📚 Pre-U CCEA Physics Formula and Theorem Quick Reference | Pre-U CCEA 物理公式定理速查手册
This quick reference guide compiles essential formulas and theorems for Pre-U CCEA Physics, covering mechanics, waves, electricity, thermal physics, and quantum phenomena. Use it as a revision tool to reinforce key concepts before the exam.
本速查手册汇编了 Pre-U CCEA 物理的核心公式与定理，涵盖力学、波动、电学、热物理和量子现象，可作为考前复习强化重点概念的工具。
1. Kinematics and Dynamics | 运动学与动力学
For motion with constant acceleration along a straight line, the following SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). They are derived from the definitions of velocity and acceleration.
对于匀加速直线运动，下列 SUVAT 方程联系了位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)，可由速度和加速度的定义推导得出。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Newton’s three laws of motion form the foundation of classical mechanics. The second law states that the net force acting on a body equals the rate of change of its momentum: F = dp/dt = ma (when mass is constant). Momentum p = mv is a vector quantity; its conservation in an isolated system is a powerful problem-solving tool.
牛顿三大运动定律是经典力学的基础。第二定律指出，作用于物体的合力等于其动量变化率：F = dp/dt = ma（质量恒定时）。动量 p = mv 是矢量；孤立系统中动量守恒是强有力的解题工具。
The impulse of a force is J = FΔt = Δp, equal to the change in momentum. In collisions, distinguish between elastic (kinetic energy conserved) and inelastic (only momentum conserved) interactions.
力的冲量为 J = FΔt = Δp，等于动量的变化量。在碰撞问题中要区分弹性碰撞（动能守恒）和非弹性碰撞（仅动量守恒）。
2. Work, Energy and Power | 功、能量与功率
Work done by a constant force is W = Fs cos θ, where θ is the angle between force and displacement. Kinetic energy is Eₖ = ½mv². Gravitational potential energy near Earth’s surface is Eₚ = mgh, with h measured from an arbitrary zero.
恒力所做的功为 W = Fs cos θ，θ 是力与位移的夹角。动能为 Eₖ = ½mv²。地表附近的重力势能为 Eₚ = mgh，h 从任意零势面量起。
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: Wₙₑₜ = ΔEₖ. In a conservative system, mechanical energy Eₘ = Eₖ + Eₚ is conserved.
动能定理指出，合力对物体做的功等于其动能的变化：Wₙₑₜ = ΔEₖ。在保守系统中，机械能 Eₘ = Eₖ + Eₚ 是守恒的。
Power is the rate of work or energy transfer: P = W/t = Fv (for a constant force in the direction of velocity). Efficiency = (useful output energy) / (total input energy) × 100%.
功率是做功或能量转移的速率：P = W/t = Fv（力与速度同向时）。效率 = (有用输出能量) / (总输入能量) × 100%。
3. Circular Motion and Gravitation | 圆周运动与万有引力
An object moving with uniform circular motion has a centripetal acceleration aₙ = v²/r = ω²r, directed toward the centre. The angular speed ω = 2π/T = 2πf, where T is the period and f the frequency.
做匀速圆周运动的物体具有向心加速度 aₙ = v²/r = ω²r，方向指向圆心。角速度 ω = 2π/T = 2πf，T 为周期，f 为频率。
The centripetal force required is F = mv²/r = mω²r. It is not a new type of force but a net force towards the centre provided by tension, gravity, friction, etc.
所需的向心力是 F = mv²/r = mω²r。它并非新型力，而是张力、重力、摩擦力等产生的指向圆心的合力。
Newton’s law of universal gravitation: F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N m² kg⁻². Gravitational field strength g = F/m = GM/r² (radial field). The gravitational potential V = -GM/r, and potential energy U = mV.
万有引力定律：F = Gm₁m₂/r²，G = 6.67 × 10⁻¹¹ N m² kg⁻²。引力场强度 g = F/m = GM/r²（径向场）。引力势 V = -GM/r，势能 U = mV。
4. Simple Harmonic Motion (SHM) | 简谐运动
SHM occurs when acceleration is directly proportional to displacement from equilibrium and always directed towards it: a = -ω²x. The displacement–time solution is x = A cos(ωt + φ) or x = A sin(ωt).
