📚 A-Level Physics: Last-Minute Revision Notes | A-Level物理：考前冲刺笔记
This set of concise revision notes covers the most essential concepts, formulas, and common pitfalls in A-Level Physics. Use it to quickly refresh your memory before the exam — each section pairs an English explanation with its Chinese equivalent, helping you solidify your understanding while getting familiar with bilingual terminology.
这套精简的考前冲刺笔记涵盖了A-Level物理中最核心的概念、公式和常见易错点。每一部分都采用英文与中文配对讲解，帮助你在考前快速回顾知识体系，同时熟悉双语术语，切实强化理解。
1. SI Units and Prefixes | 国际单位制与词头
Always express quantities in SI base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd). Derived units like the newton (N) and joule (J) can be decomposed: 1 N = 1 kg m s⁻², 1 J = 1 N m = 1 kg m² s⁻².
始终把物理量用国际单位制基本单位表示：米(m)、千克(kg)、秒(s)、安培(A)、开尔文(K)、摩尔(mol)、坎德拉(cd)。导出单位如牛顿(N)和焦耳(J)可分解：1 N = 1 kg m s⁻², 1 J = 1 N m = 1 kg m² s⁻²。
Memorise common prefixes: nano (n, 10⁻⁹), micro (µ, 10⁻⁶), milli (m, 10⁻³), centi (c, 10⁻²), kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹). Convert carefully: 5 cm = 5 × 10⁻² m, and 3.2 MHz = 3.2 × 10⁶ Hz.
牢记常用词头：纳诺(n, 10⁻⁹)、微(µ, 10⁻⁶)、毫(m, 10⁻³)、厘(c, 10⁻²)、千(k, 10³)、兆(M, 10⁶)、吉(G, 10⁹)。换算要仔细：5 cm = 5 × 10⁻² m, 3.2 MHz = 3.2 × 10⁶ Hz。
Check the homogeneity of equations: each term must have the same base units. For example, in s = ut + ½at², both ut and ½at² must reduce to metres.
检验方程的量纲一致性：每一项的基本单位必须相同。例如在 s = ut + ½at² 中，ut 和 ½at² 最终的单位都必须是米。
2. Scalars and Vectors | 标量与矢量
Scalars have magnitude only, e.g., distance, speed, mass, energy. Vectors have both magnitude and direction, e.g., displacement, velocity, acceleration, force, momentum.
标量只有大小，如距离、速率、质量、能量。矢量既有大小又有方向，如位移、速度、加速度、力、动量。
Add vectors using tip-to-tail or by resolving into perpendicular components. For a vector of magnitude F at angle θ to the horizontal, the horizontal component is F cos θ and the vertical component is F sin θ.
矢量相加使用三角形法则或分解为垂直分量来求解。若矢量大小为F，与水平方向夹角 θ，则水平分量为 F cos θ，竖直分量为 F sin θ。
Subtracting a vector is equivalent to adding its negative. To find change in velocity Δv = v₂ – v₁, draw v₁ and v₂ tail-to-tail; Δv is the vector from the head of v₁ to the head of v₂.
减去一个矢量等同于加上它的反向矢量。求速度变化量 Δv = v₂ – v₁ 时，把 v₁ 和 v₂ 尾对尾画出，Δv 就是从 v₁ 的箭头指向 v₂ 箭头的矢量。
3. Kinematics Equations | 运动学方程
For constant acceleration along a straight line, use the SUVAT equations:
v = u + at
s = ut + ½at²
v² = u² + 2as
s = (u+v)t / 2
对于直线上的匀加速运动，使用匀变速运动方程（suvat）：
v = u + at
s = ut + ½at²
v² = u² + 2as
s = (u+v)t / 2
Choose a positive direction and sign all vectors accordingly. Acceleration due to gravity near Earth’s surface is g = 9.81 m s⁻² downwards. For projectile motion, treat horizontal (constant velocity) and vertical (constant acceleration) motions independently.
选定正方向，并给所有矢量加上正确的符号。地表附近的重力加速度为 g = 9.81 m s⁻²，方向向下。处理抛体运动时，将水平（匀速）和竖直（匀加速）两个方向的运动分开处理。
Velocity–time graphs: gradient gives acceleration, area under graph gives displacement. For acceleration–time graphs, area gives change in velocity.
速度–时间图：斜率表示加速度，线下面积表示位移。加速度–时间图：线下面积表示速度变化量。
4. Dynamics and Newton’s Laws | 动力学与牛顿定律
Newton’s First Law: An object remains at rest or in uniform motion unless acted on by a resultant force. Second Law: F = ma, where F is resultant force. Third Law: action–reaction pairs act on different objects and are equal in magnitude and opposite in direction.
