📚 A-Level Physics: Mastering Formula Derivation from the Unit 3 Examiner’s Report (Jan 2021) | A-Level物理：从2021年1月单元3考官报告精通公式推导
The January 2021 International A-Level Physics Unit 3 examiner’s report provided clear feedback on a skill that many candidates find unexpectedly challenging: deriving and rearranging the correct equation for an experiment, then using it to extract physical quantities from a straight-line graph. This article focuses on the art of formula derivation as emphasised in that report, walking through worked examples, exposing common mistakes, and offering a reliable method that will boost your confidence in practical assessments.
2021年1月的国际A-Level物理单元3考官报告明确反馈了令许多考生意外的一项难点技能：为实验推导并变换出正确的方程，然后利用该方程从直线图中提取物理量。本文聚焦考官报告中强调的公式推导艺术，通过具体实例梳理步骤，揭示常见错误，并提供一套可靠的方法，帮助你在实验考试中建立信心。
1. Why Derivation Matters in Unit 3 | 为什么推导在单元3中至关重要
Unit 3 is the practical and investigative skills paper, where you are often presented with a novel experimental setup and raw data. The instruction ‘deduce the relationship’ or ‘derive an equation that would produce a straight-line graph’ is not just a test of memory – it assesses your ability to link physical principles to the real measurements you have made. The Jan 2021 report indicated that candidates who attempted to skip derivation or relied on poorly remembered standard equations often selected the wrong graph axes, leading to an invalid gradient analysis.
单元3是实验与探究技能试卷，你经常会遇到一个新颖的实验装置和原始数据。指令“推导关系”或“推导出一个可生成直线图的方程”不仅仅是对记忆力的测试——它评估你是否能将物理原理与你所做的实际测量联系起来。2021年1月的报告指出，那些试图跳过推导或依靠记不清的标准方程的考生，往往会选择错误的坐标轴，导致梯度分析无效。
Deriving the equation yourself forces you to identify which variables are controlled, which are independent, and which is the dependent variable. It also reveals the link between the gradient or intercept and the physical constant you are trying to determine. Without this step, even correct data logging will not translate into reliable conclusions.
亲自动手推导方程能迫使你识别哪些是控制变量、自变量和因变量。它还会揭示梯度或截距与你试图确定的物理常数之间的联系。缺少这一步，即使数据记录正确，也无法转化为可靠的结论。
2. The Examiner’s Core Message: Linearisation | 考官的核心信息：线性化
The Jan 2021 report stressed that many errors stemmed not from complex algebra, but from a failure to properly linearise a relationship. For example, if theory predicts y = k x², a graph of y against x will be a parabola, not a straight line. To obtain a straight line you must plot y against x², treating x² as the independent variable. In an experiment, you would calculate x² values in a new column of your results table, then plot y against those values. The slope of that line then equals k.
2021年1月的报告强调，许多错误并非源于复杂的代数，而是未能正确将关系线性化。例如，如果理论预测 y = k x²，那么 y 对 x 的图将是抛物线，而不是直线。要得到一条直线，你必须绘制 y 对 x² 的图像，将 x² 视作自变量。在实验中，你应在结果表中新列计算 x² 的值，然后绘制 y 对该列的图像。该直线的斜率就等于 k。
Candidates often misidentify which expression should be used on the horizontal axis. The examiner noted cases where students plotted T against L instead of T² against L when using a pendulum, or s against t instead of s against t² in free fall. These misplots mean that even if the graph is drawn perfectly, the extracted gradient does not give the expected constant.
考生经常错误识别应作为横轴的表达式。考官提到，有学生用单摆时绘制了 T 对 L 而非 T² 对 L 的图像，或在自由落体实验中绘制 s 对 t 而非 s 对 t²。这类绘图错误意味着，即使图像画得完美，得到的梯度也无法给出预期的常数。
3. Worked Example 1: Deriving the Equation for g by Free Fall | 实例1：自由落体测 g 的方程推导
Consider a small steel ball dropped from rest. The distance fallen s is measured for different times t using light gates or a millisecond timer. The motion equation for constant acceleration from rest is s = ut + ½ a t². Since u = 0 and a = g, this simplifies to s = ½ g t². This shows that s is directly proportional to t², not t.
