📚 IB Physics: Essentials of Measurement and Mathematical Foundations | IB物理：测量与数学基础要点
Physics is an experimental science, and every measurement carries meaning only when it is accompanied by its uncertainty and unit. This article reviews the core ideas of measurement and the mathematical tools you will use constantly in IB Physics.
物理是一门实验科学，任何测量只有带上不确定度和单位才有意义。本文系统梳理IB物理中反复使用的测量核心概念与数学工具，帮助你在计算、作图和实验分析中打下坚实基础。
1. SI Units and Prefixes | SI单位与词头
The International System of Units (SI) defines seven base quantities. In IB Physics, the most important base units are metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), and mole (mol).
国际单位制（SI）定义了七个基本量。在IB物理中最重要的是米（m）、千克（kg）、秒（s）、安培（A）、开尔文（K）和摩尔（mol）。
Prefixes change units by powers of ten. Common ones include nano (n, 10⁻⁹), micro (μ, 10⁻⁶), milli (m, 10⁻³), centi (c, 10⁻²), kilo (k, 10³), mega (M, 10⁶), and giga (G, 10⁹).
词头以十的幂次改变单位。常用词头包括纳（n，10⁻⁹）、微（μ，10⁻⁶）、毫（m，10⁻³）、厘（c，10⁻²）、千（k，10³）、兆（M，10⁶）和吉（G，10⁹）。
1 μs = 10⁻⁶ s, 1 nm = 10⁻⁹ m, 1 MW = 10⁶ W
Always convert prefixes to base units before substituting into equations.
代入方程前，务必将词头换算为基本单位。
2. Significant Figures and Scientific Notation | 有效数字与科学记数法
Significant figures reflect the precision of a measurement. For example, 3.20 cm has three significant figures, while 0.032 m has only two (leading zeros are not significant).
有效数字反映测量的精确程度。例如，3.20 cm有三位有效数字，而0.032 m只有两位（前导零不算有效）。
Scientific notation removes ambiguity: write 3.20 × 10⁻² m instead of 0.0320 m when the zero is significant.
科学记数法可消除歧义：若末尾的零有效，应写为3.20 × 10⁻² m，而不是0.0320 m。
When multiplying or dividing, the result should have the same number of significant figures as the factor with the fewest. When adding or subtracting, round to the least precise decimal place.
乘除运算时，结果的有效数字位数应与有效数字最少的因子相同；加减运算时，则按小数位最少的数据进行四舍五入。
2.5 × 3.141 = 7.9 (two significant figures)
3. Uncertainty and Error | 不确定度与误差
Uncertainty (or absolute uncertainty) is the range within which the true value is expected to lie. It is written as x ± Δx, for example, 5.0 ± 0.2 cm.
不确定度（或绝对不确定度）是真值可能落入的范围，记作 x ± Δx，例如 5.0 ± 0.2 cm。
Systematic error shifts readings consistently in one direction and affects accuracy. Random error causes scatter and affects precision.
系统误差使读数朝同一方向偏移，影响准确度；随机误差造成数据离散，影响精确度。
For a single reading, the uncertainty is often taken as half the smallest scale division. For repeated measurements, use half the range or calculate the standard deviation of the mean.
单次测量的不确定度通常取最小分度的一半；多次测量时，可用极差的一半或计算平均值的标准偏差。
Relative uncertainty = (Δx / x) × 100%
4. Propagating Uncertainties | 不确定度的传播
When adding or subtracting quantities, add absolute uncertainties: (a ± Δa) + (b ± Δb) = (a + b) ± (Δa + Δb).
加减运算时，绝对不确定度相加：(a ± Δa) + (b ± Δb) = (a + b) ± (Δa + Δb)。
When multiplying or dividing, add fractional (relative) uncertainties: if y = a × b / c, then Δy/y = Δa/a + Δb/b + Δc/c.
乘除运算时，相对不确定度相加：若 y = a × b / c，则 Δy/y = Δa/a + Δb/b + Δc/c。
For powers, multiply the relative uncertainty by the exponent: if y = aⁿ, then Δy/y = n × (Δa/a).
幂运算时，相对不确定度乘以指数：若 y = aⁿ，则 Δy/y = n × (Δa/a)。
Example: R = V/I, ΔR/R = ΔV/V + ΔI/I
5. Scalars and Vectors | 标量与矢量
A scalar has magnitude only: mass, time, energy, and speed are scalars. A vector has both magnitude and direction: displacement, velocity, force, and momentum are vectors.
