📚 Mastering Derivations in AS Physics Unit 2 (Jan 2022) | AS物理第2单元公式推导精讲（2022年1月）
The AS Physics Unit 2 paper from January 2022 demands not only recall of key equations but also the ability to derive them from fundamental principles. This article breaks down the essential derivations that frequently appear in mechanics, materials, and waves, linking each step to the reasoning expected by examiners. By mastering these derivations, students can confidently tackle the structured questions that test both mathematical fluency and conceptual understanding.
2022年1月的AS物理第2单元试卷不仅要求记住关键公式，还要求能够从基本原理推导它们。本文详细剖析力学、材料与波中频繁出现的核心推导，将每一步推理与考官期望的逻辑链连接起来。掌握这些推导后，学生便能有信心地应对那些既考察数学流畅度又考察概念理解的试题。
1. Deriving the SUVAT Equations from Definitions | 从定义出发推导匀加速运动方程
Start with the definition of acceleration: a = (v – u)/t. Rearranging gives v = u + at. This is the first SUVAT equation. Next, average velocity is defined as (u + v)/2 when acceleration is constant. Displacement s = average velocity × time, so s = ((u + v)/2) t. Substituting v = u + at into this yields s = ut + ½at². Squaring v = u + at and combining with s = ((u + v)/2) t gives v² = u² + 2as. These derivations rely solely on the definitions of acceleration and average velocity under constant acceleration.
从加速度的定义 a = (v – u)/t 开始。变形得 v = u + at，这是第一个匀加速方程。接下来，匀加速时平均速度定义为 (u + v)/2。位移 s = 平均速度 × 时间，故 s = ((u + v)/2) t。把 v = u + at 代入得到 s = ut + ½at²。将 v = u + at 平方并结合 s = ((u + v)/2) t 可得 v² = u² + 2as。这些推导完全建立在匀加速条件下加速度和平均速度的定义之上。
2. Deriving the Horizontal Range of a Projectile | 推导抛体水平射程公式
Consider a projectile launched with speed u at angle θ to the horizontal. Resolve velocity: horizontal component u cosθ, vertical component u sinθ. Time of flight T is found from vertical motion: s_y = u sinθ · T – ½g T². Setting s_y = 0 (return to same level) gives T = (2u sinθ)/g. The horizontal range R = u cosθ · T = u cosθ · (2u sinθ)/g = (u² sin2θ)/g. This assumes no air resistance and constant g. The derivation highlights the independence of horizontal and vertical motions.
考虑一物体以速率 u、与水平方向成 θ 角抛出。分解速度：水平分量 u cosθ，竖直分量 u sinθ。通过竖直运动求飞行时间 T：s_y = u sinθ · T – ½g T²。令 s_y = 0（落回原高度），得 T = (2u sinθ)/g。水平射程 R = u cosθ · T = u cosθ · (2u sinθ)/g = (u² sin2θ)/g。推导假设无空气阻力且重力加速度 g 恒定，显示了竖直运动与水平运动的独立性。
3. Deriving Conservation of Momentum from Newton’s Laws | 由牛顿定律推导动量守恒
Newton’s second law states F = dp/dt. For two interacting objects A and B, Newton’s third law says F_AB = -F_BA. Therefore, dp_A/dt = -dp_B/dt, or d(p_A + p_B)/dt = 0. Integrating with respect to time, p_A + p_B = constant, so total momentum is conserved provided no external resultant force acts. This general derivation underpins all collision and explosion calculations. For constant masses, it reduces to m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
牛顿第二定律指出 F = dp/dt。对于两个相互作用的物体 A 和 B，牛顿第三定律给出 F_AB = -F_BA。因此，dp_A/dt = -dp_B/dt，或者 d(p_A + p_B)/dt = 0。对时间积分得 p_A + p_B = 常数，所以在无外力作用时总动量守恒。这个一般性推导是所有碰撞和爆炸计算的基础。当质量不变时，可简化为 m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
4. Relating Kinetic Energy and Momentum | 动能与动量的关系推导
Kinetic energy Eₖ = ½mv² and momentum p = mv. Express v in terms of p: v = p/m. Substituting into kinetic energy gives Eₖ = ½m(p/m)² = p²/(2m). This relationship is extremely useful in problems comparing the energies of particles with equal momenta but different masses. Note that for a given momentum, kinetic energy is inversely proportional to mass. This derivation also helps analyse situations where kinetic energy is not conserved but momentum is, such as inelastic collisions.
