📚 Year 12 CAIE Physics: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 物理：公式定理速查手册
This quick reference handbook provides a concise collection of the essential formulas, laws, and theorems required for the CAIE AS-Level Physics syllabus. Each section summarises key relationships using standard notation, followed by a brief explanation in both English and Chinese to support bilingual learners. Use this guide for rapid revision and as a checklist to ensure all fundamental equations are at your fingertips before the exam.
这份速查手册汇总了CAIE AS阶段物理课程必备的核心公式、定律和定理。每个部分用标准符号列出关键关系式，并配有简明扼要的中英文解释，帮助双语学习者高效复习。请利用本指南进行快速回顾，并在考前排查是否已掌握所有基础方程。
1. Kinematics | 运动学
The equations of uniformly accelerated motion apply when acceleration is constant. They link displacement s, initial velocity u, final velocity v, acceleration a, and time t. Remember to define a positive direction and treat vectors accordingly.
匀加速直线运动方程适用于加速度恒定的情况。它们将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。务必先规定正方向，并按矢量处理符号。
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
In free fall under gravity, a is replaced by g (9.81 m s⁻²), usually acting vertically downwards. Motion in two dimensions can be resolved into perpendicular components using trigonometry, typically with horizontal projection or projection at an angle.
自由落体中加速度 a 用重力加速度 g (9.81 m s⁻²) 替代，方向竖直向下。二维运动可通过三角函数分解为相互垂直的分量，常见的有水平抛射和斜抛。
| Horizontal component | vₓ = v cosθ, sₓ = v cosθ t |
| Vertical component | vᵧ = v sinθ – gt, sᵧ = v sinθ t – ½gt² |
At the highest point, vertical velocity is momentarily zero.
在最高点，竖直速度瞬时为零。
2. Dynamics | 动力学
Newton’s laws of motion govern the relationship between force, mass, and acceleration. The net force causing acceleration is the vector sum of all forces acting on a body.
牛顿运动定律描述了力、质量和加速度之间的关系。产生加速度的合外力是作用在物体上所有力的矢量和。
F = ma
Weight is the gravitational force on a mass: W = mg. Normal contact force, tension, friction, and air resistance must be included in free-body diagrams. When connected bodies move over pulleys or on surfaces, apply F = ma to each mass separately, taking account of string tension.
重量是质量所受的重力：W = mg。受力分析图中须包括法向接触力、张力、摩擦力和空气阻力。当连接体通过滑轮或在平面上运动时，要对每个物体分别应用 F = ma，并考虑绳中张力。
Friction f is proportional to the normal reaction N in limiting equilibrium or sliding: f ≤ μN, where μ is the coefficient of friction. For static cases, frictional force adjusts up to the maximum value μₛN.
摩擦力 f 在极限平衡或滑动时与法向反力 N 成正比：f ≤ μN，μ 为摩擦系数。静摩擦情况下，摩擦力可达最大值 μₛN。
3. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application in the direction of the force. Energy can be transferred between stores as work, heat, or radiation.
力使其作用点沿力的方向移动时便做了功。能量可通过做功、传热或辐射在能量储存之间转移。
W = Fd cosθ
Kinetic energy: Ek = ½mv²
Gravitational potential energy: ΔEₚ = mgΔh
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed. Power is the rate of doing work or transferring energy.
能量守恒定律表明能量既不会创生也不会消灭，只能转化。功率是做功或能量转移的速率。
P = W / t = Fv
Efficiency is the ratio of useful output power to total input power, often expressed as a percentage.
效率是有用输出功率与总输入功率之比，常用百分数表示。
4. Momentum | 动量
Momentum is a vector quantity given by the product of mass and velocity. The principle of conservation of momentum states that in a closed system with no external forces, total momentum before collision equals total momentum after collision.
动量是矢量，等于质量与速度的乘积。动量守恒定律指出，在没有外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。
p = mv
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Impulse is the change in momentum and equals the average force multiplied by the time for which it acts. This follows from Newton’s second law in its more general form.
冲量是动量的变化量，等于平均力乘以作用时间。这源自牛顿第二定律更一般的形式。
Impulse = Δp = FΔt
Perfectly elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum only, and completely inelastic collisions result in the bodies sticking together.
完全弹性碰撞同时守恒动量和动能；非弹性碰撞只守恒动量，而完全非弹性碰撞中两物体粘合在一起。
5. Circular Motion | 圆周运动
An object moving in a circle at constant speed is accelerating because its direction changes. The centripetal acceleration points toward the centre and requires a net inward force.
