📚 Deriving Key Formulas from the AS Physics Unit 3 Insert (June 2019) | AS物理Unit 3公式推导（2019年6月）
The AS Physics Unit 3 exam tests practical skills, experimental techniques, and the ability to handle uncertainties. The June 2019 data sheet insert provided a concise set of equations for calculating absolute and percentage uncertainties, combining measurements, and analysing graphical data. While you are allowed to refer to these formulas during the exam, working through their derivations will build a deeper conceptual understanding and make it easier to apply them in unfamiliar contexts. This article walks you through the step-by-step derivation of each key formula from that insert.
AS物理第三单元考试考查实验技能、数据处理以及不确定度的处理能力。2019年6月考试提供的公式表给出了一套用于计算绝对不确定度与相对不确定度、合成测量值和分析图表数据的核心公式。虽然考试时你可以直接查阅这些公式，但亲自推导它们能够加深你对其概念的理解，并让你在陌生情境中更自如地应用它们。本文将带你一步步推导该公式表上的每一个关键公式。
1. Absolute and Percentage Uncertainty | 绝对不确定度与相对不确定度
Every measured quantity x has an absolute uncertainty Δx, which represents the range within which the true value is likely to lie. If a repeated measurement gives a mean value x̄, then the absolute uncertainty tells you the margin of error: the value is reported as x̄ ± Δx. For a single reading, Δx is typically the instrument precision or half the smallest division.
每一个被测量 x 都有绝对不确定度 Δx，它表示真实值可能存在的范围。如果重复测量的平均值为 x̄，那么绝对不确定度就给出了误差范围：测量结果应表示为 x̄ ± Δx。对于单次读数，Δx 通常取仪器精度或最小分度值的一半。
The percentage uncertainty quantifies the relative size of the uncertainty compared to the measured value. It is defined as (absolute uncertainty divided by the mean) multiplied by 100%.
相对不确定度（常用百分数表示）衡量的是不确定度相对于实测值的大小。其定义为绝对不确定度除以平均值再乘以100%。
Percentage uncertainty = (Δx / x̄) × 100%
This simple ratio allows you to compare the precision of different measurements. For example, a 1 mm uncertainty on a 50 mm length (2%) is far more significant than the same 1 mm on a 500 mm length (0.2%).
这一比值让你可以比较不同测量结果的精确度。比如，长度为50 mm时1 mm的不确定度（2%）就要比长度为500 mm时同样的1 mm不确定度（0.2%）大得多。
2. Deriving Uncertainty for Sums and Differences | 加减法的不确定度推导
When two quantities A and B are added, C = A + B. To find the worst-case uncertainty in C, consider the extreme values: the maximum possible C occurs when both A and B are at their maximum, giving Cmax = (A+ΔA) + (B+ΔB) = A+B+ΔA+ΔB. The minimum possible C occurs when both are at their minimum: Cmin = (A-ΔA) + (B-ΔB) = A+B-ΔA-ΔB. The spread is 2(ΔA+ΔB), so the absolute uncertainty in C is half the spread: ΔC = ΔA + ΔB.
当两个量 A 和 B 相加时，C = A + B。要找出 C 的最不利情况不确定度，考虑极值：当 A 和 B 都取最大值时 C 最大，Cmax = (A+ΔA) + (B+ΔB) = A+B+ΔA+ΔB；当两者都取最小值时 C 最小，Cmin = (A-ΔA) + (B-ΔB) = A+B-ΔA-ΔB。数据的散布范围为 2(ΔA+ΔB)，因此 C 的绝对不确定度是散布范围的一半：ΔC = ΔA + ΔB。
For subtraction, C = A – B, the maximum C occurs when A is largest and B is smallest: Cmax = (A+ΔA) – (B-ΔB) = A-B+ΔA+ΔB. The minimum C occurs when A is smallest and B is largest: Cmin = (A-ΔA) – (B+ΔB) = A-B-ΔA-ΔB. Again the spread is 2(ΔA+ΔB), giving the same result: ΔC = ΔA + ΔB.