当加速度与离开平衡位置的位移成正比且始终指向平衡点时发生简谐运动：a = -ω²x。位移-时间解为 x = A cos(ωt + φ) 或 x = A sin(ωt)。
The velocity is v = ±ω√(A² – x²) and the maximum speed is vₘₐₓ = ωA. Period formulas: mass–spring system T = 2π√(m/k), simple pendulum T = 2π√(L/g).
速度为 v = ±ω√(A² – x²)，最大速率 vₘₐₓ = ωA。周期公式：弹簧振子 T = 2π√(m/k)，单摆 T = 2π√(L/g)。
Energy in SHM: total mechanical energy Eₜₒₜ = ½mω²A². This is the sum of kinetic energy Eₖ = ½mω²(A² – x²) and potential energy Eₚ = ½mω²x².
简谐运动的能量：总机械能 Eₜₒₜ = ½mω²A²，它等于动能 Eₖ = ½mω²(A² – x²) 和势能 Eₚ = ½mω²x² 之和。
5. Waves and Optics | 波动与光学
Wave speed is related to frequency and wavelength: v = fλ. For a transverse wave on a string, v = √(T/μ), where T is tension and μ is mass per unit length.
波速与频率、波长的关系：v = fλ。对弦上的横波，v = √(T/μ)，其中 T 为张力，μ 为线密度。
Phase difference φ (in radians) = 2π × (path difference / λ). Constructive interference occurs when path difference = nλ; destructive when = (n + ½)λ.
相位差 φ（弧度）= 2π × (路程差 / λ)。路程差为整数倍波长时发生相长干涉，半整数倍时发生相消干涉。
Young’s double-slit fringe spacing: Δy = λD/d, where d is slit separation and D is screen distance. Diffraction grating equation: d sin θ = nλ, with n = 0, ±1, ±2, …
杨氏双缝条纹间距：Δy = λD/d，d 为缝距，D 为屏距。衍射光栅方程：d sin θ = nλ，n = 0, ±1, ±2, …
Standing waves are described by y = 2A sin(kx) cos(ωt). Nodes occur at positions where sin(kx) = 0; antinodes where |sin(kx)| = 1.
驻波可描述为 y = 2A sin(kx) cos(ωt)。波节在 sin(kx) = 0 处，波腹在 |sin(kx)| = 1 处。
6. Electric Fields | 电场
Coulomb’s law for the force between two point charges: F = (1/4πε₀) · q₁q₂/r², where ε₀ = 8.85 × 10⁻¹² F m⁻¹. The force is attractive for opposite signs, repulsive for like signs.
两点电荷间的库仑力：F = (1/4πε₀) · q₁q₂/r²，ε₀ = 8.85 × 10⁻¹² F m⁻¹。异号相吸，同号相斥。
Electric field strength is defined as E = F/q. For a point charge, E = (1/4πε₀) · Q/r². In a uniform field between parallel plates, E = V/d, where V is the potential difference and d the plate separation.
电场强度定义为 E = F/q。点电荷的电场 E = (1/4πε₀) · Q/r²。在平行板产生的匀强电场中，E = V/d，V 为电势差，d 为板间距。
Electric potential V = (1/4πε₀) · Q/r relative to zero at infinity. Potential energy of a charge q in a potential V is W = qV. The work done in moving a charge between points A and B is W = q(V_B – V_A).
电势 V = (1/4πε₀) · Q/r，以无穷远为零电势。电荷 q 在电势 V 中的电势能为 W = qV。将电荷从 A 移到 B 所做的功为 W = q(V_B – V_A)。
7. Capacitors | 电容器
Capacitance is defined as C = Q/V, measured in farads (F). For a parallel-plate capacitor, C = ε₀εᵣ A / d, where εᵣ is the relative permittivity of the dielectric.
