Physics Exam Preparation: Understanding and Applying Core Formulas | 物理备考：核心公式的理解与应用

📚 Physics Exam Preparation: Understanding and Applying Core Formulas | 物理备考：核心公式的理解与应用

Physics is often perceived as a subject of endless equations, but the real challenge lies not in memorising formulas, but in understanding the physical principles they represent and knowing when and how to apply them. This guide breaks down the most essential formulas across major topics, explaining their meaning, derivation logic, and common exam applications.

物理常常被视为一门充满公式的学科，但真正的挑战不在于背诵公式，而在于理解公式背后的物理原理，以及知道何时、如何运用它们。本指南将分主题解析最重要的核心公式，阐释其含义、推导逻辑及常见考试应用。

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1. Kinematics: Describing Motion | 运动学：描述运动

Kinematics forms the foundation of mechanics. The key is to recognise that the SUVAT equations apply only to motion with constant acceleration. Before applying them, always check whether acceleration is indeed uniform — this single step prevents countless mark deductions in exams.

运动学是力学的基础。关键在于认识到SUVAT方程组仅适用于匀加速运动。在应用之前，务必检查加速度是否确实恒定——这一步骤能避免考试中无数不必要的失分。

• Equations of motion (constant acceleration a):

  v = u + at    s = ut + ½at²    v² = u² + 2as    s = ½(u + v)t

• Each equation contains four of the five variables (u, v, a, s, t). Identify which three you know and which one you need — then choose the equation that omits the irrelevant variable.

Each equation contains four of the five variables (u, v, a, s, t). Identify which three you know and which one you need — then choose the equation that omits the irrelevant variable.

每个方程包含五个变量（u、v、a、s、t）中的四个。先确定已知哪三个量、需要求哪个量，然后选择不含无关变量的那个方程即可。

Worked principle: An object is dropped from rest. After 3 seconds, its velocity is v = 0 + (9.8)(3) = 29.4 m/s, and its displacement is s = ½(9.8)(3²) = 44.1 m. Notice how the sign conventions for up/down directions must be consistent throughout the calculation.

应用示例：物体从静止开始下落。3秒后速度为 v = 0 + (9.8)(3) = 29.4 m/s，位移为 s = ½(9.8)(3²) = 44.1 m。注意上/下方向的正负号约定须在整个计算中保持一致。

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2. Dynamics: Newton’s Laws and Momentum | 动力学：牛顿定律与动量

Newton’s second law is often quoted as F = ma, but the more fundamental form is F = Δp/Δt, the rate of change of momentum. When mass is constant, the former follows directly; when mass changes (e.g. rockets), the latter becomes essential.

牛顿第二定律常写作 F = ma，但更基本的形式是 F = Δp/Δt，即动量变化率。当质量恒定时，前者可直接由后者推出；当质量发生变化时（如火箭），则必须使用后一种形式。

F = ma     p = mv     F = Δp/Δt     Impulse = FΔt = Δp

• Conservation of momentum: total momentum before = total momentum after (in a closed system with no external forces).
• 动量守恒：在无外力作用的封闭系统中，碰撞前后总动量守恒。
• The impulse-momentum theorem explains why airbags reduce injury: increasing contact time Δt reduces average force F for the same change in momentum.
• 冲量-动量定理解释了安全气囊为何能减少伤害：在动量变化相同时，增大接触时间Δt可以减小平均作用力F。

Elastic vs inelastic collisions: In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, only momentum is conserved. In a perfectly inelastic collision, objects stick together. When a ball rebounds from a wall, the change in momentum is 2mv, not mv — a classic exam trap.

弹性碰撞与非弹性碰撞：弹性碰撞中动量与动能均守恒；非弹性碰撞中仅动量守恒。完全非弹性碰撞中物体粘合在一起运动。小球从墙壁反弹时，动量变化为2mv而非mv——这是经典考试陷阱。

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3. Work, Energy and Power | 功、能与功率

Energy is the currency of physics — it is never created or destroyed, only transformed. The work-energy theorem links force and motion: the net work done on an object equals its change in kinetic energy.

