International A-Level Physics PH03 Unit 3 Formula Derivation Guide | 国际 A-Level 物理 PH03 单元三公式推导指南

📚 International A-Level Physics PH03 Unit 3 Formula Derivation Guide | 国际 A-Level 物理 PH03 单元三公式推导指南

In Edexcel International A-Level Physics Unit 3 (PH03), practical skills and data analysis are central. Mastering the derivations of uncertainty formulas, linearisation techniques, and graphical analysis is essential for high-scoring responses. This guide walks through key formula derivations with clear steps, helping you understand how experimental uncertainties combine and how to extract physical constants from linearised graphs.

在 Edexcel 国际 A-Level 物理单元三 (PH03) 中，实验技能与数据分析是核心。掌握不确定度公式的推导、线性化技巧以及图像分析方法，是获得高分的关键。本文带你逐步推导关键公式，帮助你理解实验不确定度如何合成，以及如何从线性化图像中提取物理常数。

1. Combining Uncertainties: Addition and Subtraction | 不确定度组合：加减法

When two measured quantities A and B are added or subtracted to give Q = A + B or Q = A − B, the absolute uncertainties ΔA and ΔB combine. If the errors are independent and random, the standard formula uses the root-sum-square (RSS) method: ΔQ = √[(ΔA)² + (ΔB)²]. This arises because the variance of a sum of independent variables equals the sum of their variances.

当两个测量量 A 和 B 相加或相减得到 Q = A + B 或 Q = A − B 时，绝对不确定度 ΔA 和 ΔB 会合成。若误差是独立且随机的，标准方法采用方和根 (RSS) 法：ΔQ = √[(ΔA)² + (ΔB)²]。这是因为独立变量之和的方差等于各自方差之和。

A simpler worst-case estimate often used in quick checks is to add the absolute uncertainties linearly: ΔQ ≈ ΔA + ΔB. However, for PH03 analysis involving multiple measurements, the quadrature sum is expected unless the question specifies a ‘worst-case’ approach.

在快速检验中常用的更简单的保守估计，是直接将绝对不确定度线性相加：ΔQ ≈ ΔA + ΔB。但在 PH03 涉及多次测量的分析中，除非题目明确要求”最坏情况”，否则应使用平方和开方。

2. Combining Uncertainties: Multiplication and Division | 不确定度组合：乘除法

For Q = A × B or Q = A / B, it is the relative (fractional) uncertainties that combine simply. The general error propagation formula for independent variables yields (ΔQ/Q)² = (ΔA/A)² + (ΔB/B)², so ΔQ = Q √[(ΔA/A)² + (ΔB/B)²]. In percentage terms, the percentage uncertainty in Q is the square root of the sum of squares of the percentage uncertainties in A and B.

对于 Q = A × B 或 Q = A / B，相对（分数）不确定度的合成更为简便。独立变量的通用误差传播公式给出 (ΔQ/Q)² = (ΔA/A)² + (ΔB/B)²，因此 ΔQ = Q √[(ΔA/A)² + (ΔB/B)²]。以百分比表示，即 Q 的百分比不确定度等于 A 和 B 的百分比不确定度平方和的平方根。

To see the origin, consider Q = A / B. The partial derivatives are ∂Q/∂A = 1/B and ∂Q/∂B = −A/B². Squaring and multiplying by the variances gives (ΔQ)² = (1/B²)(ΔA)² + (A²/B⁴)(ΔB)². Dividing by Q² = A²/B² returns the fractional uncertainty formula above.

溯源来看，对 Q = A / B 求偏导得 ∂Q/∂A = 1/B，∂Q/∂B = −A/B²，平方后乘以方差可得 (ΔQ)² = (1/B²)(ΔA)² + (A²/B⁴)(ΔB)²。两边除以 Q² = A²/B² 便得到上述相对不确定度公式。

3. Uncertainty in a Power Law Q = k Aⁿ | 幂函数 Q = k Aⁿ 的不确定度

When a quantity is raised to a power n, the relative uncertainty scales by the magnitude of that power. For Q = k Aⁿ where k is a constant and ΔA is the uncertainty in A, the propagation gives ΔQ/Q = |n| (ΔA/A). This follows from taking the logarithmic differential: d(ln Q) = n d(ln A), so dQ/Q = n dA/A.

当物理量被升幂 n 时，相对不确定度会被该幂的绝对值所放大。对于 Q = k Aⁿ（k 为常数，A 的不确定度为 ΔA），传播公式为 ΔQ/Q = |n| (ΔA/A)。这源自对数微分：d(ln Q) = n d(ln A)，从而 dQ/Q = n dA/A。

In practice, if you measure a diameter d to find volume V = (4/3)π(d/2)³, the relative uncertainty in V is three times the relative uncertainty in diameter: ΔV/V = 3 (Δd/d). The factor arises because radius is proportional to diameter, and volume depends on radius cubed.

实际应用中，若测量直径 d 以求体积 V = (4/3)π(d/2)³，V 的相对不确定度便是直径相对不确定度的三倍：ΔV/V = 3 (Δd/d)。该因子之所以出现，是因为体积依赖于半径的三次方。

4. Linearisation of Non-Linear Relationships | 非线性关系的线性化

Many physical laws are not linear but can be transformed into a straight-line form. The pendulum equation T = 2π√(L/g) becomes T² = (4π²/g)L, giving a straight line through origin when plotting T² against L. For a power law y = k xⁿ, taking base-10 logarithms yields log₁₀ y = n log₁₀ x + log₁₀ k, which matches Y = m X + c with slope n and intercept log₁₀ k.

许多物理定律并非线性，但可通过变换转化为直线形式。单摆公式 T = 2π√(L/g) 化为 T² = (4π²/g)L，作 T²-L 图可得一条过原点的直线。对于幂函数 y = k xⁿ，取常用对数得 log₁₀ y = n log₁₀ x + log₁₀ k，即直线形式 Y = m X + c，其中斜率 m = n，截距 c = log₁₀ k。

Exponential relationships such as I = I₀ e⁻⁴λt are linearised by taking the natural logarithm: ln I = ln I₀ − λt. The gradient of the ln I vs t graph equals −λ. Linearisation not only simplifies the determination of constants but also makes uncertainty analysis more straightforward by allowing the use of linear regression and graphical max/min slope methods.

指数关系如 I = I₀ e⁻⁴λt 可通过取自然对数线性化：ln I = ln I₀ − λt。ln I-t 图的斜率即为 −λ。线性化不仅简化了常数的测定，

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