📚 Revision Village IB Physics Formula Booklet 2025 – Key Derivations | IB 物理：Revision Village 2025 公式手册重点公式推导
The IB Physics Formula Booklet provided by Revision Village 2025 is an indispensable tool for exam success, but true understanding comes from knowing where those formulas originate. In this article, we walk through the essential derivations that underpin the most commonly used equations in the syllabus, linking them to the core concepts of the IB physics curriculum.
Revision Village 2025 提供的 IB 物理公式手册是考试成功不可或缺的工具，但真正的理解来自于知道这些公式的来源。本文将带你逐一推导教学大纲中最常用公式的核心推导过程，并将它们与 IB 物理课程的核心概念联系起来。
1. Uniformly Accelerated Motion – The Suvat Equations | 匀加速直线运动 – Suvat 方程
The displacement–time relation s = ut + ½at² is derived from the definitions of average velocity and constant acceleration. Starting with v = u + at, and knowing that average velocity = (u+v)/2, displacement s = average velocity × time gives the result directly.
位移–时间关系 s = ut + ½at² 可以从平均速度和恒定加速度的定义推导出来。从 v = u + at 出发，平均速度 = (u+v)/2，位移 s = 平均速度 × 时间，直接得出结果。
v = u + at → s = (u+v)/2 × t = (u + u + at)/2 × t = ut + ½at²
2. Kinetic Energy from Work–Energy Theorem | 动能定理推导动能公式
The formula for kinetic energy Eₖ = ½mv² can be derived by calculating the work done by a constant net force. Substituting F = ma and using the suvat relation v² = u² + 2as, with initial speed u = 0, gives the energy stored in motion.
动能公式 Eₖ = ½mv² 可以通过计算恒定合外力做的功推导出来。代入 F = ma 并使用 suvat 关系 v² = u² + 2as，取初速 u = 0，即可得出运动储存的能量。
W = Fs = mas = m × (v² − u²)/2a × a = ½mv² (u=0)
3. Centripetal Acceleration Direction and Magnitude | 向心加速度的大小与方向
For an object moving at constant speed v in a circle of radius r, the magnitude of centripetal acceleration is a = v²/r. By considering the change in velocity vector over a small time interval Δt, using similar triangles in the velocity–radius diagram, we obtain Δv/v = Δs/r, and dividing by Δt gives a = v²/r.
对于以恒定速率 v 在半径为 r 的圆周上运动的物体，向心加速度大小为 a = v²/r。通过考虑一小段时间 Δt 内速度矢量的变化，利用速度–半径图中的相似三角形，得到 Δv/v = Δs/r，再除以 Δt 即得 a = v²/r。
Δv/v = vΔt/r → Δv/Δt = v²/r
4. Newton’s Law of Gravitation and Orbital Speed | 万有引力定律与轨道速度
Equating the gravitational force GMm/r² to the required centripetal force mv²/r for a satellite in circular orbit yields the orbital speed formula v = √(GM/r). This derivation links universal gravitation with circular motion and is a favourite in IB exam questions.
对于圆轨道卫星，令万有引力 GMm/r² 等于所需的向心力 mv²/r，可得轨道速度公式 v = √(GM/r)。这个推导将万有引力与圆周运动联系起来，是 IB 考试中的常见考点。
GMm/r² = mv²/r → v² = GM/r → v = √(GM/r)
5. Simple Harmonic Motion – Omega from Restoring Force | 简谐运动 – 由回复力得到角频率 ω
In SHM the acceleration is proportional to displacement: a = –ω²x. For a mass–spring system, Hooke’s law gives F = –kx, so ma = –kx. Equating a = –(k/m)x with the SHM condition yields ω = √(k/m). This derivation is central to Topic C1.
在简谐运动中加速度与位移成正比：a = –ω²x。对于弹簧–质量系统，胡克定律给出 F = –kx，因此 ma = –kx。比较 a = –(k/m)x 与 SHM 条件可得 ω = √(k/m)。这一推导是主题 C1 的核心。
F = −kx = ma → a = −(k/m)x = −ω²x → ω = √(k/m)
6. Ideal Gas Equation from Kinetic Model | 从分子动理论推导理想气体状态方程
By considering N molecules of mass m bouncing elastically between two walls of a cube of side L, the pressure exerted on one wall is P = (Nmv²)/3V. Using the relationship between mean kinetic energy and temperature, Eₖ = 3/2 kᴮT, we obtain PV = NkᴮT = nRT. This connects microscopic and macroscopic physics.