能量是物理学的通用货币——它既不会凭空产生，也不会凭空消失，只会相互转化。动能定理将力与运动联系起来：对物体所做的净功等于其动能的变化。

W = Fs·cosθ     KE = ½mv²     PE = mgh     P = W/t = Fv

• In energy conservation problems, always account for ALL forms of energy: kinetic, gravitational potential, elastic potential, thermal (from friction), sound, etc.
• 在能量守恒问题中，务必考虑所有形式的能量：动能、重力势能、弹性势能、摩擦产生的热能、声能等。
• Power is the rate of energy transfer. The formula P = Fv is particularly useful when dealing with vehicles: at constant power, as speed increases, the driving force decreases.
• 功率是能量传递的速率。公式P = Fv在处理车辆问题时特别有用：功率恒定时，速度增大则牵引力减小。

Common exam application — pendulum: At the highest point, PE is maximum and KE is zero. At the lowest point, PE is zero and KE is maximum. If a pendulum swings from height h, its maximum speed is v = √(2gh), independent of the mass.

常见考点——单摆：最高点时势能最大、动能为零；最低点时势能为零、动能最大。若单摆从高度h处释放，最大速度 v = √(2gh)，与质量无关。

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4. Circular Motion and Gravitation | 圆周运动与万有引力

Uniform circular motion requires a net force pointing toward the centre — the centripetal force. It is not a new force; rather, it is the role played by tension, friction, gravity, or the normal reaction in providing the necessary acceleration toward the centre.

匀速圆周运动需要指向中心的合力——向心力。它并非一种新的力，而是张力、摩擦力、重力或支持力在提供指向圆心所需加速度时所担任的角色。

a = v²/r = ω²r     F = mv²/r = mω²r     F = GMm/r²     v = √(GM/r)

Orbital speed derivation: For a satellite orbiting mass M at radius r, the gravitational force provides the centripetal force: GMm/r² = mv²/r. Cancelling m gives v² = GM/r — one of the most elegant derivations in physics, and a favourite in exam questions.

轨道速度推导：对于绕质量M、半径为r的轨道运行的卫星，万有引力提供向心力：GMm/r² = mv²/r。消去m后得到 v² = GM/r——这是物理学中最优雅的推导之一，也是考试题目的最爱。

Kepler’s third law: T² ∝ r³. This follows from equating GMm/r² = m(2π/T)²r, leading to T² = (4π²/GM)r³. A simpler way to remember it: all planets orbiting the same star have the same T²/r³ ratio.

开普勒第三定律：T² ∝ r³。可由 GMm/r² = m(2π/T)²r 推导得出 T² = (4π²/GM)r³。更简单的记忆方式：绕同一恒星运行的所有行星具有相同的 T²/r³ 比值。

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5. Simple Harmonic Motion (SHM) | 简谐运动（SHM）

SHM occurs when the restoring force is proportional to displacement and directed toward equilibrium: F = −kx. The negative sign is crucial — it indicates the force always opposes the displacement.

当回复力与位移成正比且指向平衡位置时，即为简谐运动：F = −kx。负号至关重要——它表示力总是与位移方向相反。

a = −ω²x     x = A·sin(ωt)     T = 2π√(m/k)     T = 2π√(L/g)

• Maximum speed occurs at equilibrium (x = 0): v_max = Aω.
• 最大速度出现在平衡位置（x = 0）处：v_max = Aω。
• Maximum acceleration occurs at amplitude (x = A): a_max = Aω².
• 最大加速度出现在振幅处（x = A）：a_max = Aω²。
• For a pendulum, the period depends only on length L and g — not on mass or amplitude (for small angles).
• 对于单摆，周期仅取决于摆长L和g——与质量和振幅无关（小角度条件下）。

Energy exchanges in SHM: At maximum displacement, all energy is potential: E = ½kA². At equilibrium, all energy is kinetic: E = ½mv_max². The total mechanical energy remains constant throughout the oscillation. Exam questions often ask you to find the speed at a given displacement — use ½kA² = ½kx² + ½mv².

