📚 How to Tackle AS Physics Unit 5 Application Problems Using the Insert (June 2019) | AS物理Unit5（2019年6月）插入资料应用题攻略技巧
The Unit 5 paper for AS Physics often weaves together thermodynamics, blackbody radiation, and cosmology into application‑heavy questions. Your best resource during the exam is the Insert booklet, which contains a compact set of formulas, constants, and reference data. This article shows you how to decode the Insert and apply it systematically so you can solve problems faster, avoid common pitfalls, and maximise your marks.
AS物理 Unit 5 试卷通常将热力学、黑体辐射和宇宙学融合成大量应用题。考试期间你最好的帮手就是插入资料小册子，它提供了一套紧凑的公式、常数和参考数据。本文将教你如何解读插入资料并有条理地运用它们，从而更快地解题、避开常见陷阱并最大化你的得分。
1. Getting to Know the Insert Booklet | 熟悉插入资料小册子
The June 2019 Insert (and similar booklets) is divided into clear sections: thermal physics, radiation, and cosmology. It lists the ideal gas equation pV = nRT, the Stefan–Boltzmann law L = 4πr²σT⁴, Wien’s displacement law λₘₐₓ T = 2.898 × 10⁻³ m·K, and Hubble’s law v = H₀d. Constants like the Boltzmann constant k = 1.38 × 10⁻²³ J K⁻¹ and the Hubble constant H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹ are also printed. Spend the first two minutes of the exam scanning each section so you know exactly where every formula lives.
2019年6月的插入资料（及类似的小册子）分为热物理、辐射和宇宙学几个清晰的部分。它列出了理想气体方程 pV = nRT、斯特藩–玻尔兹曼定律 L = 4πr²σT⁴、维恩位移定律 λₘₐₓ T = 2.898 × 10⁻³ m·K 和哈勃定律 v = H₀d。此外还有玻尔兹曼常数 k = 1.38 × 10⁻²³ J K⁻¹ 和哈勃常数 H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹ 等常数。考试开始时花两分钟浏览各个部分，确切记住每个公式的位置。
Notice that the Insert often provides alternative forms, e.g. pV = NkT where N is the number of molecules. Making a mental note of these variations prevents you from misapplying the gas laws when the question uses particle count instead of moles.
注意插入资料通常会给出不同形式，例如 pV = NkT，其中 N 是分子数。留意这些变体可以防止在题目使用粒子数而非摩尔数时误用气体定律。
2. Identify Given Quantities and Unknowns | 识别已知量与未知量
Before you reach for a formula, underline every numerical value with its unit in the question. For instance, a star has radius r = 3.2 × 10⁹ m and surface temperature T = 5800 K. The unknown might be its luminosity L. Write the symbols next to the numbers: r, T, L.
在动笔选公式之前，先在题目中划出每个带有单位的数值。例如，一颗恒星的半径 r = 3.2 × 10⁹ m，表面温度 T = 5800 K。未知量可能是它的光度 L。在数字旁写上对应的符号：r、T、L。
Then glance at the Insert. The presence of r and T immediately points toward the Stefan–Boltzmann law, which uses r² and T⁴. This simple matching exercise saves you from randomly hunting through equations.
然后扫一眼插入资料。出现了 r 和 T 立刻就指向斯特藩–玻尔兹曼定律，因为它用到 r² 和 T⁴。这个简单的匹配练习可以避免你在公式堆里漫无目的地寻找。
3. Selecting the Right Formula from the Insert | 从插入资料中选择正确公式
Each physics relationship in the Insert is written in standard symbols. Match the symbols you identified with the left‑hand side of an equation. If a question asks for the peak wavelength λₘₐₓ of a star’s radiation given its surface temperature T, the Insert gives Wien’s law immediately.
插入资料中的每个物理关系都用标准符号书写。将你识别出的符号与某个方程的左边进行匹配。如果题目要求根据恒星表面温度 T 求辐射的峰值波长 λₘₐₓ，插入资料立刻给出维恩定律。
λₘₐₓ T = 2.898 × 10⁻³ m·K
Make sure you understand the meaning of every symbol. For example, in the Stefan–Boltzmann law L = 4πr²σT⁴, r is the radius of the spherical blackbody emitter. If the question gives a diameter, you must first halve it. The Insert will not remind you of this.
确保理解每个符号的含义。例如，在斯特藩–玻尔兹曼定律 L = 4πr²σT⁴ 中，r 是球形黑体辐射体的半径。如果题目给出的是直径，必须先将它除以二。插入资料不会提醒你这一点。
4. Mastering Unit Conversions and Prefixes | 掌握单位换算与词头
Application problems frequently mix units: distances in pc or Mpc for cosmology, temperatures in °C that must be converted to K, and masses in grams rather than kilograms. The Insert provides constants in SI units, so all inputs must be in SI unless stated otherwise.