考虑 N 个质量为 m 的分子在边长为 L 的立方体中与器壁做弹性碰撞，施加于一个壁面的压强为 P = (Nmv²)/3V。利用平均动能与温度的关系 Eₖ = 3/2 kᴮT，得到 PV = NkᴮT = nRT。这搭建了微观与宏观物理的桥梁。
P = (1/3)ρ⟨v²⟩ , ⟨Eₖ⟩ = (3/2)kᴮT → PV = NkᴮT
7. Capacitance Charge and Discharge Equations | 电容充放电方程
For an RC circuit, applying Kirchhoff’s voltage law gives ε = IR + Q/C. Using I = dQ/dt leads to the differential equation dQ/dt = (εC – Q)/(RC). Solving yields Q = Q₀(1 – e⁻ᵗ/ᴿᶜ), from which the voltage and current decay formulas follow. The time constant τ = RC appears naturally.
对于 RC 电路，应用基尔霍夫电压定律得到 ε = IR + Q/C。利用 I = dQ/dt 导出微分方程 dQ/dt = (εC – Q)/(RC)。求解得到 Q = Q₀(1 – e⁻ᵗ/ᴿᶜ)，进而可推导出电压和电流的衰减公式。时间常数 τ = RC 自然出现。
ε = R(dQ/dt) + Q/C → Q = Cε(1−e⁻ᵗ/ᴿᶜ)
8. Magnetic Force on a Moving Charge | 运动电荷在磁场中的力
The magnitude F = qvB sinθ and direction given by the right-hand rule can be derived from the more general Lorentz force F = q(E + v × B). When only a magnetic field is present, F = qvB sinθ, where θ is the angle between v and B. This is the foundation for circular motion of charges in magnetic fields.
洛伦兹力的一般形式为 F = q(E + v × B)。当只有磁场存在时，力的大小为 F = qvB sinθ，其中 θ 是 v 与 B 的夹角，方向由右手定则确定。这是磁场中电荷圆周运动的基础。
F = qvB sinθ (v ⟂ B 时 F = qvB，向心)
9. Faraday’s Law and Induced emf in a Moving Conductor | 法拉第定律与移动导体中的感应电动势
For a conducting rod of length L moving with speed v perpendicular to a uniform magnetic field B, the emf induced is ε = BLv. This can be derived from the motional emf concept: magnetic force on electrons causes charge separation until the electrostatic force balances it, giving E = vB across length L, hence ε = vBL.
对于长度为 L 的导体棒以速度 v 垂直于匀强磁场 B 运动，感应电动势为 ε = BLv。可从动生电动势的概念推导：电子所受磁力导致电荷分离，直至静电力与之平衡，产生电场 E = vB，沿长度 L 得到 ε = vBL。
qE = qvB → E = vB → ε = EL = BLv
10. Photoelectric Effect and Einstein’s Equation | 光电效应与爱因斯坦方程
Einstein’s photoelectric equation Eₖ max = hf – Φ is derived from the photon model. A photon of energy hf is absorbed by an electron; the electron requires an energy Φ (work function) to escape the metal surface; any excess energy becomes kinetic energy of the emitted electron. The stopping potential Vₛ is related by eVₛ = Eₖ max.
爱因斯坦光电方程 Eₖ max = hf – Φ 源自光子模型。能量为 hf 的光子被电子吸收；电子需要能量 Φ（功函数）才能逃逸金属表面；剩余能量转化为发射电子的动能。遏止电压 Vₛ 满足 eVₛ = Eₖ max。
hf = Φ + Eₖ max → Eₖ max = hf – Φ
11. de Broglie Wavelength | 德布罗意波长
The wave–particle duality relation λ = h/p is a postulate that can be motivated by combining Einstein’s photon energy E = hf with the relativistic energy–momentum relation for a photon, p = E/c. Since c = fλ, we have p = hf/c = h/λ. Applying this to particles with momentum p = mv gives their associated wavelength.
波粒二象性关系 λ = h/p 是一个假设，可通过将爱因斯坦光子能量 E = hf 与光子的相对论能量–动量关系 p = E/c 结合得到启发。因为 c = fλ，故 p = hf/c = h/λ。将此推广到动量为 p = mv 的粒子，即得到其对应的波长。
p = E/c = hf/(fλ) = h/λ → λ = h/p
12. Nuclear Binding Energy per Nucleon | 原子核的结合能与平均结合能
The binding energy of a nucleus is the energy required to separate it into its individual protons and neutrons; it corresponds to the mass defect Δm = (Z mₚ + N mₙ) – mₙᵤc. Dividing by the mass number A gives the binding energy per nucleon, which explains the stability of iron‑56 and the curve of nuclear stability.
原子核的结合能是将其分离为单个质子和中子所需的能量，对应于质量亏损 Δm = (Z mₚ + N mₙ) – mₙᵤc。除以质量数 A 得到每个核子的平均结合能，这解释了铁‑56 的稳定性以及核稳定曲线。
B.E. = Δm c², B.E./A = (Δm c²)/A
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