简谐运动中的能量转化：最大位移处，所有能量为势能：E = ½kA²；平衡位置处，所有能量为动能：E = ½mv_max²。整个振动过程中机械能总量保持不变。考试常要求求出某一位移处的速度——可利用 ½kA² = ½kx² + ½mv² 求解。

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6. Thermal Physics: Ideal Gases | 热学：理想气体

The ideal gas equation links pressure, volume, temperature and the number of moles in a single framework. The key to applying it successfully is temperature — it must ALWAYS be in kelvin (K), never in degrees Celsius.

理想气体状态方程将压强、体积、温度与物质的量统一在一个框架中。成功应用它的关键是温度——必须始终使用开尔文（K），绝不能使用摄氏度。

pV = nRT     pV = NkT     pV/T = constant

• R is the molar gas constant (8.31 J·mol⁻¹·K⁻¹); k is Boltzmann’s constant (1.38 × 10⁻²³ J·K⁻¹).
• R是摩尔气体常数（8.31 J·mol⁻¹·K⁻¹）；k是玻尔兹曼常数（1.38 × 10⁻²³ J·K⁻¹）。
• In isothermal processes (constant T): p₁V₁ = p₂V₂ (Boyle’s law).
• 等温过程（T恒定）：p₁V₁ = p₂V₂（玻意耳定律）。
• In isobaric processes (constant p): V₁/T₁ = V₂/T₂ (Charles’s law).
• 等压过程（p恒定）：V₁/T₁ = V₂/T₂（查理定律）。

Kinetic theory link: The average translational kinetic energy of gas molecules is given by KE_avg = ½m⟨v²⟩ = (3/2)kT. This beautiful result shows that temperature is a direct measure of the average molecular kinetic energy — the hotter the gas, the faster its molecules move.

与分子动理论的联系：气体分子的平均平动动能由 KE_avg = ½m⟨v²⟩ = (3/2)kT 给出。这一优美结论表明，温度是分子平均动能的直接量度——气体越热，分子运动越快。

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7. Electric Fields and Circuits | 电场与电路

Coulomb’s law governs the force between point charges, while electric field strength is defined as force per unit positive charge. In circuits, Ohm’s law and the power equations form the foundation of almost every calculation.

库仑定律描述点电荷间的相互作用力，电场强度定义为每单位正电荷所受的力。在电路中，欧姆定律和功率公式几乎是所有计算的基础。

F = kQ₁Q₂/r²     E = F/q     V = IR     P = VI = I²R = V²/R

• Electric field strength from a point charge: E = kQ/r² (note: this describes the field at a point, distinct from the uniform field E = V/d between parallel plates).
• 点电荷产生的电场强度：E = kQ/r²（注意：这描述的是空间某点的场强，与平行板间的匀强电场 E = V/d 不同）。
• In circuits, the terminal voltage of a cell with EMF ε and internal resistance r is V = ε − Ir. When current flows through a battery, some voltage is lost across the internal resistance.
• 在电路中，电动势为ε、内阻为r的电池的路端电压为 V = ε − Ir。当电流通过电池时，部分电压降落在内阻上。

Series vs parallel: In series, resistances add: R_total = R₁ + R₂ + R₃. In parallel, reciprocals add: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃. Understanding this distinction is essential for analysing complex circuit diagrams in exam papers.

串联与并联：串联时电阻相加：R_total = R₁ + R₂ + R₃；并联时倒数相加：1/R_total = 1/R₁ + 1/R₂ + 1/R₃。理解这一区别是分析考试试卷中复杂电路图的基础。

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8. Magnetic Fields and Electromagnetic Induction | 磁场与电磁感应

Magnetic forces act on moving charges — never on stationary ones. The force on a current-carrying wire is given by F = BIL·sinθ, and on a moving charge by F = qvB·sinθ. When θ = 90°, sinθ = 1 and the force is maximum.