考虑一个小钢球从静止下落。利用光闸或毫秒计时器测量不同时间 t 下落的距离 s。从静止开始的匀加速运动方程为 s = ut + ½ a t²。由于 u = 0 且 a = g，可简化为 s = ½ g t²。这表明 s 与 t² 成正比，而非与 t 成正比。
s = ½ g t²
To linearise, we compare with y = m x + c. Here y is s, x is t², the gradient m = ½ g, and the intercept c = 0 (the line passes through the origin). Therefore, by plotting s against t², you obtain a straight line whose gradient equals ½ g. Doubling this gradient gives the experimental value of g.
为了线性化，我们与标准直线方程 y = m x + c 比较。此处 y 为 s，x 为 t²，斜率 m = ½ g，截距 c = 0（直线过原点）。因此，通过绘制 s 对 t² 的图像，可得到一条直线，其斜率等于 ½ g。将该斜率乘以 2 即可得到 g 的实验值。
The Jan 2021 report confirmed that candidates who correctly derived this linearised form were far more successful in calculating the correct gradient and uncertainty. Those who simply plotted s against t often assumed g was the gradient itself, resulting in a value roughly twice as large.
2021年1月的报告证实，那些正确推导出这一线性形式的考生，在计算正确的斜率和不确定度方面要成功得多。那些直接绘制 s 对 t 图像的考生常常假定 g 就是斜率，结果得到的值大约偏高一倍。
4. Using the Straight-Line Equation to Extract g | 利用直线方程提取 g
Once you have plotted s vs t², select two well-separated points on the best-fit line to calculate the gradient: gradient = Δs / Δ(t²). You must use the values of t², not t. If the line does not pass through the origin due to a small systematic error (e.g., timing delay), you can still find the gradient, but the intercept should be small. The examiner noted that students often forced their line through (0,0) without considering whether the data genuinely support that.
绘制出 s 对 t² 的图像后，在最佳拟合直线上选取两个相距较远的点计算斜率：斜率 = Δs / Δ(t²)。你必须使用 t² 的值，而非 t。假如因为微小的系统误差（例如计时延迟）使得直线不经过原点，你仍然可以得到斜率，但截距应该很小。考官指出，学生常常迫使直线通过 (0,0)，而未考虑数据是否真正支持这一点。
The experimental value of g is then g = 2 × gradient. When calculating percentage uncertainty, remember to double the gradient’s absolute uncertainty, as g = 2m. If the gradient and its uncertainty are m ± Δm, then g = 2m ± 2Δm. The report highlighted that many candidates forgot to propagate the factor of 2 correctly.
那么 g 的实验值为 g = 2 × 斜率。在计算百分不确定度时，记得要将斜率的绝对不确定度加倍，因为 g = 2m。如果斜率及其不确定度为 m ± Δm，那么 g = 2m ± 2Δm。报告强调，许多考生忘记正确传播系数 2。
5. Worked Example 2: Deriving Young’s Modulus from a Stretching Wire | 实例2：从金属丝拉伸推导杨氏模量
In a typical Young’s modulus experiment, a long wire is loaded with forces F, and the extension ΔL is measured using a vernier scale. The original length L and cross-sectional area A (calculated from wire diameter d as A = π d² / 4) are constants. The definition of Young’s modulus E is E = stress / strain = (F/A) / (ΔL/L). Rearranging: ΔL = (L / (A E)) × F. Substituting A gives:
在典型的杨氏模量实验中，给一条长金属丝加载力 F，利用游标尺测量伸长量 ΔL。原始长度 L 和横截面积 A（由金属丝直径 d 计算，A = π d² / 4）均为常数。杨氏模量 E 的定义为 E = 应力 / 应变 = (F/A) / (ΔL/L)。整理得：ΔL = (L / (A E)) × F。代入 A 得到：
ΔL = (4 L / (π d² E)) F
This is the equation ready for linearisation. Compare with y = m x + c: y is ΔL, x is F, gradient m = 4 L / (π d² E), and intercept c = 0 (assuming no slack). Therefore, plot ΔL against F. The gradient allows us to find E:
这就是可以线性化的方程。与 y = m x + c 对比：y 为 ΔL，x 为 F，斜率 m = 4 L / (π d² E)，截距 c = 0（假设不存在松弛）。因此，绘制 ΔL 对 F 的图像。通过斜率可以求得 E：
E = 4 L / (π d² × gradient)
The examiner’s report flagged that many students failed to square the diameter correctly or used radius instead of diameter. Others wrote the final expression for E incorrectly, omitting the factor of 4 or misplacing L. Deriving step by step from the definition minimises such errors.