牛顿第一定律：不受合外力作用时，物体保持静止或匀速直线运动。第二定律：F = ma，F 为合外力。第三定律：作用力与反作用力作用在不同物体上，大小相等、方向相反。
Free-body diagrams are essential: isolate the body, draw all forces (weight, normal reaction, tension, friction, applied forces) and use F = ma along each axis.
受力分析图至关重要：隔离物体，画出所有力（重力、支持力、张力、摩擦力、外力），并沿各轴向使用 F = ma 列方程。
Friction f ≤ μR, where μ is the coefficient of friction and R is the normal reaction. For an object on an inclined plane at angle θ, component of weight down the slope = mg sin θ, normal reaction R = mg cos θ.
摩擦力 f ≤ μR，μ 为摩擦系数，R 为支持力。倾斜角为 θ 的斜面上，重力沿斜面分量为 mg sin θ，支持力 R = mg cos θ。
When a body moves in a lift or accelerating system, apparent weight = m(g ± a). The direction of acceleration determines whether the reading increases or decreases.
物体在电梯等加速系统中时，视重 = m(g ± a)。加速度方向决定读数增大还是减小。
5. Work, Energy and Power | 功、能与功率
Work done W = F s cos θ, where F is force, s displacement, θ angle between force and displacement. Work is measured in joules (J). Energy is the capacity to do work.
功 W = F s cos θ，F 为力，s 为位移，θ 为力与位移的夹角。功的单位是焦耳(J)。能量是做功的本领。
Kinetic energy Ek = ½mv², gravitational potential energy Ep = mgh. Conservation of mechanical energy: in absence of resistive forces, (Ek + Ep) is constant.
动能 Ek = ½mv²，重力势能 Ep = mgh。在无阻力的情况下，机械能守恒：(Ek + Ep) 为常数。
Power P = W/t = Fv when force and velocity are parallel. Efficiency = (useful power output / total power input) × 100%.
功率 P = W/t；当力与速度同向时，P = Fv。效率 = (有用输出功率 / 总输入功率) × 100%。
For springs obeying Hooke’s Law F = kx, elastic potential energy = ½kx². Area under force–extension graph equals work done.
弹簧遵循胡克定律时 F = kx，弹性势能 = ½kx²。力–伸 长图线下的面积等于做功。
6. Momentum and Collisions | 动量与碰撞
Linear momentum p = mv. Impulse = change in momentum = FΔt. Impulse equals area under a force–time graph.
线动量 p = mv。冲量 = 动量变化 = FΔt。冲量等于力–时间图下的面积。
Principle of conservation of momentum: In an isolated system, total momentum before an interaction equals total momentum after. For two objects: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
动量守恒定律：在孤立系统中，相互作用前的总动量等于作用后的总动量。对两个物体：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
Elastic collisions: kinetic energy is conserved. Inelastic collisions: some kinetic energy is transformed into other forms, and objects may stick together (perfectly inelastic).
弹性碰撞：动能守恒。非弹性碰撞：部分动能转化为其他形式的能量；物体可能粘在一起（完全非弹性碰撞）。
Newton’s third law and conservation of momentum both arise from the symmetry of forces. In explosions, initial momentum is zero, so the vector sum of momenta of fragments is zero.
牛顿第三定律与动量守恒都源于力的对称性。在爆炸中，初始动量为零，因此各个碎片动量的矢量和为零。
7. Circular Motion and Gravitation | 圆周运动与引力
Uniform circular motion: an object moving in a circle at constant speed has centripetal acceleration a = v²/r = ω²r directed towards the centre. Angular speed ω = 2π/T.
匀速圆周运动：物体以恒定速率沿圆周运动时，具有指向圆心的向心加速度 a = v²/r = ω²r。角速度 ω = 2π/T。
Centripetal force F = mv²/r = mω²r. It is not a new type of force — it is provided by tension, friction, gravitational attraction or a component of normal reaction.
向心力 F = mv²/r = mω²r。它并非一种新型力，而是由张力、摩擦力、万有引力或支持力的分量提供。
Newton’s law of gravitation: F = G m₁ m₂ / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻². Gravitational field strength g = F/m. Near a planet’s surface, g = GM/R², where M is mass and R is radius of the planet.
牛顿万有引力定律：F = G m₁ m₂ / r²，G = 6.67 × 10⁻¹¹ N m² kg⁻²。引力场强度 g = F/m。在行星表面附近，g = GM/R²，M 为行星质量，R 为行星半径。
For satellites in circular orbits, gravitational force provides centripetal force: GMm/r² = mv²/r, yielding v = √(GM/r). Geostationary satellites have period T = 24 hours and orbit above the equator.