磁场力作用于运动电荷——对静止电荷不产生作用。载流导线所受安培力为 F = BIL·sinθ，运动电荷所受洛伦兹力为 F = qvB·sinθ。当θ = 90°时，sinθ = 1，力达到最大值。

F = BIL·sinθ     F = qvB·sinθ     Φ = BA·cosθ     ε = −N·ΔΦ/Δt

Faraday’s law states that the induced EMF equals the rate of change of magnetic flux linkage. The negative sign (Lenz’s law) reminds us that the induced current opposes the change producing it — energy conservation in electromagnetic form.

法拉第定律表明，感应电动势等于磁通链的变化率。负号（楞次定律）提醒我们：感应电流的方向总是阻碍引起它的磁通量变化——这是能量守恒在电磁学中的体现。

Right-hand rules: Use the right-hand grip rule for the field around a current-carrying wire, and Fleming’s left-hand rule for the force on a conductor in a magnetic field. Exam candidates often confuse the two — practise until the choice becomes automatic.

右手定则：用右手螺旋定则判断载流导线周围的磁场方向，用弗莱明左手定则判断磁场中导体所受力的方向。考生常将两者混淆——需要反复练习直至能自动做出正确选择。

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9. Waves and Optics | 波动与光学

The wave equation v = fλ connects speed, frequency and wavelength, while interference phenomena provide some of the most visually striking demonstrations of wave nature.

波动方程 v = fλ 将波速、频率和波长联系起来，而干涉现象则是对波动性的最直观演示之一。

v = fλ     n = c/v     n₁·sinθ₁ = n₂·sinθ₂     d·sinθ = nλ

• Snell’s law, n₁·sinθ₁ = n₂·sinθ₂, governs refraction. The refractive index n is the ratio of the speed of light in vacuum to its speed in the medium.
• 斯涅尔定律 n₁·sinθ₁ = n₂·sinθ₂ 描述折射规律。折射率n是光在真空中的速度与在介质中速度的比值。
• Young’s double-slit experiment: fringe spacing x = λD/d, where D is the slit-to-screen distance and d is the slit separation.
• 杨氏双缝实验：条纹间距 x = λD/d，其中D为缝到屏的距离，d为双缝间距。
• For diffraction gratings, d·sinθ = nλ gives the angles of bright maxima, where d is the grating spacing.
• 对于衍射光栅，d·sinθ = nλ 给出各级亮纹的角度，其中d为光栅常数。

Total internal reflection: When light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle, all light is reflected back. The critical angle is found from sinθ_c = n₂/n₁ (when n₂ corresponds to the rarer medium). This principle powers optical fibres and is a frequent exam topic.

全反射：当光从光密介质射向光疏介质且入射角大于临界角时，光全部被反射回来。临界角由 sinθ_c = n₂/n₁ 求得（n₂对应光疏介质）。该原理是光纤技术的核心，也是高频考点。

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10. Quantum Physics and the Photoelectric Effect | 量子物理与光电效应

The photoelectric effect provided the crucial evidence for the particle nature of light. Einstein’s equation establishes that light energy comes in discrete packets (photons), and that the maximum kinetic energy of emitted electrons depends on the frequency of light, not its intensity.