物体以恒定速率做圆周运动时，由于方向改变而具有加速度。向心加速度指向圆心，需要一个指向中心的合力。
v = ωr
a = v²/r = ω²r
F = mv²/r = mω²r
The angular speed ω is measured in rad s⁻¹. The period T for one complete revolution is T = 2π/ω. For banking, the horizontal component of the normal reaction or friction provides the centripetal force.
角速度 ω 的单位是 rad s⁻¹。旋转一周的周期 T = 2π/ω。对于弯道倾斜，法向反力或摩擦力的水平分量提供向心力。
6. Gravitational Fields | 万有引力场
Newton’s law of gravitation describes the attractive force between two point masses. The gravitational field strength g at a point is the force per unit mass exerted on a small test mass placed there.
牛顿万有引力定律描述了两个质点之间的吸引力。某点的引力场强度 g 等于放置在该处的小检验物体所受的力与其质量的比值。
F = Gm₁m₂ / r²
g = GM / r²
Near Earth’s surface, g ≈ 9.81 N kg⁻¹. In a radial field, field lines point toward the centre of a mass. For a satellite in circular orbit, gravitational force provides the centripetal force; equating GMm/r² = mv²/r leads to orbital speed v = √(GM/r) and period T² ∝ r³ (Kepler’s third law).
在地表附近，g ≈ 9.81 N kg⁻¹。在径向场中，场线指向质量中心。对于圆形轨道上的卫星，万有引力提供向心力；令 GMm/r² = mv²/r 可得轨道速度 v = √(GM/r)，以及周期 T² ∝ r³（开普勒第三定律）。
7. Materials | 材料力学
Hooke’s law states that the extension x of a spring is proportional to the applied load F, provided the elastic limit is not exceeded. Beyond this limit, permanent deformation occurs.
胡克定律指出，只要不超过弹性极限，弹簧的伸长量 x 与负载 F 成正比。超过弹性极限则产生永久变形。
F = kx
Stress, strain, and the Young modulus characterise the mechanical properties of materials independent of their dimensions. Stress is force per unit cross-sectional area, and strain is extension per original length.
应力、应变和杨氏模量用来描述材料与尺寸无关的力学性质。应力是单位横截面积上的力，应变是伸长量与原长之比。
σ = F / A
ε = ΔL / L₀
Young modulus E = σ / ε
The area under a force–extension graph gives the work done (strain energy). Elastic strain energy stored in a stretched spring is ½Fx = ½kx².
力–伸长图下的面积表示所做的功（应变能）。拉伸弹簧储存的弹性应变能为 ½Fx = ½kx²。
8. Waves | 波
Progressive waves transfer energy without transferring matter. The wave equation links speed v, frequency f, and wavelength λ for all periodic waves.
行波传递能量而不传递物质。波速方程将速度 v、频率 f 和波长 λ 联系到所有周期波中。
v = fλ
Transverse waves (e.g. electromagnetic waves) oscillate perpendicular to the direction of propagation; longitudinal waves (e.g. sound) oscillate parallel to the direction. The phase difference between two points on a wave is 2πΔx/λ radians.
横波（如电磁波）的振动方向与传播方向垂直；纵波（如声波）的振动方向与传播方向平行。波上两点之间的相位差为 2πΔx/λ 弧度。
Intensity I is power per unit area and, for a spherical wave, I ∝ 1/r². For a wave, I ∝ A², where A is the amplitude.
强度 I 是单位面积的功率，对于球面波，I ∝ 1/r²。对于波动，I ∝ A²，A 为振幅。
When two waves superpose, the principle of superposition applies. Constructive interference occurs when the path difference is nλ; destructive when it is (n+½)λ. Stationary waves form from two identical travelling waves in opposite directions, with nodes (zero amplitude) and antinodes (maximum amplitude).
两列波叠加时遵循叠加原理。当波程差为 nλ 时发生相长干涉，为 (n+½)λ 时发生相消干涉。驻波由两列沿相反方向传播的相同行波形成，具有波节（振幅为零）和波腹（振幅最大）。
9. Electric Fields | 电场
An electric field exerts a force on a charged particle. The field strength E at a point is the force per unit positive charge. For a uniform field between parallel plates, E is constant.
电场对带电粒子施加力。某点的电场强度 E 等于单位正电荷所受的力。对于平行板间的匀强电场，E 为常数。
E = F / q
Uniform field: E = V / d
Coulomb’s law gives the force between two point charges in a vacuum; it is analogous to Newton’s law of gravitation but can be attractive or repulsive.