卫星在圆周轨道上时，万有引力提供向心力：GMm/r² = mv²/r，可得 v = √(GM/r)。地球同步轨道卫星周期为24小时，位于赤道上空。
8. Simple Harmonic Motion | 简谐运动
SHM condition: acceleration a is directly proportional to displacement x from equilibrium and always directed toward equilibrium: a = -ω²x.
简谐运动的条件：加速度 a 与离开平衡位置的位移 x 成正比，且始终指向平衡位置：a = -ω²x。
Solutions are sinusoidal: x = A sin(ωt) or x = A cos(ωt). Velocity v = ± ω √(A² – x²). Maximum speed v_max = ωA at equilibrium.
位移解是正弦或余弦函数：x = A sin(ωt) 或 x = A cos(ωt)。速度 v = ± ω √(A² – x²)。最大速率 v_max = ωA，出现在平衡位置。
Period T = 2π/ω. For a mass–spring system, T = 2π √(m/k). For a simple pendulum with small amplitude, T = 2π √(l/g).
周期 T = 2π/ω。对弹簧振子，T = 2π √(m/k)。对小振幅单摆，T = 2π √(l/g)。
Energy in SHM interchanges between kinetic and potential. Total energy E = ½m ω²A² = ½kA² for spring. At amplitude A, energy is all potential; at equilibrium, all kinetic.
简谐运动中的能量在动能和势能间转换。总能量 E = ½m ω²A²（弹簧振子为 ½kA²）。在振幅 A 处，能量全部为势能；在平衡位置，全部为动能。
Damping reduces amplitude over time. Light damping slightly increases period; critical damping brings the system to equilibrium in the shortest time without oscillation; heavy damping gives a slow return.
阻尼使振幅随时间减小。弱阻尼略增周期；临界阻尼使系统在最短时间回到平衡位置且无振荡；过阻尼则缓慢返回平衡位置。
9. Waves: Properties and Superposition | 波的性质与叠加
Wave speed v = fλ, where f is frequency and λ is wavelength. Transverse waves: oscillation perpendicular to direction of energy transfer. Longitudinal waves: oscillation parallel to energy transfer.
波速 v = fλ，f 为频率，λ 为波长。横波：振动方向垂直于能量传递方向。纵波：振动方向平行于能量传递方向。
Superposition: when two or more waves meet, the resultant displacement is the vector sum of individual displacements. Stationary waves form when two identical progressive waves travel in opposite directions.
叠加原理：两列或多个波相遇时，合位移等于各波位移的矢量和。驻波由两列完全相同但相向传播的行波叠加而成。
In stationary waves, nodes have zero displacement, antinodes have maximum amplitude. Distance between adjacent nodes is λ/2. Harmonics on a string fixed at both ends: λ = 2L/n, f_n = n(v/2L).
驻波中，波节处位移恒为零，波腹处振幅最大。相邻波节间距为 λ/2。两端固定的弦线上的谐波：λ = 2L/n，f_n = n(v/2L)。
Diffraction: waves spread out after passing through a gap. Significant diffraction occurs when the gap width is comparable to the wavelength. Interference is constructive when path difference = nλ, destructive when path difference = (n + ½)λ.
衍射：波通过狭缝后扩展。当缝宽与波长相近时衍射显著。干涉：当波程差为 nλ 时出现相长干涉，为 (n + ½)λ 时出现相消干涉。
Young’s double-slit fringe spacing Δy = λD/d, where d is slit separation, D distance to screen. Coherent sources are essential for stable interference patterns.
杨氏双缝干涉条纹间距 Δy = λD/d，d 为双缝间距，D 为缝屏距离。相干光源是产生稳定干涉图样的必要条件。
10. Electric Fields and Capacitors | 电场与电容
Electric field strength E = F/q (force per unit positive charge). For a uniform field between parallel plates separated by distance d with potential difference V, E = V/d.
电场强度 E = F/q（单位正电荷所受的力）。对于相距 d、电势差为 V 的平行板间的匀强电场，E = V/d。
Coulomb’s law: force between two point charges, F = k Q₁ Q₂ / r², where k = 1/(4πε₀). The direction is along the line joining the charges; like charges repel, opposite attract.
库仑定律：两点电荷间作用力 F = k Q₁ Q₂ / r²，k = 1/(4πε₀)。方向沿电荷连线；同号相斥、异号相吸。
Capacitance C = Q/V. For a parallel-plate capacitor, C = ε₀A/d. Energy stored = ½QV = ½CV² = Q²/(2C).