应用题经常混合单位：宇宙学中距离用 pc 或 Mpc，温度用 °C 必须转换为 K，质量用克而不是千克。插入资料中的常数是 SI 单位，因此所有输入量都必须是 SI 单位，除非另有说明。
For instance, Hubble’s law v = H₀d expects d in metres. If the question gives 150 Mpc, convert to metres: 1 pc = 3.09 × 10¹⁶ m, so 150 Mpc = 150 × 10⁶ × 3.09 × 10¹⁶ m = 4.64 × 10²⁴ m. Write the conversion explicitly to avoid power‑of‑ten mistakes.
例如，哈勃定律 v = H₀d 要求 d 以米为单位。如果题目给出 150 Mpc，就要转换为米：1 pc = 3.09 × 10¹⁶ m，所以 150 Mpc = 150 × 10⁶ × 3.09 × 10¹⁶ m = 4.64 × 10²⁴ m。把转换过程写清楚，避免 10 的幂次出错。
Also watch out for centimetre–metre conversions when dealing with wavelength. The Insert’s Wien constant is in m·K, so any λ must be in metres.
同样，处理波长时要注意厘米与米的换算。插入资料中的维恩常数单位是 m·K，因此任何波长 λ 都必须采用米。
5. Using Standard Form and Significant Figures | 科学记数法与有效数字
The numbers in the Insert are expressed in standard form, and your calculator will often give very large or very small numbers. Always present final answers in standard form, matching the least number of significant figures given in the question data.
插入资料中的数值都用科学记数法表示，而你的计算器常常给出非常大或非常小的数字。最终答案始终要用科学记数法呈现，并与题目所给数据中最少的有效数字位数保持一致。
For example, if the stellar radius is given as 6.96 × 10⁸ m (3 s.f.) and temperature as 5778 K (4 s.f.), the calculated luminosity should be rounded to 3 s.f. Using the Insert’s constants, keep all digits during calculation and round only at the end.
例如，若恒星半径给定为 6.96 × 10⁸ m（3位有效数字），温度为 5778 K（4位有效数字），计算得出的光度就应该四舍五入到3位有效数字。使用插入资料中的常数时，计算过程中保留所有位数，只在最后一步舍入。
6. Handling Graphs and Data Tables | 处理图表与数据表
Some questions provide a graph of, say, λₘₐₓ against 1/T, or a table of distance and recessional velocity. The Insert gives the theoretical relationship; your job is to link the graph’s gradient or intercept to the Insert’s equation.
有些题目会提供图表，例如 λₘₐₓ 随 1/T 变化的图像，或一张距离和退行速度的数据表。插入资料给出了理论关系；你的任务是把图的斜率或截距与插入资料中的公式联系起来。
If a graph plots v against d, the gradient equals H₀. Always read the gradient from the best‑fit line, not a single data point, and express it with the correct units: s⁻¹ if d is in metres and v in m s⁻¹.
如果图像绘制了 v 对 d 的关系，其斜率就等于 H₀。永远根据最佳拟合线读取斜率，而不是单个数据点，并且要用正确的单位表示：如果 d 的单位是米，v 的单位是米每秒，那么 H₀ 的单位就是 s⁻¹。
7. Thermal Physics Application Example | 热物理应用题示例
Consider a problem: ‘A sealed cylinder of volume 0.020 m³ contains helium at 2.0 × 10⁵ Pa and 300 K. Calculate the number of moles of helium.’ The Insert shows pV = nRT, with R = 8.31 J mol⁻¹ K⁻¹. All quantities are in SI, so simply rearrange: n = pV / (RT).
考虑这样一个问题：“一个体积为 0.020 m³ 的密封气缸内装有氦气，压强为 2.0 × 10⁵ Pa，温度为 300 K。计算氦气的摩尔数。”插入资料给出 pV = nRT，R = 8.31 J mol⁻¹ K⁻¹。所有量都是 SI 单位，所以只需变形：n = pV / (RT)。
Plug in: n = (2.0 × 10⁵ × 0.020) / (8.31 × 300) = 4000 / 2493 ≈ 1.6 mol (2 s.f.). Notice that the Insert gives R to 3 s.f., but the volume and pressure are given to 2 s.f., so our answer is appropriately rounded.
代入数值：n = (2.0 × 10⁵ × 0.020) / (8.31 × 300) = 4000 / 2493 ≈ 1.6 mol（2位有效数字）。留意插入资料中 R 给出3位有效数字，但体积和压强只给了2位，所以我们的答案也相应舍入到2位。
8. Blackbody Radiation and Wien’s Law | 黑体辐射与维恩定律
‘The Sun’s surface temperature is about 5800 K. Estimate the peak wavelength of its radiation.’ The Insert gives λₘₐₓ T = 2.898 × 10⁻³ m·K, so λₘₐₓ = (2.898 × 10⁻³) / 5800 = 5.0 × 10⁻⁷ m (500 nm). This is visible light, a common sanity check.