考官报告指出，许多学生没有正确地对直径进行平方，或者使用了半径而非直径。还有些人将最终的 E 表达式写得不对，遗漏了系数 4 或放错了 L 的位置。从定义出发逐步推导可以最大程度减少这类错误。
6. Handling Compound Variables and Units | 复合变量与单位的处理
When experiments involve quantities like cross-sectional area, density, or multiple terms, treat the entire combination that remains constant as a single constant. For example, in the resistivity experiment, R = ρ L / A. If you vary the length L of a wire and measure resistance R, you keep A constant. The equation becomes R = (ρ / A) L, which is y = m x + c with slope m = ρ / A. You do not need to separate ρ and A until after finding the slope; then calculate ρ = gradient × A.
当实验涉及横截面积、密度等量时，将保持不变的整个组合视为单一常数。例如，在电阻率实验中，R = ρ L / A。若你改变导线的长度 L 并测量电阻 R，则 A 保持不变。方程变为 R = (ρ / A) L，即 y = m x + c，斜率 m = ρ / A。在求得斜率之前，你无需分离 ρ 和 A；然后计算 ρ = 斜率 × A。
The Jan 2021 report observed that candidates often confused what to keep constant and attempted to plot R against 1/A or other variations, leading to unnecessary complexity. Always ask yourself: which variables am I changing deliberately? Which am I measuring? And which are inherent to the apparatus and remain fixed? Your graph axes must reflect the variables you change and measure, with the gradient containing the fixed constants.
2021年1月的报告发现，考生常常混淆应保持不变的量，试图绘制 R 对 1/A 或其他变体，造成不必要的复杂性。总是问自己：哪些变量是我有意改变的？哪些是我测量的？哪些是装置固有的并保持固定？你的坐标轴必须反映你改变与测量的变量，而梯度应包含那些固定常数。
7. Common Mistakes Highlighted in the Jan 2021 Report | Jan21报告中强调的常见错误
The examiner listed several recurring issues that directly undermined derivation marks. One was treating a non-linear relationship as linear without transformation. For instance, when verifying the relationship for a vibrating string, f ∝ √T, some students plotted f against T instead of f² against T or f against √T. Another was confusing direct proportionality with linear proportionality – an equation like y = k/x + c is not a straight line if plotted against x; one would need to plot y against 1/x, but then the line would still have an intercept if c ≠ 0.
考官列出了直接损害推导分数的若干反复出现的问题。其中一个是将非线性关系当作线性关系处理而不进行变换。例如，在验证弦振动关系 f ∝ √T 时，一些学生绘制了 f 对 T 的图像，而非 f² 对 T 或 f 对 √T。另一个是混淆正比与线性关系——像 y = k/x + c 这样的方程，如果对 x 作图并不是一条直线；需要绘制 y 对 1/x 的图像，但如果 c ≠ 0，该直线仍然存在截距。
Additional errors included: using incorrect derived units for the gradient (e.g., quoting g in m s⁻¹ instead of m s⁻²); failing to convert diameter from mm to m before calculating area, causing huge order-of-magnitude errors in E; and not distinguishing between the wire’s extension ΔL and its original length L in the modulus derivation. The report strongly recommends writing down the basic theoretical equation first, then rearranging and substituting with the experimental symbols exactly as they appear in your results table.
其他错误还包括：斜率的导出单位使用不正确（例如，将 g 的单位写作 m s⁻¹ 而非 m s⁻²）；在计算面积之前未能将直径从 mm 转换为 m，导致 E 值出现巨大的数量级误差；在模量推导中未能区分导线的伸长量 ΔL 与其原始长度 L。报告强烈建议先写下基本的理论方程，然后按照结果表中出现的实验符号进行重组和代换。
8. A Step-by-Step Strategy for Exam Success | 考试成功的分步策略
Based on the examiner’s guidance, adopt this systematic approach whenever you are asked to find a linear relationship. First, identify the physical principle and write the fundamental equation (e.g., s = ut + ½ a t²). Second, impose any initial conditions or constants that apply in your experiment (u=0, a=g). Third, rearrange to isolate the measured dependent variable on the left. Fourth, compare with y = m x + c and decide exactly what should be plotted on each axis – write these calculations as new column headings in your experimental plan. Finally, express the gradient and intercept in terms of the desired constants.
