PART I · PAPER 1 STYLE第一部分 · 第一卷风格Short structured · calculator · 28 marks短结构题 · 可用计算器 · 28 分
Short Structured Items短结构题
Show all working in the space below each question. Marks are awarded for correct method as well as final answers. For every sound or light problem, decide "approach or recede" first and state the sign convention before substituting. Give numerical answers to an appropriate number of significant figures.在每题下方空白处写出全部解题过程。方法分（method marks）与最终答案同等重要。每道声波或光的题目，先判断"接近还是远离"，代入前先写明符号约定（sign convention）。数值答案保留适当的有效数字。
A car sounds a steady horn while driving along a straight road at constant speed. A pedestrian stands beside the road as the car drives toward, level with, and then away from them.一辆汽车在直路上以恒定速度行驶并持续鸣笛。一名行人站在路旁，汽车先朝其驶来、与其平齐、再驶离。
(a)Using the idea of wavefront spacing, explain why the observed frequency is higher than the emitted frequency while the car approaches.用波前间距的概念，解释为何汽车接近时观察到的频率高于发射频率。[2]
(b)Describe how the pitch heard by the pedestrian changes over the whole passage, and state what is heard at the instant the car is level with them.描述行人在整个经过过程中所听音调如何变化，并说明汽车与其平齐瞬间听到什么。[2]
Q2MEDIUMPaper 1light shift: line displacement to speed光频移：谱线位移求速度[6 marks]
A hydrogen line has a rest wavelength of $434.0\ \mathrm{nm}$ in the laboratory. In the spectrum of a star it is observed at $434.3\ \mathrm{nm}$.某氢谱线在实验室中的静止波长为 $434.0\ \mathrm{nm}$。在一颗恒星的光谱中观测到该线为 $434.3\ \mathrm{nm}$。
(a)State whether the line is redshifted or blueshifted, and hence whether the star is approaching or receding.说明该线是红移还是蓝移，并由此判断恒星是接近还是远离。[2]
(b)Using $\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$, calculate the radial speed of the star.用 $\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$ 计算恒星的径向速率。[3]
(c)State one reason why this low-speed formula is justified for this star.说明对这颗恒星采用该低速公式合理的一个理由。[1]
Q3HARDPaper 1HL ONLYmoving source, both signs波源运动，两种符号[6 marks]
A train sounds a horn of frequency $480\ \mathrm{Hz}$ and moves along a straight track at a constant $25\ \mathrm{m\,s^{-1}}$. A stationary observer stands beside the track. Take the speed of sound as $340\ \mathrm{m\,s^{-1}}$.一列火车鸣响频率为 $480\ \mathrm{Hz}$ 的汽笛，沿直轨以恒定速率 $25\ \mathrm{m\,s^{-1}}$ 行驶。一名静止观察者站在轨道旁。取声速为 $340\ \mathrm{m\,s^{-1}}$。
(a)Calculate the frequency heard while the train approaches the observer.计算火车接近观察者时所听到的频率。[2]
(b)Calculate the frequency heard after the train has passed and is receding.计算火车经过并远离后所听到的频率。[2]
(c)The rise on approach is larger than the fall on recession. Explain this asymmetry by referring to the form of the moving-source equation.接近时的升高量大于远离时的降低量。结合波源运动方程的形式解释这一不对称。[2]
Q4HARDPaper 1HL ONLYmoving observer then combined观察者运动再组合[6 marks]
A loudspeaker emits a steady tone of frequency $600\ \mathrm{Hz}$. Take the speed of sound as $340\ \mathrm{m\,s^{-1}}$.一只扬声器发出频率为 $600\ \mathrm{Hz}$ 的稳定音调。取声速为 $340\ \mathrm{m\,s^{-1}}$。