“太阳表面温度约为 5800 K。估算其辐射的峰值波长。”插入资料给出 λₘₐₓ T = 2.898 × 10⁻³ m·K，因此 λₘₐₓ = (2.898 × 10⁻³) / 5800 = 5.0 × 10⁻⁷ m（500 nm）。这位于可见光范围，是一个常见的合理性检验。
When comparing two stars, you can use the ratio form: λₘₐₓ,₁ / λₘₐₓ,₂ = T₂ / T₁. This avoids inserting the constant repeatedly and is handy for multiple‑choice questions.
比较两颗恒星时，可使用比例形式：λₘₐₓ,₁ / λₘₐₓ,₂ = T₂ / T₁。这样可以避免反复代入常数，对选择题尤其方便。
A typical pitfall is using Celsius instead of kelvin. The Wien constant uses kelvin; 27 °C must become 300 K before substituting.
一个典型陷阱是代入的是摄氏度而非开尔文。维恩常数适用于开尔文温度；27 °C 必须先转换成 300 K 再代入公式。
9. Cosmology and Redshift Problems | 宇宙学与红移问题
Hubble’s law is given as v = H₀d. If a galaxy’s spectral line is redshifted from λ₀ = 656.3 nm to λ = 660.0 nm, first find Δλ = 3.7 nm. The Insert may also give the redshift formula z = Δλ / λ₀ ≈ v / c (for v ≪ c). Combine these to find the recession speed and then distance.
哈勃定律写作 v = H₀d。如果一个星系的谱线从 λ₀ = 656.3 nm 红移到 λ = 660.0 nm，首先算出 Δλ = 3.7 nm。插入资料还可能提供红移公式 z = Δλ / λ₀ ≈ v / c（当 v ≪ c 时）。联合这两个式子，先求出退行速度，再求出距离。
v = c × (Δλ / λ₀)
Using c = 3.00 × 10⁸ m s⁻¹, v ≈ 3.00 × 10⁸ × (3.7 / 656.3) ≈ 1.69 × 10⁶ m s⁻¹. Then d = v / H₀, with H₀ from the Insert. Always check whether the question expects the answer in Mpc; convert metres to Mpc by dividing by 3.09 × 10²².
取 c = 3.00 × 10⁸ m s⁻¹，v ≈ 3.00 × 10⁸ × (3.7 / 656.3) ≈ 1.69 × 10⁶ m s⁻¹。然后 d = v / H₀，其中 H₀ 取自插入资料。始终检查题目是否要求答案以 Mpc 为单位；把米转换为 Mpc 需除以 3.09 × 10²²。
The Insert often lists H₀ in s⁻¹, so d will come out in metres. A quick unit check – v in m s⁻¹ divided by s⁻¹ gives m – confirms the algebra.
插入资料中 H₀ 通常以 s⁻¹ 列出，因此算出的 d 单位是米。做一个快速的单位检查——m s⁻¹ 除以 s⁻¹ 得到 m——便可确认代数运算无误。
10. Cross‑Checking with Alternative Insert Forms | 利用插入资料的变形公式交叉检验
The Insert may give two forms of the same law. For instance, the ideal gas equation appears as both pV = nRT and pV = NkT. If a problem provides the number of molecules, use pV = NkT to save conversion steps. After solving, cross‑check by converting N to n and trying the other form – the answers must agree.
插入资料中同一定律可能给出两种形式。例如，理想气体方程既写作 pV = nRT，也写作 pV = NkT。如果题目提供了分子数，就用 pV = NkT 来省去转换步骤。解完后，可以把 N 转换成 n，用另一种形式检验——答案必须一致。
Similarly, you may find L = σAT⁴ alongside L = 4πr²σT⁴. The first is for any blackbody of surface area A; the second is for a sphere. Choose the form that matches the given information to avoid unnecessary area calculations.
类似地，你可能会看到 L = σAT⁴ 和 L = 4πr²σT⁴ 并存。前者适用于任何表面积为 A 的黑体，后者适用于球体。选择与题给信息匹配的形式可以避免不必要的面积计算。
11. Common Pitfalls When Using the Insert | 使用插入资料时的常见陷阱
A frequent mistake is substituting diameter for radius in L = 4πr²σT⁴. The factor 4πr² uses radius; if the question gives diameter, r = d/2. Another trap is forgetting to square or raise to the fourth power when applying Stefan–Boltzmann or Wien’s law in rearrangement.
一个常见错误是在 L = 4πr²σT⁴ 中将直径当作半径代入。因子 4πr² 用的是半径；如果题目给出直径，务必先求 r = d/2。另一个陷阱是在变形应用斯特藩–玻尔兹曼定律或维恩定律时忘记平方或四次方。
Also check whether the temperature is uniform. The Stefan–Boltzmann law assumes the entire surface is at the same T. If the question gives core and surface temperatures, use the surface value for radiation calculations.
还要检查温度是否均匀。斯特藩–玻尔兹曼定律假定整个表面温度相同。如果题目同时给出了核心温度和表面温度，辐射计算应该使用表面温度。
12. Putting It All Together: A Mock Exam Strategy | 综合应用：模拟考场策略
Read the question carefully and circle the target quantity. Then open the Insert and locate the relevant section. Write the chosen formula exactly as it appears in the Insert. Substitute values with units
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