📚 AS Physics: Unit 2 Insert Jan 19 Formula Derivation | AS物理：Unit 2 公式推导（2019年1月插页）
The January 2019 AS Physics Unit 2 insert provides a concise reference of essential equations for mechanics, materials, waves and electricity. This article walks through the derivations of those formulas, showing how each one arises from first principles or simple definitions. By tracing the logic behind the equations, you can deepen your understanding and apply them with confidence in exams.
2019 年 1 月的 AS 物理 Unit 2 公式表汇集了力学、材料、波和电学中的核心方程。本文会逐一展示这些公式的推导过程，解释它们是如何从基本原理或定义出发得到的。理清背后的逻辑，不仅能帮助你更好地记忆，还能让你在考试中更灵活地运用它们。
1. Equations of Motion | 运动学方程
Acceleration is defined as the rate of change of velocity: a = (v – u) / t. Rearranging gives the first equation, v = u + at. To find displacement, consider a velocity–time graph: the area under the line is a trapezium. For constant acceleration, area = average velocity × time = ((u + v) / 2) × t. Substitute v = u + at to obtain s = ut + ½at². Eliminating t between v = u + at and the displacement formula yields v² = u² + 2as.
加速度定义为速度的变化率：a = (v – u) / t。整理后得到第一个方程 v = u + at。要求位移，可借助速度‑时间图像：图线下的面积是一个梯形。对于匀加速运动，面积 = 平均速度 × 时间 = ((u + v) / 2) × t。将 v = u + at 代入，便得到 s = ut + ½at²。在 v = u + at 和位移公式之间消去 t，就得到 v² = u² + 2as。
2. Newton’s Second Law and Momentum | 牛顿第二定律与动量
Newton’s second law states that the net force equals the rate of change of momentum: F = Δp / Δt. Momentum p = mv, so Δp = m(v – u) when mass is constant. Substituting gives F = m(v – u) / Δt = ma. This is the familiar F = ma. The insert also lists impulse as Δp = FΔt, which follows directly from the definition of force.
牛顿第二定律指出，合外力等于动量的变化率：F = Δp / Δt。动量 p = mv，因此质量不变时 Δp = m(v – u)。代入后得到 F = m(v – u) / Δt = ma，也就是我们熟悉的 F = ma。公式表中还列出了冲量 Δp = FΔt，这直接从力的定义得出。
3. Work, Energy and Power | 功、能量和功率
Work done by a constant force is W = Fs cosθ; when force and displacement are parallel, W = Fs. To derive kinetic energy, start from the work–energy principle and the equation v² = u² + 2as. Multiply both sides by ½m: ½mv² – ½mu² = mas = Fs = work done. Hence the gain in kinetic energy is ½mv² – ½mu². Gravitational potential energy near Earth’s surface is simply work done against gravity: GPE = mgh. Power is the rate of doing work, P = W / t. If a force moves at constant speed v, then P = Fs / t = Fv.
恒力做功为 W = Fs cosθ；当力与位移同向时，W = Fs。动能的推导从功能原理和公式 v² = u² + 2as 开始。两边同乘 ½m：½mv² – ½mu² = mas = Fs = 功。因此动能的增加量为 ½mv² – ½mu²。地表附近的重力势能就是克服重力所做的功：GPE = mgh。功率是做功的快慢，P = W / t。若力以恒定速度 v 运动，则 P = Fs / t = Fv。
4. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能
Within the elastic limit, extension Δx is proportional to applied force: F = kΔx, where k is the spring constant. The work done to stretch a spring is stored as elastic potential energy. On a force–extension graph, this energy equals the area under the line, which is a triangle: E = ½FΔx. Substituting F = kΔx gives E = ½k(Δx)².
在弹性限度内，伸长量 Δx 与施加的力成正比：F = kΔx，k 为劲度系数。拉伸弹簧所做的功储存为弹性势能。在力‑伸长量图像中，这部分能量等于图线下三角形的面积：E = ½FΔx。代入 F = kΔx 即得 E = ½k(Δx)²。
5. Stress, Strain and Young Modulus | 应力、应变和杨氏模量
Tensile stress is force per unit area: σ = F / A. Tensile strain is extension per original length: ε = ΔL / L. Young modulus E is the ratio of stress to strain within the linear region: E = σ / ε = (F / A) / (ΔL / L). This can be written as F = (EA / L)ΔL, which is analogous to Hooke’s law with an effective spring constant k = EA / L.
