PART I · PAPER 1 STYLE第一部分 · 第一卷风格Short structured · calculator · 30 marks短结构题 · 可用计算器 · 30 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. Quote $r$ as the distance from the centre, not the height above the surface, and keep extra figures in intermediate steps. 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。$r$ 取到中心的距离，而非距表面的高度；中间步骤多保留几位。数值答案保留适当的有效数字。
Two identical spheres, each of mass $8.0\times 10^{3}\ \mathrm{kg}$, are placed with their centres $5.0\ \mathrm{m}$ apart.两个完全相同的球体，每个质量 $8.0\times 10^{3}\ \mathrm{kg}$，中心相距 $5.0\ \mathrm{m}$ 放置。
(a)Calculate the gravitational force of attraction between the two spheres.计算两球之间的引力。[2]
(b)The centre-to-centre separation is now tripled. State the new force, and explain in one sentence why it changes by this factor.现将中心间距增至三倍。写出新的引力，并用一句话解释它为何按此倍率变化。[2]
Q2MEDIUMPaper 1field strength g = GM/r²场强度 g = GM/r²[4 marks]
Mars has mass $M = 6.4\times 10^{23}\ \mathrm{kg}$ and radius $R = 3.4\times 10^{6}\ \mathrm{m}$.火星质量 $M = 6.4\times 10^{23}\ \mathrm{kg}$，半径 $R = 3.4\times 10^{6}\ \mathrm{m}$。
(a)Calculate the gravitational field strength at the surface of Mars.计算火星表面的引力场强度。[2]
(b)State the field strength at a point one planet-radius above the surface (so $r = 2R$), justifying the factor you use.写出距表面一个行星半径处（即 $r = 2R$）的场强度，并说明所用的倍率依据。[2]
Q3HARDPaper 1orbital speed, Kepler's third law轨道速度与开普勒第三定律[6 marks]
A satellite moves in a circular orbit of radius $r = 8.0\times 10^{6}\ \mathrm{m}$ around a planet of mass $M = 6.0\times 10^{24}\ \mathrm{kg}$.一卫星沿半径 $r = 8.0\times 10^{6}\ \mathrm{m}$ 的圆轨道绕质量 $M = 6.0\times 10^{24}\ \mathrm{kg}$ 的行星运行。
(a)By equating the gravitational force to the centripetal force, show that the orbital speed is $v = \sqrt{GM/r}$.通过令引力等于向心力，证明轨道速度为 $v = \sqrt{GM/r}$。[2]
(b)Calculate the orbital speed and the orbital period of the satellite.计算该卫星的轨道速度与轨道周期。[3]
(c)A second satellite orbits the same planet at four times this radius. Using Kepler's third law, state by what factor its period is larger.第二颗卫星以四倍半径绕同一行星运行。用开普勒第三定律写出其周期增大的倍数。[1]
A planet has mass $M = 6.0\times 10^{24}\ \mathrm{kg}$ and radius $R = 6.4\times 10^{6}\ \mathrm{m}$. A probe of mass $1500\ \mathrm{kg}$ rests on its surface.某行星质量 $M = 6.0\times 10^{24}\ \mathrm{kg}$，半径 $R = 6.4\times 10^{6}\ \mathrm{m}$。一个质量 $1500\ \mathrm{kg}$ 的探测器停在其表面。
(a)Calculate the gravitational potential at the surface, and state why it is negative.计算表面处的引力势，并说明它为何为负。[3]
(b)Calculate the gravitational potential energy of the probe while it sits on the surface.计算探测器停在表面时的引力势能。[2]
(c)Define an equipotential surface and state how much work is done in moving the probe along one such surface.定义等势面，并说明沿该面移动探测器所做的功为多少。[3]
Q5HARDPaper 1HL ONLYescape speed逃逸速度[8 marks]
A planet has mass $M = 6.0\times 10^{24}\ \mathrm{kg}$ and radius $R = 6.4\times 10^{6}\ \mathrm{m}$. Air resistance and the planet's rotation may be ignored.某行星质量 $M = 6.0\times 10^{24}\ \mathrm{kg}$，半径 $R = 6.4\times 10^{6}\ \mathrm{m}$。空气阻力与行星自转可忽略。
(a)By setting the total energy of a projectile to zero, derive the expression $v_{esc} = \sqrt{2GM/r}$ for the escape speed.令抛体的总能量为零，推导逃逸速度表达式 $v_{esc} = \sqrt{2GM/r}$。[3]
