量纲、估算与不确定度Units, estimation & uncertainty
这些方法在 Section 1 中反复出现，常穿插在力学或电学情境中。本组选题每题 2–6 分，重点防止单位换算、百分比误差和微分符号失分。五题共 20 分；自动评分用于练习，不代替正式分步评分。These methods recur in Section 1, often within mechanics or electricity. This selection spans 2–6 marks per question, targeting unit conversions, percentage uncertainties and derivative signs. Total: 20 marks; automatic practice scoring is not an official marking scheme.
1量纲匹配与数量级Dimensions and orders of magnitude
把每个量拆成 M、L、T 的幂，同一种基本量的指数必须相等。估算先声明几何尺寸和材料参数，再逐步计算。Express each quantity as powers of M, L and T and equate each exponent. For estimates, state dimensions and material assumptions before calculating.
[E] = ML²T⁻²; [ρ] = ML⁻³; [g] = LT⁻²; m = σA量纲只能确定幂次，不能确定无量纲系数。Dimensions determine powers, but cannot determine a dimensionless coefficient.
2百分比不确定度与极限斜率Percentage uncertainty and limiting gradients
乘除与幂次的最大相对不确定度按绝对贡献相加。图像法用穿过误差范围的最陡与最缓直线；平方和开方则需要独立随机标准差的假设。For a worst-case bound, add the absolute fractional contributions of products, quotients and powers. On graphs, use steepest and shallowest allowed lines; quadrature instead requires independent random standard uncertainties.
y = Axᵖzᑫ ⇒ Δy/|y| ≈ |p|Δx/|x| + |q|Δz/|z|小误差的一阶近似；也可直接代入最大、最小值检查。A first-order approximation for small errors; direct extrema provide a check.
3变化率、投影与曲线下面积Rates, projections and areas
求瞬时变化率用微分；求累计量用积分。角度参与微分时必须使用弧度，积分还需要明确起止时刻。Differentiate for instantaneous rates and integrate for accumulated quantities. Use radians in angular derivatives and specify integration limits.
v = dx/dt; I = dQ/dt; Q = ∫ I dt图像的斜率是变化率，图像下面积是累计量。A gradient gives a rate; an area gives an accumulated quantity.
| 子题型Sub-type | 识别特征Recognise | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 量纲幂次Dimension powers | d = kEᵅρᵝgᵞd = kEᵅρᵝgᵞ | 匹配三种指数Match three exponents | 2020 Q1(m) |
| 费米估算Fermi estimate | EstimateEstimate | 面积 × 面密度Area × grammage | 2019 Q1(c) |
| 不确定度Uncertainty | ±± | 比值与极限斜率Ratio and limiting gradients | 2015 Q1(a) |
| 积分Integration | I(t)I(t) | 电流图像的面积Area under current | 2024 Q1(n) |
| 关联变化率Related rates | dθ/dtdθ/dt | 先列几何关系再微分Geometry then differentiation | 2021 Q1(i) |