(a)The speaker is stationary and a runner moves directly toward it at $18\ \mathrm{m\,s^{-1}}$. Calculate the frequency the runner hears.扬声器静止，一名跑者以 $18\ \mathrm{m\,s^{-1}}$ 直接朝它跑去。计算跑者听到的频率。[2]
(b)Now the speaker is carried toward the runner at $18\ \mathrm{m\,s^{-1}}$ while the runner still moves toward it at $18\ \mathrm{m\,s^{-1}}$. Calculate the frequency the runner now hears.现在扬声器以 $18\ \mathrm{m\,s^{-1}}$ 朝跑者运动，而跑者仍以 $18\ \mathrm{m\,s^{-1}}$ 朝它运动。计算此时跑者听到的频率。[2]
(c)Both speeds are $18\ \mathrm{m\,s^{-1}}$, yet the moving-observer and moving-source contributions are not equal. State the physical reason sound treats the two cases differently.两个速率都是 $18\ \mathrm{m\,s^{-1}}$，但观察者运动与波源运动的贡献并不相等。说明声波对这两种情形区别对待的物理原因。[2]
In a distant galaxy the hydrogen line of rest wavelength $656.3\ \mathrm{nm}$ is observed at $660.0\ \mathrm{nm}$. Take $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$.在一个遥远星系中，静止波长为 $656.3\ \mathrm{nm}$ 的氢线被观测为 $660.0\ \mathrm{nm}$。取 $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$。
(a)Define redshift and blueshift in terms of the change in observed wavelength.用观测波长的变化定义红移与蓝移。[2]
(b)Calculate the recession speed of this galaxy.计算该星系的退行速率。[2]
(c)The Andromeda galaxy is observed to be blueshifted. State what this tells you about its motion, and why it does not contradict the general redshift of distant galaxies.观测到仙女座星系为蓝移。说明这反映其怎样的运动，以及为何这并不与遥远星系普遍红移相矛盾。[2]
PART II · PAPER 1B / DATA ANALYSIS第二部分 · 第一卷 B / 数据分析Graphs · data · uncertainties · 22 marks图像 · 数据 · 不确定度 · 22 分
Graph and Data Questions图像与数据题
These items reward careful reading of gradients and correct handling of uncertainties. Read a gradient from a best-fit line or from widely separated points, never from a single reading. Quote uncertainties to one significant figure and round the value to match.这些题考查对斜率的细致读取以及对不确定度的正确处理。斜率应从最佳拟合直线或相距较远的点读取，绝不用单个读数。不确定度保留 1 位有效数字，并使数值的末位与之对齐。
Q6HARDPaper 1Bspectral data: speed vs shift谱线数据：速度对频移[10 marks]
An astronomer records the observed wavelength of the same hydrogen line (rest wavelength $\lambda_0 = 486.1\ \mathrm{nm}$) in four galaxies, and computes the wavelength shift $\Delta\lambda = \lambda_{\text{obs}} - \lambda_0$ for each:一位天文学家在四个星系中记录同一条氢线（静止波长 $\lambda_0 = 486.1\ \mathrm{nm}$）的观测波长，并对每个算出波长移动 $\Delta\lambda = \lambda_{\text{obs}} - \lambda_0$：
Galaxy星系
P
Q
R
S
$\lambda_{\text{obs}}\ /\ \mathrm{nm}$
$488.5$
$491.0$
$493.4$
$495.9$
$\Delta\lambda\ /\ \mathrm{nm}$
$2.4$
$4.9$
$7.3$
$9.8$
(a)Starting from $\frac{\Delta\lambda}{\lambda_0} \approx \frac{v}{c}$, show that a graph of recession speed $v$ against $\Delta\lambda$ should be a straight line through the origin, and state what the gradient represents.从 $\frac{\Delta\lambda}{\lambda_0} \approx \frac{v}{c}$ 出发，证明退行速度 $v$ 对 $\Delta\lambda$ 的图应为过原点的直线，并说明斜率代表什么。[3]
(b)Calculate the recession speed of galaxy S, and confirm it is consistent with the gradient $c/\lambda_0$ acting on its $\Delta\lambda$.计算星系 S 的退行速率，并验证它与梯度 $c/\lambda_0$ 作用于其 $\Delta\lambda$ 的结果一致。[3]