动能 Eₖ = ½mv²，动量 p = mv。用动量表示速度：v = p/m。代入动能表达式得 Eₖ = ½m(p/m)² = p²/(2m)。这一关系在比较动量相同但质量不同的粒子的能量时非常有用。注意，在动量一定的情况下，动能与质量成反比。该推导也有助于分析动量守恒但动能不守恒的情形，如非弹性碰撞。
5. Elastic Potential Energy from Hooke’s Law | 从胡克定律推导弹性势能
Hooke’s law states that force F = kx, where k is the spring constant and x is extension. The work done stretching a spring is the area under the force–extension graph. Because the relationship is linear, the average force is ½F = ½kx. Work done = average force × extension = (½kx)x = ½kx². This stored work is the elastic potential energy Eₚ = ½kx², provided the elastic limit is not exceeded. The derivation uses the definition of work and the linear force law.
胡克定律指出 F = kx，其中 k 为劲度系数，x 为伸长量。拉伸弹簧所做的功等于力-伸长图下的面积。由于关系是线性的，平均力为 ½F = ½kx。功 = 平均力 × 伸长量 = (½kx)x = ½kx²。储存的功即弹性势能 Eₚ = ½kx²，前提是未超出弹性限度。该推导利用了功的定义和线性力律。
6. Young Modulus, Stress and Strain | 杨氏模量、应力与应变关系推导
Stress σ is defined as force per unit cross-sectional area: σ = F/A. Strain ε is the fractional change in length: ε = ΔL/L. For a wire obeying Hooke’s law, F = (EA/L) ΔL, where E is the Young modulus. Rearranging: F/A = E (ΔL/L). Hence, σ = Eε. This linear relationship holds within the limit of proportionality. The constant E is a material property, derived from the gradient of the stress–strain graph. Knowing this derivation allows calculation of extension: ΔL = FL/(AE).
应力 σ 定义为单位横截面积上的力：σ = F/A。应变 ε 是长度的相对变化量：ε = ΔL/L。对于遵守胡克定律的细线，有 F = (EA/L) ΔL，其中 E 为杨氏模量。重新整理：F/A = E (ΔL/L)。因此，σ = Eε。该线性关系在比例极限内成立。常数 E 是材料属性，可由应力-应变图的斜率求得。掌握此推导便能计算伸长量：ΔL = FL/(AE)。
7. Deriving the Double-Slit Fringe Spacing Formula | 双缝干涉条纹间距公式推导
For Young’s double-slit experiment, consider two coherent sources separated by a distance a. At a point P on a distant screen at distance D, the path difference is approximately a sinθ, where θ is the angle subtended at the slits. For constructive interference, a sinθ = nλ. For small angles, sinθ ≈ tanθ = x/D, where x is the distance from the central maximum. Thus, a (x/D) = nλ. The fringe separation Δx = x_n+1 – x_n = λD/a. This derivation assumes D >> a and small angles, yielding the well‑known formula w = λD/s, where w is fringe width and s the slit separation.
对于杨氏双缝实验，考虑两个相距为 a 的相干光源。在远处屏幕上的 P 点，光程差近似为 a sinθ，其中 θ 为在双缝处的张角。相长干涉时 a sinθ = nλ。在小角度下，sinθ ≈ tanθ = x/D，其中 x 为到中央明纹的距离。故 a (x/D) = nλ。条纹间距 Δx = x_{n+1} – x_n = λD/a。该推导假设 D >> a 且角度很小，从而得到著名的公式 w = λD/s，其中 w 为条纹宽度，s 为缝间距。
8. Single-Slit Diffraction Minima | 单缝衍射暗纹位置推导
Consider a single slit of width a. Divide the slit into two halves. For a direction making an angle θ, the path difference between waves from the top of the top half and the top of the bottom half is (a/2) sinθ. Destructive interference occurs when this path difference is λ/2, giving (a/2) sinθ = λ/2, or a sinθ = λ. In general, minima occur when a sinθ = nλ, where n = ±1, ±2, … This explains why the central maximum is twice as wide as subsidiary maxima. The small‑angle approximation gives the angular position of the first minimum as θ ≈ λ/a.