标量只有大小：质量、时间、能量和速率都是标量。矢量既有大小又有方向：位移、速度、力和动量都是矢量。
Distance is the total path length, while displacement is the straight-line change in position with direction.
路程是路径总长度，位移是从起点到终点的有向直线距离。
Speed = |velocity|, but average speed and average velocity are generally different.
6. Vector Resolution and Addition | 矢量的分解与合成
To add vectors, resolve each into perpendicular components using trigonometry, then add components separately.
进行矢量合成时，先用三角函数把每个矢量分解为垂直分量，然后分别对分量求和。
For a vector A at angle θ to the x-axis:
对于与x轴夹角为θ的矢量A：
Aₓ = A cos θ, Aᵧ = A sin θ
Resultant magnitude = √(ΣAₓ² + ΣAᵧ²), direction = tan⁻¹(ΣAᵧ / ΣAₓ)
When two vectors are perpendicular, the magnitude of the resultant is found by the Pythagorean theorem.
当两个矢量垂直时，合矢量大小可用勾股定理求得。
7. Graphing and Linearization | 绘图与直线化
Graphs in IB Physics should have the independent variable on the x-axis and the dependent variable on the y-axis. Always include units and error bars.
IB物理绘图中，自变量放在x轴，因变量放在y轴，并始终标注单位与误差棒。
The slope of a straight-line graph often represents a physical quantity. For example, in a displacement-time graph, the slope gives velocity.
直线图的斜率通常代表一个物理量。例如，在位移-时间图中，斜率表示速度。
Many non-linear relationships can be linearized by changing variables. For T = 2π√(L/g), plot T² against L to obtain a straight line with slope 4π²/g.
许多非线性关系可以通过变量替换转化为线性。对于 T = 2π√(L/g)，可作 T² 与 L 的图，得到斜率为 4π²/g 的直线。
y ∝ xⁿ → plot log y against log x, slope = n
8. Trigonometry in Physics | 物理中的三角函数
Trigonometric functions connect angles and side ratios in right-angled triangles:
三角函数把直角三角形中的角与边之比联系起来：
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Small-angle approximations are useful in wave and pendulum contexts: for θ in radians, sin θ ≈ θ, tan θ ≈ θ, and cos θ ≈ 1 − θ²/2.
小角近似在波动和单摆问题中非常有用：当θ以弧度表示时，sin θ ≈ θ，tan θ ≈ θ，cos θ ≈ 1 − θ²/2。
In projectile motion, the initial velocity components are v₀ₓ = v₀ cos θ and v₀ᵧ = v₀ sin θ.
在抛体运动中，初速度分量为 v₀ₓ = v₀ cos θ，v₀ᵧ = v₀ sin θ。
9. Logarithms and Exponentials | 对数与指数
Logarithms are essential for handling quantities that span many orders of magnitude, such as sound intensity or radioactive decay.
对数用于处理跨越多个数量级的物理量，比如声强或放射性衰变。
The decibel scale is defined as β = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W m⁻².
分贝标度定义为 β = 10 log₁₀(I/I₀)，其中 I₀ = 10⁻¹² W m⁻²。
Exponential decay follows the form N = N₀ e⁻λᵗ, which can be linearized by taking natural logarithms:
指数衰减满足 N = N₀ e⁻λᵗ，两侧取自然对数可将其线性化：
ln N = ln N₀ − λt
Thus, a graph of ln N versus t gives a straight line with slope −λ.
因此，作 ln N 与 t 的图，得到斜率为 −λ 的直线。
10. Estimation and Order-of-Magnitude | 估算与数量级
Estimation skills are tested in IB Physics through Fermi problems. You should be able to estimate quantities using reasonable assumptions and one or two significant figures.
IB物理通过费米问题考查估算能力。你需要基于合理假设，用一两位有效数字估算物理量。
For example, estimate the number of seconds in a year: 365 × 24 × 3600 ≈ 3 × 10⁷ s.
例如，估算一年有多少秒：365 × 24 × 3600 ≈ 3 × 10⁷ 秒。
Order-of-magnitude calculations help you judge whether an answer is physically sensible and are vital in multiple-choice questions that ask “approximately”.
数量级计算能帮助你判断答案是否物理上合理，在要求“约等于”的选择题中尤为重要。
Mass of an atom ≈ 10⁻²⁶ kg, diameter of nucleus ≈ 10⁻¹⁴ m
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