(c)For galaxy S, both $\lambda_{\text{obs}}$ and $\lambda_0$ have an absolute uncertainty of $\pm 0.1\ \mathrm{nm}$. Calculate the percentage uncertainty in $\Delta\lambda$ for galaxy S.对星系 S，$\lambda_{\text{obs}}$ 与 $\lambda_0$ 的绝对不确定度均为 $\pm 0.1\ \mathrm{nm}$。计算星系 S 中 $\Delta\lambda$ 的百分比不确定度。[2]
(d)State what trend in the data, together with independent distance measurements, would provide evidence for an expanding universe.说明数据中怎样的趋势，连同独立的距离测量，能为宇宙膨胀提供证据。[2]
Q7HARDPaper 1BHL ONLYmoving source: linearise the data波源运动：数据线性化[12 marks]
A siren of fixed frequency $f = 500\ \mathrm{Hz}$ is driven directly toward a stationary microphone at several speeds $v_s$. The observed frequency $f'$ is recorded. Take the speed of sound as $340\ \mathrm{m\,s^{-1}}$.一只固定频率 $f = 500\ \mathrm{Hz}$ 的警笛以若干速率 $v_s$ 直接朝静止麦克风运动。记录观测频率 $f'$。取声速为 $340\ \mathrm{m\,s^{-1}}$。
$v_s\ /\ \mathrm{m\,s^{-1}}$
$10$
$20$
$30$
$40$
$f'\ /\ \mathrm{Hz}$
$515$
$531$
$548$
$567$
(a)Write down the moving-source equation for an approaching source, and explain in one line why a direct plot of $f'$ against $v_s$ is not a straight line.写出波源接近的运动方程，并用一行说明为何 $f'$ 对 $v_s$ 的直接作图不是直线。[2]
(b)Show algebraically that $\frac{1}{f'} = \frac{1}{f} - \frac{v_s}{f v}$, so that a plot of $\frac{1}{f'}$ against $v_s$ is linear. State the gradient and the intercept in terms of $f$ and $v$.用代数证明 $\frac{1}{f'} = \frac{1}{f} - \frac{v_s}{f v}$，故 $\frac{1}{f'}$ 对 $v_s$ 的图为直线。用 $f$ 与 $v$ 写出斜率与截距。[3]
(c)Using the first and last data points, calculate $\frac{1}{f'}$ for $v_s = 10\ \mathrm{m\,s^{-1}}$ and $v_s = 40\ \mathrm{m\,s^{-1}}$, and hence find the gradient of the linearised graph.用首末两组数据，计算 $v_s = 10\ \mathrm{m\,s^{-1}}$ 与 $v_s = 40\ \mathrm{m\,s^{-1}}$ 时的 $\frac{1}{f'}$，由此求线性化图的斜率。[4]
(d)Use your gradient to obtain a value for the speed of sound $v$, given $f = 500\ \mathrm{Hz}$, and comment on whether it agrees with the stated $340\ \mathrm{m\,s^{-1}}$.已知 $f = 500\ \mathrm{Hz}$，用你的斜率求声速 $v$ 的值，并评论它是否与给定的 $340\ \mathrm{m\,s^{-1}}$ 一致。[3]
PART III · PAPER 2 STYLE第三部分 · 第二卷风格Extended structured · calculator · 30 marks长结构题 · 可用计算器 · 30 分
Extended Structured Problems长结构问题
Set up each problem with a clear statement of which body moves and the sign convention you adopt. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer.每题先明确写出哪个物体运动以及你采用的符号约定。长题中方法分占比最大；中间值多保留几位，仅在最终答案处取舍有效数字。
A factory whistle emits a tone of frequency $660\ \mathrm{Hz}$. The speed of sound in still air is $340\ \mathrm{m\,s^{-1}}$.一只工厂汽笛发出频率为 $660\ \mathrm{Hz}$ 的音调。静止空气中声速为 $340\ \mathrm{m\,s^{-1}}$。
(a)The whistle is mounted on a vehicle moving directly toward a stationary cyclist at $30\ \mathrm{m\,s^{-1}}$. Calculate the frequency the cyclist hears.汽笛装在以 $30\ \mathrm{m\,s^{-1}}$ 直接朝静止骑车者运动的车辆上。计算骑车者听到的频率。[2]
(b)Instead the whistle is stationary and the cyclist rides directly toward it at $15\ \mathrm{m\,s^{-1}}$. Calculate the frequency the cyclist hears.改为汽笛静止，骑车者以 $15\ \mathrm{m\,s^{-1}}$ 直接朝它骑去。计算骑车者听到的频率。[2]