电容 C = Q/V。平行板电容器的电容 C = ε₀A/d。储存的能量 = ½QV = ½CV² = Q²/(2C)。
In RC charging, V = V₀(1 – e^{-t/RC}); during discharging, V = V₀ e^{-t/RC}. The time constant τ = RC gives the time for the charge/voltage to fall to 37% of its initial value, or rise to 63% of the final value.
RC电路充电时，V = V₀(1 – e^{-t/RC})；放电时，V = V₀ e^{-t/RC}。时间常数 τ = RC，表示电荷或电压衰减至初始值的 37% 或上升至最终值的 63% 所需时间。
11. Magnetic Fields and Electromagnetic Induction | 磁场与电磁感应
Force on a current-carrying conductor in a magnetic field: F = BIL sin θ, where B is magnetic flux density, I current, L length of conductor, θ angle between conductor and B field. Fleming’s left-hand rule gives direction.
磁场对电流的作用力：F = BIL sin θ，B 为磁通量密度，I 为电流，L 为导体长度，θ 为导体与磁场方向夹角。方向由弗莱明左手定则确定。
Force on a moving charge: F = Bqv sin θ. For a charged particle moving perpendicular to a uniform magnetic field, it follows a circular path: radius r = mv/(Bq), period T = 2πm/(Bq).
运动电荷在磁场中所受的力：F = Bqv sin θ。带电粒子垂直于匀强磁场入射时，做匀速圆周运动：半径 r = mv/(Bq)，周期 T = 2πm/(Bq)。
Magnetic flux Φ = BA cos θ, where θ is angle between B and area normal. Faraday’s law: induced emf ε = -N (ΔΦ/Δt). Lenz’s law: induced current flows in a direction that opposes the change causing it.
磁通量 Φ = BA cos θ，θ 为磁场方向与面积法线的夹角。法拉第电磁感应定律：感应电动势 ε = -N (ΔΦ/Δt)。楞次定律：感应电流的方向总是阻碍引起感应的变化。
For a conductor of length L moving at speed v perpendicular to field B, motional emf = B L v. Transformers: V_p/V_s = N_p/N_s, and for ideal transformer power in = power out.
长度为 L 的导体在磁场 B 中以速度 v 垂直切割磁感线时，动生电动势 = B L v。变压器：V_p/V_s = N_p/N_s，理想变压器输入功率等于输出功率。
12. Quantum and Nuclear Physics | 量子与核物理
Photon energy E = hf = hc/λ, where h is Planck’s constant (6.63 × 10⁻³⁴ J s). Electromagnetic radiation has both wave and particle properties — the photoelectric effect shows photon nature.
光子能量 E = hf = hc/λ，h 为普朗克常数 (6.63 × 10⁻³⁴ J s)。电磁辐射具有波粒二象性——光电效应体现了粒子性。
Photoelectric effect: emission of electrons from a metal surface when light of frequency above the threshold frequency f₀ hits it. Maximum kinetic energy of photoelectrons: Ek_max = hf – Φ, where Φ = hf₀ is work function.
光电效应：当入射光频率大于截止频率 f₀ 时，金属表面发射电子。光电子的最大动能：Ek_max = hf – Φ，Φ = hf₀ 为逸出功。
Atomic energy levels: electrons exist in discrete energy states; transitions between levels produce emission or absorption spectra with photon energy equal to the energy difference. E₂ – E₁ = hf.
原子能级：电子处于分立的能态；能级间的跃迁产生发射光谱或吸收光谱，光子能量等于能级差。E₂ – E₁ = hf。
Nuclear structure: nucleus consists of protons (Z) and neutrons (N), mass number A = Z + N. Strong nuclear force holds nucleons together. Radioactive decay: alpha (α, helium nucleus), beta (β⁻, electron and antineutrino; β⁺, positron and neutrino), gamma (γ, photon).
原子核结构：核由质子(Z)和中子(N)组成，质量数 A = Z + N。强核力束缚核子。放射性衰变：α（氦核）、β⁻（电子和反中微子）、β⁺（正电子和中微子）、γ（光子）。
Mass–energy equivalence: ΔE = Δm c². Binding energy per nucleon peaks at iron-56. Nuclear fission and fusion release energy when products have higher binding energy per nucleon.
质能等价：ΔE = Δm c²。比结合能在铁-56处最大。核裂变和聚变释放能量，因为产物具有更高的比结合能。
Activity A = λN, where λ is decay constant. Exponential decay law: N = N₀ e^{-λt}, half-life T_{½} = ln 2 /λ. After n half-lives, fraction remaining = (½)ⁿ.
活度 A = λN，λ 为衰变常量。指数衰变律：N = N₀ e^{-λt}，半衰期 T_{½} = ln 2 /λ。经过 n 个半衰期，剩余比例为 (½)ⁿ。
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