库仑定律给出真空中两点电荷之间的作用力；它类似于万有引力定律，但可以是吸引力或排斥力。
F = kQq / r², where k = 1/(4πε₀)
For a point charge Q, the radial field strength is E = kQ/r². The motion of charged particles in uniform electric fields is analysed by resolving forces and using equations of kinematics.
对于点电荷 Q，径向场强为 E = kQ/r²。带电粒子在匀强电场中的运动通过力的分解和运动学方程进行分析。
10. Current Electricity | 电流
Electric current I is the rate of flow of charge. The potential difference V between two points is the work done per unit charge. Resistance R is defined by the ratio V/I, while Ohm’s law requires resistance to be constant at constant temperature.
电流 I 是电荷流动的速率。两点之间的电势差 V 是移动单位电荷所做的功。电阻 R 由比值 V/I 定义，而欧姆定律要求电阻在恒温下保持恒定。
I = ΔQ / Δt
V = W / q
R = V / I
Resistance depends on the material’s resistivity ρ, length L, and cross-sectional area A: R = ρL/A. The resistivity changes with temperature; for a metallic conductor, resistance increases as temperature rises.
电阻取决于材料的电阻率 ρ、长度 L 和横截面积 A：R = ρL/A。电阻率随温度变化；金属导体的电阻随温度升高而增大。
Electrical power transferred to a component is P = IV = I²R = V²/R. Energy transferred is E = IVt.
传给元件的电功率为 P = IV = I²R = V²/R。传递的能量为 E = IVt。
11. D.C. Circuits | 直流电路
Kirchhoff’s laws enable the analysis of complex circuits. The first law (current law) states that the sum of currents entering a junction equals the sum leaving. The second law (voltage law) states that the sum of e.m.f.s around any closed loop equals the sum of p.d.s.
基尔霍夫定律可用于分析复杂电路。第一定律（电流定律）指出，流入节点的电流总和等于流出电流总和。第二定律（电压定律）指出，绕任意闭合回路的电动势之和等于电势差之和。
ΣIin = ΣIout
Σε = ΣIR
In series circuits, current is the same through all components and total resistance RT = R₁ + R₂ + … The p.d. splits in proportion to resistance. In parallel circuits, the p.d. across each branch is the same and the total resistance is given by 1/RT = 1/R₁ + 1/R₂ + … The currents in branches add to the total current.
串联电路中各处电流相同，总电阻 RT = R₁ + R₂ + …，电压按电阻比例分配。并联电路中各支路两端电压相同，总电阻由 1/RT = 1/R₁ + 1/R₂ + … 给出，支路电流之和等于总电流。
A potential divider uses two resistors in series to provide a fraction of the input voltage. A variable potential divider (potentiometer) allows an adjustable output voltage.
分压器利用两个串联电阻提供输入电压的一部分。可变分压器（电位器）可以调节输出电压。
Vout = Vin × R₂/(R₁ + R₂)
12. Nuclear Physics & Radioactivity | 核物理与放射性
The atomic nucleus contains nucleons: protons and neutrons. The proton number (atomic number) Z determines the element; the nucleon number (mass number) A is the total number of nucleons. Isotopes have the same Z but different A.
原子核由核子（质子和中子）组成。质子数（原子序数）Z 决定元素种类；核子数（质量数）A 是核子总数。同位素具有相同的 Z 但不同的 A。
Radioactive decay is a random process described by the decay constant λ. The activity A is the number of decays per second, measured in becquerels (Bq).
放射性衰变是一个随机过程，由衰变常数 λ 描述。活度 A 是每秒衰变次数，单位为贝克勒尔 (Bq)。
A = λN
The exponential decay law and half-life t½ link time and the number of undecayed nuclei.
指数衰变律和半衰期 t½ 将时间与未衰变核子数联系起来。
N = N₀e–λt
t½ = ln 2 / λ
Alpha (α), beta (β⁻, β⁺), and gamma (γ) radiations have different ionising abilities and penetration ranges. The unified atomic mass unit u is 1/12 of the mass of a carbon-12 atom, and E = Δmc² relates mass defect to nuclear binding energy.
α、β⁻、β⁺ 和 γ 辐射具有不同的电离能力和穿透范围。统一原子质量单位 u 是碳-12原子质量的 1/12，E = Δmc² 将质量亏损与核结合能联系起来。
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