对于减法 C = A – B，当 A 最大而 B 最小时 C 取最大值：Cmax = (A+ΔA) – (B-ΔB) = A-B+ΔA+ΔB；当 A 最小而 B 最大时 C 取最小值：Cmin = (A-ΔA) – (B+ΔB) = A-B-ΔA-ΔB。同样散布范围为 2(ΔA+ΔB)，因此结果不变：ΔC = ΔA + ΔB。
ΔC = ΔA + ΔB (for both C = A + B and C = A – B)
This simple addition of absolute uncertainties is a worst-case estimate. It assumes that the errors in A and B are perfectly correlated in the direction that maximises C. In practice, if the errors are independent, a root-sum-square combination may be used, but the exam insert typically presents this arithmetic addition rule for AS level.
不确定度绝对值直接相加是一种最保守的估计。它假设 A 和 B 的误差方向完全一致，从而使 C 取到极大值。实际操作中，若误差相互独立，可采用平方和开根号的方法，但AS阶段考试给出的公式表通常使用这种算术相加规则。
3. Deriving Uncertainty for Products and Quotients | 乘除法的不确定度推导
For a product C = A × B, the maximum value of C, ignoring the very small term ΔAΔB, is Cmax ≈ (A+ΔA)(B+ΔB) = AB + AΔB + BΔA + ΔAΔB ≈ AB + AΔB + BΔA. The minimum value, again neglecting ΔAΔB, is Cmin ≈ AB – AΔB – BΔA. The absolute uncertainty in C is then approximately half the spread: ΔC ≈ AΔB + BΔA. Dividing by C = AB gives the relative uncertainty:
对于乘积 C = A × B，略去高阶小量 ΔAΔB，C 的最大值近似为 Cmax ≈ (A+ΔA)(B+ΔB) = AB + AΔB + BΔA + ΔAΔB ≈ AB + AΔB + BΔA。最小值 Cmin ≈ AB – AΔB – BΔA。因此 C 的绝对不确定度大约为 ΔC ≈ AΔB + BΔA。除以 C = AB 即得相对不确定度：
ΔC/C = (AΔB + BΔA) / (AB) = ΔA/A + ΔB/B
This shows that the relative (or percentage) uncertainties add when quantities are multiplied. For the quotient C = A / B, we can use a similar approach. The maximum C is approximately (A+ΔA)/(B-ΔB), which can be expanded for small uncertainties: (A/B) × (1+ΔA/A) × (1+ΔB/B) ≈ (A/B)(1 + ΔA/A + ΔB/B). The minimum C gives (A/B)(1 – ΔA/A – ΔB/B). Hence the relative uncertainty in C is again ΔA/A + ΔB/B.
这表明乘法运算时相对（或百分比）不确定度是相加的。对于除法 C = A / B，可用类似方法处理。C 的最大值近似为 (A+ΔA)/(B-ΔB)，对微小不确定度展开：(A/B)×(1+ΔA/A)×(1+ΔB/B) ≈ (A/B)(1 + ΔA/A + ΔB/B)。最小值则为 (A/B)(1 – ΔA/A – ΔB/B)。因此 C 的相对不确定度同样是 ΔA/A + ΔB/B。
For C = A × B or C = A / B: %eC = %eA + %eB
In other words, the percentage uncertainty of a product or quotient is simply the sum of the individual percentage uncertainties. This rule is extremely useful for propagating errors through complex equations.
换句话说，乘积或商值的百分比不确定度等于各个因子百分比不确定度的代数和。这条规则在复杂公式中进行误差传递时非常实用。
4. Deriving Uncertainty for Powers | 幂函数的不确定度推导
Consider a quantity C = Aⁿ, where n is a constant (which may be fractional or negative). If A has a small relative uncertainty ε = ΔA/A, then we can write A ± ΔA as A(1 ± ε). Using the binomial expansion (1 ± ε)ⁿ ≈ 1 ± nε for small ε, the maximum value of C is approximately Aⁿ (1 + nε) and the minimum is Aⁿ (1 – nε). The absolute uncertainty is therefore Aⁿ × nε = n Aⁿ⁻¹ ΔA.