(b)Calculate the escape speed from the surface of this planet.计算从该行星表面逃逸的速度。[2]
(c)State why the escape speed does not depend on the mass of the escaping object, and show that the escape speed equals $\sqrt{2}$ times the circular orbital speed at the same radius.说明逃逸速度为何与逃逸物体的质量无关，并证明逃逸速度等于同一半径处圆轨道速度的 $\sqrt{2}$ 倍。[3]
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 intercepts, and correct handling of uncertainties. Read gradients from the line or from widely spaced points, never from a single pair of values. Quote uncertainties to one significant figure and round the value to match.这些题考查对斜率与截距的细致读取以及对不确定度的正确处理。斜率应从直线或相距较远的点读取，绝不用单组数值相除。不确定度保留 1 位有效数字，并使数值的末位与之对齐。
Q6HARDPaper 1Blog T vs log r (Kepler) + uncertaintylog T 对 log r（开普勒）与不确定度[12 marks]
Four moons orbit a planet in circular orbits. For each, the orbital radius $r$ and period $T$ are measured and the logarithms (base 10, with $r$ in $\mathrm{m}$ and $T$ in $\mathrm{s}$) are tabulated:四颗卫星沿圆轨道绕一行星运行。对每颗测出轨道半径 $r$ 与周期 $T$，并列出其对数（以 10 为底，$r$ 以 $\mathrm{m}$、$T$ 以 $\mathrm{s}$ 为单位）：
$\lg(r/\mathrm{m})$
$6.90$
$7.00$
$7.30$
$7.60$
$\lg(T/\mathrm{s})$
$3.85$
$4.00$
$4.45$
$4.90$
(a)Starting from $T^{2} = \dfrac{4\pi^{2}}{GM}r^{3}$, show that a graph of $\lg T$ against $\lg r$ is a straight line, and state the value its gradient should take.从 $T^{2} = \dfrac{4\pi^{2}}{GM}r^{3}$ 出发，证明 $\lg T$ 对 $\lg r$ 的图为直线，并写出其斜率应取的值。[3]
(b)Calculate the gradient of the line from the data and comment on whether it agrees with Kepler's third law.由数据计算该直线的斜率，并评论它是否与开普勒第三定律相符。[3]
(c)The vertical intercept of the line (where $\lg r = 0$) is $\lg c = -6.50$. Use it to determine the mass $M$ of the planet.该直线在纵轴的截距（即 $\lg r = 0$ 处）为 $\lg c = -6.50$。用它求行星质量 $M$。[4]
(d)State one experimental advantage of plotting the logarithmic graph rather than $T^{2}$ against $r^{3}$ directly.写出绘制对数图而非直接作 $T^{2}$ 对 $r^{3}$ 图的一个实验优势。[2]
A probe descending toward a planet measures the gravitational field strength $g$ at several distances $r$ from the planet's centre. The data are tabulated against $1/r^{2}$:一探测器在向某行星下降途中，测量距行星中心若干距离 $r$ 处的引力场强度 $g$，将数据对 $1/r^{2}$ 列表：
$1/r^{2}\ /\ 10^{-15}\,\mathrm{m^{-2}}$
$1.60$
$2.50$
$4.44$
$10.0$
$g\ /\ \mathrm{N\,kg^{-1}}$
$0.64$
$1.00$
$1.78$
$4.00$
(a)Explain why a graph of $g$ against $1/r^{2}$ is expected to be a straight line through the origin, and state what its gradient represents.解释为何 $g$ 对 $1/r^{2}$ 的图应为过原点的直线，并说明其斜率代表什么。[3]
(b)Calculate the gradient of the line and hence determine the mass $M$ of the planet.计算该直线的斜率，由此求行星质量 $M$。[4]
(c)State the relationship between the field strength $g$ and the gravitational potential $V_g$, and explain what feature of a graph of $V_g$ against $r$ would give $g$ at a point.写出场强度 $g$ 与引力势 $V_g$ 的关系，并说明 $V_g$ 对 $r$ 图上的哪个特征能给出某点的 $g$。[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 diagram and state the assumption that the planet is spherically symmetric and acts as a point mass at its centre. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer.每题先画清晰示意图，并写明"行星球对称、等效为中心处点质量"的假设。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 communications satellite is to be placed in a geostationary orbit around Earth, so that its orbital period equals one Earth day, $T = 8.64\times 10^{4}\ \mathrm{s}$. Earth has mass $M = 6.0\times 10^{24}\ \mathrm{kg}$ and radius $R = 6.4\times 10^{6}\ \mathrm{m}$.一颗通信卫星将被送入绕地球的地球静止轨道，使其轨道周期等于一个地球日，$T = 8.64\times 10^{4}\ \mathrm{s}$。地球质量 $M = 6.0\times 10^{24}\ \mathrm{kg}$，半径 $R = 6.4\times 10^{6}\ \mathrm{m}$。
