): the frequency of the sound the man hears.

$v$ ($\text{m s}^{-1}$): the speed of sound in air.

$u$ ($\text{m s}^{-1}$): the speed of the man on the wheel.

$\varphi$ ($^\circ$): the angle between the path of the man and the line from him to $X$.

$u_0$ ($\text{m s}^{-1}$): the part of his speed along the line to $X$.

Relevant theory

When the train approaches you, the pitch of the sound you hear from the horn will not be that relevant to the natural frequency f0 of the horn, but a pitch corresponding to a frequency higher than that of f0.

but will suddenly drop to a pitch corresponding to a lower frequency than that of f0, when the horn of the train passes you.

Hence Doppler effect is the apparent change of frequency (pitch) of the sound from a source as heard by an observer when there is any relative motion between the observer and the source emitting the sound.

Suppose the source emitting sound of frequency f₀ is stationary while the observer is approaching the source with a velocity u₀.

Reading the options

The options are combinations of (A), (B) and (C).

We test each statement on its own.

For example, think of a school bell that rings steadily. A boy runs straight towards the bell. The rings reach him more often than when he stands still.

Now he runs straight away from the bell. The rings reach him less often.

If he runs round the bell on a circle, then at the moment he runs across the line to the bell, the rings reach him at the usual rate.

At any other place on the circle he runs a little towards the bell or a little away from it. That part of his running is what counts.

So what matters is the part of the man's speed along the line from him to the speaker.

Reasoning and calculation

$u_0 = u\cos\varphi$

$f = \left(\dfrac{v + u_0}{v}\right)f_0$

$f = \left(\dfrac{v - u_0}{v}\right)f_0$

$f = f_0$

The pitch is highest when $u_0$ is largest on the way in, and lowest when $u_0$ is largest on the way out.

$u_0 = u\cos\varphi$ is largest, equal to $u$, when the man moves straight along the line, with $\varphi = 0^\circ$.

$5$

$L = \sqrt{4^2 + 3^2}$

$0.8$

$\cos\varphi = \dfrac{4}{5}$

$0.8u$

$u_0 = 0.8u$

$d / pow(d * d + r * r, 0.5)$

The original pitch is heard when the man is at positions $Q$ and $S$.

At $Q$ the man moves down and at $S$ he moves up. The line to $X$ is level, so both movements are across it. None of his speed is along the line, so $u_0 = 0$ and he hears $f_0$.

A

The highest pitch is heard when the man is at position $P$.

At $P$ the man moves to the right, but the line from $P$ to $X$ slants down, so just $0.8u$ of his speed is along it. Between $Q$ and $P$ he heads straight at $X$ and the whole of $u$ is along the line. The pitch is higher there than at $P$.

B

The lowest pitch is heard when the man is at position $R$.

At $R$ the man moves to the left, but the line from $R$ to $X$ slants up, so just part of his speed is along it. Between $Q$ and $R$ he moves straight away from $X$ and the whole of $u$ is along the line. The pitch is lower there than at $R$.

C

Statement (A) is true, because across the line there is no change.

Statement (B) is false, because the highest pitch comes between $Q$ and $P$, where he moves straight at $X$.

Statement (C) is false, because the lowest pitch comes between $Q$ and $R$, where he moves straight away from $X$.

So (A) is the one true statement.

Why the other options fail

At $P$ the man does not head straight at $X$, so (B) is false. He does that between $Q$ and $P$.

At $R$ the man does not move straight away from $X$, so (C) is false. He does that between $Q$ and $R$.

At $Q$ and $S$ the man moves across the line to the speaker, so he hears $f_0$ and (A) is true. (B) and (C) are false.

(A) is true, but (B) and (C) are false. The highest and lowest pitches come between $Q$ and $P$ and between $Q$ and $R$.

Conclusion

Only statement (A) is true. (B) and (C) are false.

The answer is (1).

The quotes give the rules for a man who moves along the line to the speaker. For a man who moves at an angle to that line, we used the part of his speed along the line, $u_0 = u\cos\varphi$.

Teacher's note

Ask one thing: how fast is the man coming closer to the speaker, or going away from it, along the line joining them?

Coming closer gives a higher pitch, going away gives a lower pitch, and moving across the line gives no change. The extreme pitches are where he moves straight at the speaker or straight away from it.

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