📚 A-Level Physics Unit 3 Insert (January 2020): Formula Derivations Explained | A-Level物理Unit 3 (2020年1月) 核心公式推导
The Edexcel International A-Level Physics Unit 3 (WPH13/01) examination assesses practical skills, and the January 2020 Insert provided a reference sheet containing key equations for several standard experiments. Understanding the derivations behind these formulas is essential not only for solving problems but also for justifying experimental procedures and analysing uncertainties. This article unpacks the core derivations that appeared in that Insert, offering clear, step‑by‑step explanations for each relationship.
Edexcel International A-Level物理单元3（WPH13/01）考试考查实验技能，2020年1月的附页给出了一张含有多个标准实验关键公式的参考表。理解这些公式背后的推导，不仅对解题至关重要，也有助于合理解释实验步骤和分析不确定度。本文将逐一解读该附页中出现的核心推导，为每一条关系提供清晰、逐步的说明。
1. Free Fall and the Acceleration of Gravity: s = ½ gt² | 自由落体与重力加速度：s = ½ gt²
e is negligible, the acceleration a is constant and equal to g. The relevant SUVAT equation is s = ut + ½ a t². Since the initial velocity u = 0, this simplifies to s = ½ g t².
在典型的自由落体实验中，一个小而密的物体（通常是钢球）从静止释放，通过已知高度 s 下落。假设空气阻力可忽略，加速度 a 恒定且等于 g。适用的运动学公式为 s = ut + ½ a t²。由于初速度 u = 0，简化为 s = ½ g t²。
e t, then plotting s against t². Because s = (½ g) t², the graph is a straight line through the origin with gradient ½ g. Hence g = 2 × gradient. The uncertainty in g follows from the scatter of points and the gradient uncertainty.
为求出 g，将方程改写为 g = 2s / t²。更可靠的方法是采集多组 s 和相应的下落时间 t，然后作 s 对 t² 的图像。由于 s = (½ g) t²，该图为过原点的直线，斜率为 ½ g。因此 g = 2 × 斜率。g 的不确定度可由散点分布以及斜率的不确定度得到。
2. Resistivity of a Wire: ρ = RA / L | 导线电阻率：ρ = RA / L
llic wire is directly proportional to its length L and inversely proportional to its cross‑ as R = ρ L / A, where ρ is the resistivity of the material. For a wire with circular cross‑ / 4. Substituting gives R = 4 ρ L / (π d²), which can be rearranged to ρ = R π d² / (4 L).
均匀金属导线的电阻 R 与其长度 L 成正比，与横截面积 A 成反比，归纳为 R = ρ L / A，其中 ρ 为材料的电阻率。对直径为 d 的圆形截面导线，面积 A = π d² / 4。代入得 R = 4 ρ L / (π d²)，重排得 ρ = R π d² / (4 L)。
m errors, R is measured for different L, and a graph of R against L is plotted. From R = (ρ / A) L, the gradient m = ρ / A, so ρ = m A. If A is known, ρ can be calculated directly from the gradient.
实验中，R 通过伏特计和安培计测得（R = V / I），d 用千分尺测量。为减少随机误差的影响，可测量不同长度 L 对应的 R，并作 R-L 图。由 R = (ρ / A) L，斜率 m = ρ / A，因此 ρ = m A。若面积 A 已知，即可直接从斜率求出 ρ。
3. Young Modulus from Stress and Strain: E = F L / (A ΔL) | 由应力应变求杨氏模量：E = F L / (A ΔL)
: E = (F / A) / (ΔL / L) = F L / (A ΔL). Here F is the applied force, A the original cross‑sectional area, L the original length, and ΔL the extension produced.
杨氏模量 E 定义为拉应力与拉应变之比：E = (F / A) / (ΔL / L) = F L / (A ΔL)。式中 F 为外加力，A 为原始横截面积，L 为原始长度，ΔL 为产生的伸长量。
scope. For a wire of diameter d, A = π d² / 4, resulting in E = 4 F L / (π d² ΔL). To obtain a reliable value, a graph of F (or mass m) against ΔL is often drawn. Since F = (E A / L) ΔL, the gradient is E A / L, giving E = gradient × L / A. The uncertainty budget originates from the measurement of d, L, and the precision of the extension readings.