(c)Now both move directly toward each other: the whistle at $30\ \mathrm{m\,s^{-1}}$ and the cyclist at $15\ \mathrm{m\,s^{-1}}$. Calculate the frequency the cyclist hears.现在两者直接相向运动：汽笛以 $30\ \mathrm{m\,s^{-1}}$、骑车者以 $15\ \mathrm{m\,s^{-1}}$。计算骑车者听到的频率。[3]
(d)By considering the wavefronts emitted in one period $T = 1/f$, derive the moving-source result $f' = \frac{fv}{v - v_s}$ for an approaching source.考虑一个周期 $T = 1/f$ 内发出的波前，推导波源接近的结果 $f' = \frac{fv}{v - v_s}$。[3]
(e)State, with a reason, whether replacing the sound source with a light source moving at the same speeds would require you to distinguish "source moving" from "observer moving".说明若把声源换成以相同速率运动的光源，是否仍需区分"波源运动"与"观察者运动"，并给出理由。[2]
Q9HARDPaper 2reflected-wave devices (factor of 2)反射波装置（因子 2）[10 marks]
A police radar gun emits microwaves of frequency $24.0\ \mathrm{GHz}$ at a car driving directly toward it. The reflected beam returns shifted up by $3.6\ \mathrm{kHz}$. Take $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$.警用雷达枪向迎面驶来的汽车发射频率为 $24.0\ \mathrm{GHz}$ 的微波。反射波返回时频率升高 $3.6\ \mathrm{kHz}$。取 $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$。
(a)Explain why the reflected beam undergoes two Doppler shifts, and hence why $\frac{\Delta f}{f} \approx \frac{2v}{c}$ rather than $\frac{v}{c}$.解释为何反射波经历两次多普勒频移，由此说明为何是 $\frac{\Delta f}{f} \approx \frac{2v}{c}$ 而非 $\frac{v}{c}$。[3]
(b)Calculate the speed of the car in $\mathrm{m\,s^{-1}}$ and in $\mathrm{km\,h^{-1}}$.计算汽车的速率，分别以 $\mathrm{m\,s^{-1}}$ 与 $\mathrm{km\,h^{-1}}$ 表示。[3]
(c)A medical probe emits ultrasound of frequency $2.0\ \mathrm{MHz}$ into tissue, where the speed of sound is $1540\ \mathrm{m\,s^{-1}}$. Red blood cells move directly toward the probe at $0.30\ \mathrm{m\,s^{-1}}$. Calculate the frequency shift of the returned echo.一只医用探头向组织发射频率为 $2.0\ \mathrm{MHz}$ 的超声，组织中声速为 $1540\ \mathrm{m\,s^{-1}}$。红细胞以 $0.30\ \mathrm{m\,s^{-1}}$ 直接朝探头运动。计算返回回波的频移。[2]
(d)State and explain one consequence of forgetting the factor of $2$ when reporting the measured speed.说明并解释在报出测得速率时漏掉因子 $2$ 的一个后果。[2]
A calcium line of rest wavelength $397.0\ \mathrm{nm}$ is observed in the spectrum of a galaxy at $405.0\ \mathrm{nm}$. Take $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$.在某星系光谱中观测到静止波长为 $397.0\ \mathrm{nm}$ 的钙线位于 $405.0\ \mathrm{nm}$。取 $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$。
(a)Calculate the wavelength shift and state whether the galaxy is approaching or receding.计算波长移动并说明该星系是接近还是远离。[2]
(b)Calculate the radial speed of the galaxy, and express it as a fraction of $c$.计算该星系的径向速率，并以 $c$ 的分数表示。[3]
(c)Explain how observations of many such galaxies provide evidence for an expanding universe.解释对许多此类星系的观测如何为宇宙膨胀提供证据。[2]
(d)State one reason why the value of $v$ found from $v \approx c\,\Delta\lambda/\lambda$ should be treated with caution for very distant galaxies.说明为何由 $v \approx c\,\Delta\lambda/\lambda$ 求得的 $v$ 值对极遥远星系应谨慎对待的一个理由。[1]