电容定义为 C = Q/V，单位法拉 (F)。平行板电容器的电容为 C = ε₀εᵣ A / d，εᵣ 为介质的相对介电常数。
Energy stored in a charged capacitor: W = ½QV = ½CV² = ½ Q²/C. This energy resides in the electric field between the plates.
电容器储存的能量：W = ½QV = ½CV² = ½ Q²/C，该能量储存在两极板间的电场中。
During charge or discharge through a resistor R, the time constant is τ = RC. The charge, voltage and current decay or grow exponentially: Q = Q₀ e^(-t/RC), V = V₀ e^(-t/RC), I = I₀ e^(-t/RC).
通过电阻 R 充放电时，时间常数为 τ = RC。电荷、电压和电流按指数规律衰减或增长：Q = Q₀ e^(-t/RC)、V = V₀ e^(-t/RC)、I = I₀ e^(-t/RC)。
8. Current Electricity and Circuits | 电流与电路
Current is the rate of flow of charge: I = Q/t. Ohm’s law states that V = IR for an ohmic conductor at constant temperature. Resistance depends on geometry and resistivity: R = ρL/A.
电流是电荷流动的速率：I = Q/t。欧姆定律指出，对恒温下的欧姆导体有 V = IR。电阻取决于几何尺寸和电阻率：R = ρL/A。
Power dissipated in a resistor: P = IV = I²R = V²/R. In a series circuit, current is the same and voltages add. In a parallel circuit, voltage is the same and currents add.
电阻消耗的功率：P = IV = I²R = V²/R。串联电路中电流相同、电压相加；并联电路中电压相同、电流相加。
Kirchhoff’s laws: (1) sum of currents entering a junction equals the sum leaving: ΣIᵢₙ = ΣIₒᵤₜ. (2) The sum of emfs around any closed loop equals the sum of potential drops: Σε = ΣIR.
基尔霍夫定律：(1) 流入节点的电流之和等于流出电流之和：ΣIᵢₙ = ΣIₒᵤₜ。(2) 回路中电动势之和等于电势降落之和：Σε = ΣIR。
A real cell has internal resistance r, limiting the terminal pd: V = ε – Ir. Maximum power is transferred to a load when load resistance equals internal resistance.
实际电池具有内阻 r，端电压为 V = ε – Ir。当负载电阻等于内阻时可获得最大功率输出。
9. Magnetic Fields and Electromagnetic Induction | 磁场与电磁感应
A magnetic field exerts a force on a moving charge: F = qvB sin θ (Lorentz force). For a current-carrying wire segment, F = BIL sin θ, where θ is the angle between current and field.
磁场对运动电荷的作用力为 F = qvB sin θ（洛伦兹力）。对通电导线则是 F = BIL sin θ，θ 为电流与磁场的夹角。
Magnetic flux through an area A is Φ = BA cos θ, where θ is the angle between the field and the normal to the area. The unit is weber (Wb).
穿过面积 A 的磁通量为 Φ = BA cos θ，θ 为磁场与面积法线的夹角，单位韦伯 (Wb)。
Faraday’s law of induction: magnitude of induced emf equals the rate of change of flux linkage: ε = -N dΦ/dt. Lenz’s law gives the minus sign: the induced current opposes the change causing it.
法拉第电磁感应定律：感应电动势的大小等于磁链变化率：ε = -N dΦ/dt。负号体现楞次定律——感应电流总是阻碍引起它的变化。
For a conductor moving perpendicular to a uniform field, motional emf is ε = BLv. An ideal transformer satisfies Vₚ/Vₛ = Nₚ/Nₛ and Iₚ/Iₛ = Nₛ/Nₚ, conserving power.
当导体垂直切割匀强磁场时，动生电动势为 ε = BLv。理想变压器满足 Vₚ/Vₛ = Nₚ/Nₛ 和 Iₚ/Iₛ = Nₛ/Nₚ，且功率守恒。
10. Thermal Physics and Ideal Gases | 热物理与理想气体
Temperature in kelvin T(K) = θ(°C) + 273.15. Specific heat capacity c relates energy change to temperature change: ΔQ = mcΔθ. Specific latent heat L relates energy to phase change at constant temperature: ΔQ = mL.