在典型实验中，一根细长的金属丝通过添加已知质量被拉伸，伸长量用游标尺或读数显微镜测量。对于直径为 d 的丝，A = π d² / 4，从而 E = 4 F L / (π d² ΔL)。为获得可靠值，通常作作用力 F（或质量 m）对 ΔL 的图像。因 F = (E A / L) ΔL，斜率为 E A / L，故 E = 斜率 × L / A。不确定度预算来源于直径 d、长度 L 的测量以及伸长读数的精度。
4. Internal Resistance of a Cell: ε = V + Ir | 电池内阻：ε = V + Ir
rranging gives V = ε − I r. This is a linear equation of the form y = mx + c, where V is the dependent variable, I is the independent variable, the gradient m = −r, and the y‑intercept c = ε.
对电动势为 ε、内阻为 r 的电源，外接负载 R 时的路端电压 V 满足 ε = I (R + r) = I R + I r = V + I r。重排得 V = ε − I r。这是一个形如 y = mx + c 的线性方程，V 为因变量，I 为自变量，梯度 m = −r，截距 c = ε。
By varying the external resistance and recording corresponding values of V and I, a V‑ inst R is drawn, from ε = I (R + r) we obtain 1/I = (1/ε) R + r/ε. The gradient is 1/ε and the intercept is r/ε, enabling both ε and r to be determined. Both methods appear in the Insert and are frequently examined.
通过改变外电阻并记录相应的 V 和 I 值，可绘制 V-I 图。梯度为负，其绝对值即为 r，纵轴截距即为 ε。另一种方法是作 1/I 对 R 的图像，由 ε = I (R + r) 可得 1/I = (1/ε) R + r/ε。梯度为 1/ε，截距为 r/ε，由此可同时得到 ε 和 r。这两种方法都出现在附页中，是常考内容。
5. Simple Pendulum and the Determination of g: T = 2π √L/g | 单摆与重力加速度测定：T = 2π √L/g
riod T of a simple pendulum of length L is independent of amplitude and is given by T = 2π √L / g. Squaring both sides yields T² = 4π² L / g, which can be rearranged to g = 4π² L / T².
在小角度摆动（通常小于约10°）下，摆长为 L 的单摆周期 T 与振幅无关，且由 T = 2π √L / g 给出。两边平方得 T² = 4π² L / g，重排可得 g = 4π² L / T²。
h is a straight line through the origin with gradient m = 4π² / g. Thus g = 4π² / m. Timings for multiple oscillations (e.g., 20 swings) are used to reduce the random uncertainty in T. The overall percentage uncertainty in g can be estimated from Δg/g = ΔL/L + 2 ΔT/T.
实验中，测量不同摆长 L 对应的周期 T，并作 T²-L 图。该方程具有 T² = (4π² / g) L 的形式，因此图像为过原点的直线，斜率 m = 4π² / g。于是 g = 4π² / m。通常测量多次摆动（例如20次）的时间以减小 T 的随机不确定度。g 的总百分比不确定度可由 Δg/g = ΔL/L + 2 ΔT/T 估算。
6. Speed of Sound in Air Using a Resonance Tube: c = 4 f (l + 0.3d) | 共振管测定空气声速：c = 4 f (l + 0.3d)
A closed‑ e length l of the air column satisfies l + e = n λ / 4, where n is an odd integer (1, 3, 5… ion e is approximately 0.3 times the diameter d of the tube, leading to the practical formula c = 4 f (l + 0.3d).
闭端共振管中，当空气柱长度 l 满足 l + e = n λ / 4（n 为奇数1,3,5…）时会形成驻波，其中 e 为端部修正。对于基频 f，n = 1，则有 l + e = λ / 4。由 c = f λ 知 λ = c / f，因此 c = 4 f (l + e)。端部修正 e 约等于管直径 d 的0.3倍，于是得到实用公式 c = 4 f (l + 0.3d)。
ed to vary l. The first position of resonance gives l₁, and further resonances give l₂, l₃… . Alternatively, if d is measured, c can be obtained from a single resonance using the end‑correction formula. Both approaches are valid and may be referenced in the Insert.