拉伸应力是单位面积上的力：σ = F / A。拉伸应变是伸长量与原长之比：ε = ΔL / L。杨氏模量 E 是线弹性范围内应力与应变的比值：E = σ / ε = (F / A) / (ΔL / L)。该式也可写成 F = (EA / L)ΔL，与胡克定律相似，等效劲度系数为 k = EA / L。
6. Wave Equation | 波动方程
The speed of a wave is the distance travelled by a wavefront per unit time. In one period T, a wave advances by one wavelength λ. Therefore wave speed v = λ / T. Since frequency f = 1 / T, this becomes v = fλ. This relationship holds for all types of wave, provided the medium does not change.
波速是单位时间内波前移动的距离。在一个周期 T 内，波前进一个波长 λ。因此波速 v = λ / T。又因为频率 f = 1 / T，于是 v = fλ。这一关系适用于所有类型的波，只要介质不变。
7. Refraction and Total Internal Reflection | 折射与全内反射
Snell’s law relates the angles of incidence and refraction when a wave crosses a boundary: n₁ sinθ₁ = n₂ sinθ₂, where n is refractive index. It can be derived from the ratio of wave speeds: sinθ₁ / sinθ₂ = v₁ / v₂ = n₂ / n₁. For light travelling from a denser to a rarer medium, total internal reflection occurs at the critical angle c when sin c = n₂ / n₁. If the rarer medium is air (n₂ ≈ 1), this simplifies to sin c = 1 / n.
斯涅耳定律描述波穿过界面时入射角与折射角的关系：n₁ sinθ₁ = n₂ sinθ₂，其中 n 为折射率。它可以从波速比推出：sinθ₁ / sinθ₂ = v₁ / v₂ = n₂ / n₁。光从光密介质射向光疏介质时，当入射角达到临界角 c，满足 sin c = n₂ / n₁。若光疏介质为空气（n₂ ≈ 1），公式简化为 sin c = 1 / n。
8. Double-Slit Interference | 双缝干涉
Young’s double‑slit experiment produces an interference pattern. For constructive interference, the path difference between waves from the two slits must be an integer multiple of wavelengths: d sinθ = nλ, where d is slit separation. For small angles, sinθ ≈ tanθ = x / D, where x is the fringe separation from the central maximum and D is the slit‑to‑screen distance. Substituting gives x = nλD / d. The fringe spacing (distance between adjacent bright fringes) is Δx = λD / d.
杨氏双缝实验产生干涉图样。相长干涉的条件是，两缝光波的波程差为波长的整数倍：d sinθ = nλ，其中 d 为缝间距。对于小角度，sinθ ≈ tanθ = x / D，x 为距离中央明纹的偏移，D 为缝到屏的距离。代入后得到 x = nλD / d。相邻明纹的间距为 Δx = λD / d。
9. Ohm’s Law, Power and Resistance | 欧姆定律、功率和电阻
Ohm’s law states that the current through a conductor is proportional to the potential difference across it: V = IR. The power dissipated in a resistor is the rate at which electrical energy is transferred: P = IV. Using V = IR, this becomes P = I²R or P = V² / R. For resistors in series, the total resistance R = R₁ + R₂ + … because the same current passes through each. For resistors in parallel, the reciprocal rule 1/R = 1/R₁ + 1/R₂ follows from the fact that the total current is the sum of branch currents.
欧姆定律指出，通过导体的电流与导体两端的电势差成正比：V = IR。电阻中消耗的电功率是电能转化的速率：P = IV。利用 V = IR，又可写成 P = I²R 或 P = V² / R。串联电阻的总阻值 R = R₁ + R₂ + …，因为每个电阻上流过相同的电流。并联电阻的倒数法则 1/R = 1/R₁ + 1/R₂ 源于总电流等于各分支电流之和。
10. EMF and Internal Resistance | 电动势和内阻
A real power supply has an internal resistance r. The electromotive force (EMF) ε is the energy supplied per unit charge. When a current I flows, the terminal potential difference V is less than ε due to the voltage drop across r: V = ε – Ir. The circuit equation is ε = I(R + r), where R is the external load. This can be rearranged to V = ε – Ir and is often investigated by measuring V for different currents.
实际电源具有内阻 r。电动势 ε 是每单位电荷获得的能量。当电流 I 通过时，由于内阻上的电压降，路端电压 V 会比 ε 小：V = ε – Ir。电路方程为 ε = I(R + r)，其中 R 为外负载。该式整理即为 V = ε – Ir，实验中常通过测量不同电流下的 V 来研究。
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