开氏温度 T(K) = θ(°C) + 273.15。比热容 c 联系能量变化与温度变化：ΔQ = mcΔθ。比潜热 L 联系能量与恒温下的相变：ΔQ = mL。
The ideal gas equation: pV = nRT (R = 8.31 J K⁻¹ mol⁻¹) or pV = NkT (k = 1.38 × 10⁻²³ J K⁻¹). The kinetic theory model gives pV = ⅓Nmv²rms, and mean translational kinetic energy = (3/2)kT.
理想气体状态方程：pV = nRT（R = 8.31 J K⁻¹ mol⁻¹）或 pV = NkT（k = 1.38 × 10⁻²³ J K⁻¹）。根据分子动理论，pV = ⅓Nmv²rms，平均平动动能 = (3/2)kT。
The first law of thermodynamics links internal energy change ΔU, heat added to the system Q, and work done on the system W: ΔU = Q + W. For an isothermal process ΔU = 0, for adiabatic Q = 0.
热力学第一定律将内能变化 ΔU、系统吸热 Q 和外界对系统做功 W 联系起来：ΔU = Q + W。等温过程 ΔU = 0，绝热过程 Q = 0。
11. Quantum Phenomena and Photons | 量子现象与光子
Photon energy is E = hf = hc/λ, with Planck’s constant h = 6.63 × 10⁻³⁴ J s. The photoelectric effect is described by Einstein’s equation: hf = Φ + Kₘₐₓ, where Φ is the work function and Kₘₐₓ = ½mv²ₘₐₓ is the maximum kinetic energy of emitted electrons.
光子能量为 E = hf = hc/λ，普朗克常量 h = 6.63 × 10⁻³⁴ J s。光电效应由爱因斯坦方程描述：hf = Φ + Kₘₐₓ，Φ 为逸出功，Kₘₐₓ = ½mv²ₘₐₓ 为出射电子的最大动能。
The threshold frequency is f₀ = Φ/h. Stopping potential Vₛ relates to maximum kinetic energy via eVₛ = Kₘₐₓ. Photocurrent saturates when all emitted electrons are collected.
截止频率为 f₀ = Φ/h。遏止电势 Vₛ 与最大动能的关系为 eVₛ = Kₘₐₓ。当所有逸出电子都被收集时，光电流达到饱和。
Electron energy levels in atoms are quantised. The energy of a photon emitted or absorbed in a transition between levels is ΔE = E₂ – E₁ = hf. For hydrogen-like atoms, Eₙ = -13.6 / n² eV (n = 1,2,3…).
原子中电子能级是量子化的。两能级间跃迁发射或吸收的光子能量为 ΔE = E₂ – E₁ = hf。类氢原子的能级为 Eₙ = -13.6 / n² eV（n = 1,2,3…）。
The de Broglie wavelength of a particle with momentum p is λ = h/p. This wave–particle duality is fundamental to quantum mechanics.
动量为 p 的粒子的德布罗意波长为 λ = h/p。这种波粒二象性是量子力学的基础。
12. Nuclear Physics | 核物理
The nucleus is characterised by atomic number Z (protons) and mass number A (protons + neutrons). Isotopes have the same Z but different A.
原子核由原子序数 Z（质子数）和质量数 A（质子+中子）表征。同位素具有相同的 Z 但不同的 A。
Radioactive decay follows the exponential law: N = N₀ e^(-λt), where λ is the decay constant. Activity A = λN has units becquerel (Bq). Half-life T₁/₂ = ln 2 / λ.
放射性衰变遵循指数规律：N = N₀ e^(-λt)，λ 为衰变常数。活度 A = λN 的单位是贝可勒尔 (Bq)。半衰期 T₁/₂ = ln 2 / λ
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