实验中，将已知频率 f 的音叉置于开口端上方，通过调节水面改变 l。第一次共振位置给出 l₁，随后的共振给出 l₂、l₃……由差值 Δl = l₂ − l₁ = λ / 2 可在无需端部修正的情况下求得波长：λ = 2 Δl，因此 c = f × 2 Δl。另一种方法，若测量了直径 d，则可用含端部修正的公式从单次共振求出 c。两种方法均有效，附页中可能提及。
7. Viscosity by Stokes’ Law: Terminal Velocity Equation | 斯托克斯定律测粘度：终端速度方程
e liquid, r is the radius of the sphere, and v is its velocity. At terminal velocity, the net force is zero: weight − buoyancy = drag. Hence (4/3)π r³ ρ₁ g − (4/3)π r³ ρ₂ g = 6π η r v, where ρ₁ is the density of the sphere and ρ₂ is the density of the liquid.
小球在黏性液体中下落时，受到由斯托克斯定律给出的阻力 Fd = 6π η r v，式中 η 为液体的动力黏度，r 为小球半径，v 为下落速度。在终端速度下，合力为零：重力减去浮力等于阻力。因此 (4/3)π r³ ρ₁ g − (4/3)π r³ ρ₂ g = 6π η r v，其中 ρ₁ 为球的密度，ρ₂ 为液体的密度。
(9 v). If the terminal velocity v is measured by timing the sphere over a known distance once the motion is uniform, and r and the densities are known, η can be calculated. The Insert may provide this formula directly in the form η = 2r²g(ρs−ρl)/9v, where ρs and ρl denote sphere and liquid densities.
化简得 (4/3)π r³ g (ρ₁ − ρ₂) = 6π η r v。约去公因子 π 和 r（r ≠ 0）得 (4/3) r² g (ρ₁ − ρ₂) = 6 η v。解出 η：η = 2 r² g (ρ₁ − ρ₂) / (9 v)。若通过小球在匀速段通过已知距离的时间来测量终端速度 v，且 r 和密度已知，便可计算 η。附页可能直接以 η = 2r²g(ρs−ρl)/9v 的形式给出该公式，其中 ρs 和 ρl 分别表示球和液体的密度。
8. Propagation of Uncertainties for Derived Quantities | 导出物理量的不确定度传递
ion (z = x + y or z = x − y), the absolute uncertainties add: Δz = Δx + Δy. For multiplication or division (z = x y or z = x / y), the fractional (or percentage) uncertainties add: Δz / z = Δx / x + Δy / y. For a power relationship (z = xⁿ), the fractional uncertainty becomes Δz / z = |n| Δx / x.
当一个物理量由测量变量计算得出时，其不确定度由各输入量的不确定度决定。对于加减运算（z = x + y 或 z = x − y），绝对不确定度相加：Δz = Δx + Δy。对于乘除运算（z = x y 或 z = x / y），相对（或百分比）不确定度相加：Δz / z = Δx / x + Δy / y。对于幂函数（z = xⁿ），相对不确定度变为 Δz / z = |n| Δx / x。
age uncertainties can be combined to find the overall uncertainty in g. Similarly, for resistivity ρ = RA / L, the uncertainty is built from ΔR, ΔA (which depends on Δd, since A = π d²/4, giving ΔA/A = 2 Δd/d), and ΔL. The Insert often expects students to derive or use such uncertainty relations to comment on the largest source of error.
以单摆方程 g = 4π² L / T² 为例。g 的相对不确定度为 Δg / g = ΔL / L + 2 ΔT / T。若 L 测量至 ±1 mm，T 至 ±0.01 s，可将百分比不确定度组合得出 g 的总不确定度。类似地，对于电阻率 ρ = RA / L，不确定度由 ΔR、ΔA（依赖于 Δd，因 A = π d²/4，故 ΔA/A = 2 Δd/d）和 ΔL 构成。附页常期望学生推导或使用这样的不确定度关系，以判断最大误差来源。
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