考虑宽度为 a 的单缝。将单缝分为上下两半。对与法线成 θ 角的方向，从上半部顶端和下半部顶端发出的光波之间的光程差为 (a/2) sinθ。当该光程差等于 λ/2 时产生相消干涉，故 (a/2) sinθ = λ/2，即 a sinθ = λ。一般地，暗纹出现在 a sinθ = nλ 处，其中 n = ±1, ±2, …。这解释了为什么中央明纹宽度是次极大明纹的两倍。利用小角度近似可得第一暗纹角位置 θ ≈ λ/a。
9. Standing Wave Wavelength on a Stretched String | 弦上驻波波长与弦长关系推导
For a string fixed at both ends, the boundary conditions require nodes at the ends. The fundamental mode has one antinode: distance between nodes = λ/2, so length L = λ/2, giving λ = 2L. The second harmonic has an extra node in the middle, so L = λ, giving λ = L. In general, L = n (λ_n/2), so λ_n = 2L/n, where n = 1, 2, 3, … Combining with the wave equation v = fλ and v = √(T/μ) yields the resonant frequencies f_n = n/(2L) √(T/μ). This derivation is essential for understanding musical instruments and sonometer experiments.
对于两端固定的弦，边界条件要求两端为波节。基频有一个波腹：波节间距 = λ/2，所以 L = λ/2，得 λ = 2L。第二谐波在中间多一个波节，故 L = λ，得 λ = L。一般地，L = n (λ_n/2)，所以 λ_n = 2L/n，其中 n = 1, 2, 3, …。结合波动方程 v = fλ 和 v = √(T/μ) 可得共振频率 f_n = n/(2L) √(T/μ)。这一推导对理解乐器和弦振动实验至关重要。
10. Critical Angle and Total Internal Reflection | 临界角与全内反射推导
Snell’s law relates incidence and refraction: n₁ sinθ₁ = n₂ sinθ₂. When light passes from a denser medium (refractive index n₁) to a less dense medium (n₂ < n₁), the angle of refraction reaches 90° at the critical angle θ_c. Setting θ₂ = 90°, sinθ₂ = 1, gives n₁ sinθ_c = n₂ × 1, so sinθ_c = n₂/n₁. If the incident angle exceeds θ_c, total internal reflection occurs. This derivation explains fiber optics and the sparkle of diamonds, and is a standard requirement on exam papers.
斯涅耳定律给出入射与折射的关系：n₁ sinθ₁ = n₂ sinθ₂。当光从光密介质（折射率 n₁）进入光疏介质（n₂ < n₁）时，在临界角 θ_c 处折射角恰好为 90°。令 θ₂ = 90°，sinθ₂ = 1，得 n₁ sinθ_c = n₂ × 1，故 sinθ_c = n₂/n₁。若入射角大于 θ_c，则发生全内反射。该推导解释了光纤和钻石闪耀的原理，是试卷中常考的标准要求。
11. Resistivity and the Resistance of a Wire | 电阻率与导线电阻公式推导
Resistance R of a uniform wire is directly proportional to its length L and inversely proportional to its cross-sectional area A: R = ρL/A, where ρ is resistivity. This can be derived from the microscopic model: for a conductor, current density J = I/A, and electric field E = V/L. Ohm’s law in microscopic form is J = σE, with conductivity σ = 1/ρ. Substituting, I/A = (1/ρ)(V/L), giving V/I = ρL/A = R. This derivation connects macroscopic measurements to the material’s intrinsic property. In the January 2022 paper, candidates might have been asked to find ρ from the gradient of an R vs L graph.
均匀导线的电阻 R 与其长度 L 成正比，与横截面积 A 成反比：R = ρL/A，其中 ρ 为电阻率。这可以从微观模型推导：对于导体，电流密度 J = I/A，电场 E = V/L。欧姆定律的微观形式为 J = σE，σ = 1/ρ。代入得 I/A = (1/ρ)(V/L)，整理得 V/I = ρL/A = R。该推导将宏观测量与材料内在属性联系起来。在2022年1月的试卷中，可能要求学生通过 R–L 图线的斜率求 ρ。
12. Deriving the Formula for EMF and Internal Resistance | 电源电动势与内阻公式推导
For a real power supply, terminal voltage V is less than the electromotive force ε due to the internal resistance r. When a current I flows, energy conservation gives ε = I(R + r), where R is the external load. Thus, V = ε – Ir. In an experiment, measuring V and I for different loads yields a straight line with equation V = -rI + ε. The gradient is -r and the y-intercept is ε. This is often tested by asking students to interpret graphs or derive the lost volts. The derivation is a direct application of Kirchhoff’s voltage law to a simple series circuit.
对于实际电源，由于内阻 r 的存在，端电压 V 小于电动势 ε。当电流 I 流过时，能量守恒给出 ε = I(R + r)，其中 R 为外负载。故 V = ε – Ir。在实验中，针对不同负载测量 V 和 I，可得到一条符合 V = -rI + ε 的直线。斜率为 -r，截距为 ε。试题常要求解释图线或推导内阻压降。该推导是基尔霍夫电压定律在简单串联电路中的直接应用。
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