(a)State the two conditions a satellite must satisfy for its orbit to be geostationary.写出卫星轨道为地球静止轨道须满足的两个条件。[2]
(b)Show that the orbital radius is given by $r = \left(\dfrac{GMT^{2}}{4\pi^{2}}\right)^{1/3}$ and calculate its value.证明轨道半径为 $r = \left(\dfrac{GMT^{2}}{4\pi^{2}}\right)^{1/3}$，并计算其数值。[4]
(c)Calculate the altitude of the satellite above Earth's surface.计算卫星相对地表的高度。[2]
(d)Calculate the orbital speed of the geostationary satellite.计算地球静止卫星的轨道速度。[2]
(e)Calculate the gravitational field strength of Earth at the satellite's orbit, and compare it with the surface value.计算地球在卫星轨道处的引力场强度，并与表面值比较。[2]
Q9HARDPaper 2HL ONLYorbit energy, raising an orbit轨道能量与抬升轨道[10 marks]
A satellite of mass $2500\ \mathrm{kg}$ is in a circular orbit of radius $r_{1} = 7.0\times 10^{6}\ \mathrm{m}$ around Earth ($M = 6.0\times 10^{24}\ \mathrm{kg}$). It is later moved to a higher circular orbit of radius $r_{2} = 1.4\times 10^{7}\ \mathrm{m}$.一颗质量 $2500\ \mathrm{kg}$ 的卫星处于绕地球（$M = 6.0\times 10^{24}\ \mathrm{kg}$）半径 $r_{1} = 7.0\times 10^{6}\ \mathrm{m}$ 的圆轨道上。其后被移至半径 $r_{2} = 1.4\times 10^{7}\ \mathrm{m}$ 的更高圆轨道。
(a)Show that the total energy of a satellite in a circular orbit of radius $r$ is $E = -\dfrac{GMm}{2r}$.证明半径 $r$ 圆轨道上卫星的总能量为 $E = -\dfrac{GMm}{2r}$。[3]
(b)Calculate the total orbital energy of the satellite in the lower orbit $r_{1}$.计算卫星在较低轨道 $r_{1}$ 上的总轨道能量。[2]
(c)Calculate the energy that must be supplied to move the satellite from orbit $r_{1}$ to orbit $r_{2}$.计算把卫星从轨道 $r_{1}$ 移到轨道 $r_{2}$ 须提供的能量。[3]
(d)The orbital speed in the higher orbit is smaller than in the lower orbit. State and briefly explain this apparent paradox: energy is added yet the satellite moves more slowly.较高轨道的轨道速度比较低轨道更小。说明并简述这一看似矛盾之处：补充了能量，卫星却运动得更慢。[2]
Q10HARDPaper 2HL ONLYescape speed, bound vs unbound energy逃逸速度、束缚与非束缚能量[8 marks]
A probe of mass $1200\ \mathrm{kg}$ is launched vertically from the surface of a rocky planet of mass $M = 4.9\times 10^{24}\ \mathrm{kg}$ and radius $R = 6.1\times 10^{6}\ \mathrm{m}$, with an initial speed of $1.0\times 10^{4}\ \mathrm{m\,s^{-1}}$. Ignore air resistance and rotation.一个质量 $1200\ \mathrm{kg}$ 的探测器以初速率 $1.0\times 10^{4}\ \mathrm{m\,s^{-1}}$ 从质量 $M = 4.9\times 10^{24}\ \mathrm{kg}$、半径 $R = 6.1\times 10^{6}\ \mathrm{m}$ 的岩质行星表面竖直发射。忽略空气阻力与自转。
(a)Calculate the escape speed from the surface of this planet.计算从该行星表面逃逸的速度。[2]
(b)By evaluating the total mechanical energy of the probe at launch, determine whether it is bound or unbound, and state what its subsequent motion will be.通过计算探测器发射时的总机械能，判断它是束缚的还是非束缚的，并说明其后续运动。[3]
(c)If the probe is bound, calculate the maximum distance it reaches from the planet's centre using conservation of energy.若探测器为束缚态，用能量守恒计算它到行星中心的最大距离。[3]