光电效应为光的粒子性提供了关键证据。爱因斯坦方程确立了光能是以不连续的量子（光子）形式存在的，并且逸出电子的最大动能取决于光的频率，而非光的强度。

E = hf     KE_max = hf − Φ     hf = Φ + KE_max

• Φ is the work function: the minimum energy needed to eject an electron from the metal surface.
• Φ为逸出功：从金属表面逸出一个电子所需的最小能量。
• Threshold frequency f₀ = Φ/h: below this frequency, no electrons are emitted regardless of intensity.
• 截止频率 f₀ = Φ/h：低于此频率时，无论光强多大都不会有电子逸出。
• Increasing intensity increases the number of photoelectrons, but NOT their individual kinetic energy — a distinction tested repeatedly in exams.
• 增大光强会增加光电子数量，但不会增加单个电子的动能——这一区别在考试中被反复考查。

Wave-particle duality: De Broglie proposed that all matter has a wavelength λ = h/p = h/(mv). This explains why the electron microscope can resolve much finer detail than an optical microscope — electrons with sufficient speed have wavelengths far shorter than visible light.

波粒二象性：德布罗意提出所有物质都有波长 λ = h/p = h/(mv)。这解释了为何电子显微镜比光学显微镜有更高的分辨率——速度足够快的电子拥有远短于可见光的波长。

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11. Exam Strategy: From Formula to Full Marks | 考试策略：从公式到满分

Knowing formulas is necessary but not sufficient. Top-scoring candidates develop a systematic approach to solving physics problems that minimises errors and maximises partial credit even when the final answer is wrong.

知道公式是必要条件但非充分条件。高分考生会建立系统的解题流程，以最小化错误并最大化过程分——即使在最终答案不正确的情况下。

Step 1 — List what you know and what you need. Write down all given quantities with their symbols and units. Convert to base SI units first (e.g. km → m, g → kg, °C → K).

第一步——列出已知量和待求量。将所有已知量及其符号、单位写下来。先将所有量转换为国际单位制基本单位（如km→m、g→kg、°C→K）。

Step 2 — Select the relevant formula. Identify which physical principle applies. Check conditions: Is acceleration constant? Is the system isolated? Is temperature in kelvin?

第二步——选择相关公式。判断适用哪个物理原理。检查条件：加速度是否恒定？系统是否孤立？温度是否使用开尔文？

Step 3 — Solve symbolically first. Rearrange algebraically before substituting numbers. This reduces arithmetic errors and earns method marks even if a calculation mistake occurs.

第三步——先代数求解。先进行代数变形再代入数值。这可以减少计算错误，即使计算有误也能获得方法分。

Step 4 — Check the answer. Verify units match the physical quantity you sought. Check the magnitude: is it sensible? A car travelling at 500 m/s, or an orbital radius smaller than the Earth’s radius, signals an error.

第四步——检查答案。确认单位与所求物理量一致。检查数量级是否合理：车速500 m/s，或轨道半径小于地球半径，都说明计算有误。

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12. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Success in physics exams often comes down to avoiding repeated errors. Here are the most frequently observed mistakes and the habits that prevent them.

物理考试能否成功，往往取决于能否避免反复出现的错误。以下是最常见的失误及其预防习惯。

Error | 错误	Prevention | 预防方法
Using °C instead of K in gas laws | 气体定律中误用摄氏度	Always convert: T(K) = T(°C) + 273.15 | 始终换算：T(K) = T(°C) + 273.15
Forgetting the negative sign in SHM and induced EMF | 忘记简谐运动和感应电动势中的负号	Understand the physical meaning of the negative sign | 理解负号的物理含义
Wrong direction for momentum change in rebounds | 反弹问题中动量变化方向弄错	Set a convention and use Δp = mv_final − mv_initial | 设定正方向，使用Δp = mv_final − mv_initial
Confusing mass and weight (N vs kg) | 混淆质量与重量（N和kg）	Weight = mg, always in newtons | 重量 = mg，单位永远是牛顿
Using v = u + at when a is not constant | 当加速度不恒定时仍使用运动学公式	Check validity conditions before applying equations | 应用公式前检查适用条件

Developing the habit of writing down units at every step transforms vague intuition into rigorous calculation. A final answer without units is, in most exam marking schemes, incomplete — even if the number is correct.

养成每一步都书写单位的习惯，能将模糊的直觉转变为严谨的计算。在大多数评分标准中，没有单位的最终答案是不完整的——即使数字完全正确。

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