根据考官的指导，每当你被要求找出线性关系时，采用以下系统方法。首先，识别物理原理并写出基本方程（例如 s = ut + ½ a t²）。其次，施加实验中适用的任何初始条件或常数（u=0，a=g）。第三，重新整理，将测量到的因变量单独放在左边。第四，与 y = m x + c 比较，并确定每个坐标轴上应绘制的具体内容——将这些计算作为新列的标题写在实验计划中。最后，用目标常数表示斜率和截距。
Practise this with several classic practicals: the pendulum (T² vs L, gradient = 4π²/g), Boyle’s law (p vs 1/V, gradient = constant), capacitor discharge (ln V vs t, gradient = -1/RC), and the radioactive decay analogue (ln count rate vs t). In each case, always note which column you would add to your table of readings. The Jan 2021 report showed that candidates who habitually created a ‘processed data’ column before plotting performed significantly better.
用几个经典实验练习这个方法：单摆（T² 对 L，斜率 = 4π²/g）、玻意耳定律（p 对 1/V，斜率为常数）、电容器放电（ln V 对 t，斜率 = -1/RC）以及放射性衰变的模拟（ln 计数率 对 t）。在每种情况下，都要标注你将添加到读数表中的新列。2021年1月的报告显示，那些在绘图前习惯性创建“处理后数据”列的考生表现明显更好。
9. Incorporating Uncertainty Analysis into Your Derivation | 将不确定度分析融入推导
Derivation is not isolated from uncertainties. When you work out the relationship E = 4 L / (π d² × gradient), you must also be ready to calculate the percentage uncertainty in E. This typically combines the percentage uncertainty in L, twice the percentage uncertainty in d (since d is squared), and the percentage uncertainty in the gradient. The examiner reported that many students attempted to substitute raw uncertainties directly without using the derived formula for propagation, leading to gross over- or underestimation.
推导并非与不确定度无关。当你得出关系式 E = 4 L / (π d² × 斜率) 时，你还必须准备好计算 E 的百分不确定度。这通常需要结合 L 的百分不确定度、d 的百分不确定度的两倍（因为 d 被平方）以及斜率的百分不确定度。考官报告指出，许多学生试图直接代入原始不确定度，而不采用推导出的传播公式，导致严重高估或低估。
Use the standard rule: if a quantity P is in the numerator to power 1, its percentage uncertainty adds once; if raised to power n, it adds n times. In the denominator, it still adds but with the same sign. Treat the gradient as a single measured quantity with its own relative uncertainty. This systematic linking of derivation to uncertainty is essential for the high-scoring practical paper.
使用标准规则：若一个量 P 在分子中且幂次为 1，其百分不确定度加一次；若幂次为 n，则加 n 次。对于分母中的量，加法相同，符号不变。将斜率视为一个单独的测量量，具有其自身的相对不确定度。这种将推导与不确定度系统联系的做法，对于高分的实验试卷至关重要。
10. Summary and Key Takeaways | 总结与关键要点
The January 2021 examiner’s report for International A-Level Physics Unit 3 makes it clear: formula derivation is a skill that must be practised analytically, not memorised rotely. Always start from core physics equations, linearise carefully, and confirm that your chosen axes will yield the constant you need. Common pitfalls like squaring errors, unit conversion oversight, and misidentification of variables can be avoided with a disciplined stepwise method.
2021年1月的国际 A-Level 物理单元3考官报告明确表明：公式推导是一项必须通过分析练习而非死记硬背来掌握的技能。务必从核心物理方程开始，仔细进行线性化，并确认你所选的坐标轴能够给出你需要的常数。只要运用严谨的逐步方法，诸如平方错误、单位换算疏忽和变量误认等常见陷阱均可避免。
Remember the mantra: ‘Compare with y = m x + c, decide what is y, what is x, then the gradient tells me the story.’ By embedding this habit, you will turn derivation from a hurdle into a reliable tool for accuracy, exactly as the examiner recommends.
记住这条法则：“与 y = m x + c 比较，确定 y 是什么、x 是什么，然后斜率会揭示一切。”通过培养这一习惯，你就能将推导从一道障碍转变为提升准确度的可靠工具，这正是考官的期望。
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