考虑函数 C = Aⁿ，其中 n 为常数（可以是分数或负数）。若 A 的相对不确定度 ε = ΔA/A 很小，可将 A ± ΔA 写为 A(1 ± ε)。利用二项式展开近似 (1 ± ε)ⁿ ≈ 1 ± nε（当 ε 很小时成立），C 的最大值约为 Aⁿ(1 + nε)，最小值约为 Aⁿ(1 – nε)。因此绝对不确定度为 Aⁿ × nε = n Aⁿ⁻¹ ΔA。
Dividing by C = Aⁿ gives the relative uncertainty directly: ΔC/C ≈ n ε = n (ΔA/A). Hence the percentage uncertainty in C is |n| times the percentage uncertainty in A (the absolute value ensures the rule works for any real n).
除以 C = Aⁿ 即得相对不确定度：ΔC/C ≈ n ε = n (ΔA/A)。因此 C 的百分比不确定度为 |n| 乘以 A 的百分比不确定度（取绝对值确保对任何实数 n 都适用）。
%eC = |n| × %eA for C = Aⁿ
This rule reveals that powers amplify the relative uncertainty: squaring a measurement doubles the percentage uncertainty, while taking the square root halves it. It also explains why measurements with high powers need to be particularly precise.
这一规则表明，幂运算会放大相对不确定度：求平方会使百分比不确定度翻倍，而开平方根则会减半。这也解释了为何高次幂的测量需要特别精确。
5. Uncertainty in Mean Values from Repeated Measurements | 重复测量的平均值不确定度
When you take several readings of the same quantity, random errors can be reduced by calculating the arithmetic mean x̄ = (Σxi)/N. The spread of the data gives an indication of the remaining uncertainty. At AS level, a straightforward method provided on the insert is to use half the range: the absolute uncertainty in the mean is estimated as (largest reading – smallest reading) / 2. This works well for small sets of data and is quick to apply.
当你对同一物理量进行多次测量时，随机误差可以通过计算算术平均值 x̄ = (Σxi)/N 来减小。数据的离散程度则反映了剩余的不确定度。在AS阶段，公式表提供了一种直接的方法——使用半距法：平均值的绝对不确定度估计为（最大读数 − 最小读数）/ 2。这种方法适用于小样本数据，且计算快捷。
Δxmean = (xmax – xmin) / 2
For example, if five measurements of a length are 12.1, 12.3, 12.0, 12.2, and 12.1 cm, the mean is 12.14 cm, and the range is 0.3 cm, so the uncertainty is ±0.15 cm. The final result is quoted as 12.14 ± 0.15 cm. This method is deliberately simple and gives a conservative estimate of the random error.
例如，若五次长度测量值为 12.1, 12.3, 12.0, 12.2, 12.1 cm，平均值为 12.14 cm，极差为 0.3 cm，则不确定度为 ±0.15 cm。最终结果可表示为 12.14 ± 0.15 cm。这一方法有意简化，能够保守地估计随机误差。
6. Using Max-Min Lines for Gradient Uncertainty | 使用最大最小梯度线求不确定度
In many practical investigations, you plot a straight-line graph and determine the gradient m of the line of best fit. The uncertainty in m arises because the data points are scattered. To quantify this, you draw two additional lines: the steepest reasonable line (maximum gradient mmax) and the shallowest reasonable line (minimum gradient mmin) that still pass through most of the error bars. These are often called worst-fit lines.
许多实验探究中，你需要绘制直线图并确定最佳拟合线的斜率 m。由于数据点存在分散性，梯度 m 也因此具有不确定度。量化这一不确定度的方法是额外绘制两条直线：一条为仍能通过大多数误差棒的、最陡的合理直线（最大梯度 mmax），另一条为最平缓的合理直线（最小梯度 mmin）。这两条线通常被称为最不利拟合线。
The absolute uncertainty in the gradient is then taken as half the difference between these two extreme gradients:
梯度绝对不确定度则取这两个极端梯度差的一半：
Δm = (mmax – mmin) / 2
This method assumes that the true gradient could lie anywhere between the two extremes with equal likelihood. The percentage uncertainty is calculated as (Δm / m) × 100%. The insert may simply remind you that ‘uncertainty in gradient = (max gradient – min gradient)/2’ without further derivation, but you can see it follows from the same half-range logic used for a single variable.
该方法假设真实梯度以相同的可能性落在两个极值之间的任意位置。百分比不确定度则按 (Δm / m) × 100% 计算。公式表可能直接写出“梯度不确定度 = (最大梯度 − 最小梯度)/2”，尽管未给出推导，但你能看出它源于与
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