📚 How to Memorise Core Physics Formulas: A Complete Revision System | 物理备考：如何牢记核心公式
Physics exams do not test how many equations you can recite; they test whether you can select, apply and rearrange the right equation under time pressure. That means the goal of formula revision is not memorisation for its own sake but fast, accurate retrieval. This guide sets out a complete system: understand each formula as a sentence about the physical world, check it with dimensional analysis, organise formulas into families, space your practice, and rehearse retrieval so that the equation arrives in your hand the way a phone number arrives when you dial it.
物理考试考查的不是你能背出多少个公式，而是你能不能在时间压力下选对公式、用对公式、并正确地变形。所以公式复习的目标不是”背下来”本身，而是快速、准确地调取。本文给出一个完整体系：把每个公式理解成一句关于物理世界的陈述，用量纲分析做校验，把公式按”家族”归类，安排间隔复习，并反复训练调取过程——让公式在需要时像拨号时想起电话号码一样自然浮现。
1. Why Formulas Slip Out of Memory | 为什么公式总是记不住
Most students lose formulas for four predictable reasons. First, they revise passively: reading a formula list feels productive but creates almost no retrieval strength. Second, similar-looking formulas interfere with each other, so F = Gm₁m₂/r² and F = (1/4πε₀)Q₁Q₂/r² collapse into one blurry memory. Third, symbols are memorised without meaning, so V = W/Q and V = IR live in separate mental boxes instead of being understood as two ways of describing the same quantity. Fourth, students rely on the data booklet and never build the internal speed needed to answer in 90 seconds per mark.
大多数学生丢公式，原因无非四种。第一，被动复习：看公式表很有”学习感”，但几乎不产生调取强度。第二，形近公式互相干扰，于是 F = Gm₁m₂/r² 和 F = (1/4πε₀)Q₁Q₂/r² 在记忆里糊成一团。第三，符号脱离了意义，V = W/Q 和 V = IR 被放进两个独立的抽屉，而不是被理解成描述同一个物理量的两种方式。第四，过度依赖公式表，从未建立”每题每分 90 秒”所需要的内在速度。
The fix is to attack all four at once: give every formula a meaning, a unit check, a family, and a retrieval schedule.
解决办法是四条同时下手：给每个公式一个意义、一个单位校验、一个所属家族、一个复习时间表。
2. Read Every Formula as a Sentence | 把每个公式读成一句话
A formula is a compressed sentence. If you can expand it into plain language, you can usually rebuild it from scratch. Acceleration a = Δv/Δt reads as “acceleration is how quickly velocity changes”. Density ρ = m/V reads as “how much mass is packed into each unit of volume”. Resistance R = V/I reads as “how many volts are needed to push one ampere through”. Once the sentence exists, the symbols become almost automatic.
公式是被压缩的句子。能把它展开成大白话，通常就能从零把它重建出来。加速度 a = Δv/Δt 读作”加速度是速度变化的快慢”。密度 ρ = m/V 读作”单位体积里塞了多少质量”。电阻 R = V/I 读作”要推动 1 安培的电流需要多少伏特”。句子一旦成立，符号几乎会自动跟上。
Proportionality reading is even more powerful for exam questions, because it lets you answer ratio questions without exact numbers:
用比例关系来读公式，对考试更管用，因为它让你不必算出具体数值就能回答比值题：
- F = ma: force is proportional to acceleration when mass is fixed. | F = ma：质量不变时，力与加速度成正比。
- P = V²/R: power is proportional to the square of the voltage. | P = V²/R：功率与电压的平方成正比。
- g = GM/r²: gravitational field strength falls as the inverse square of distance. | g = GM/r²：重力场强度随距离的平方反比衰减。
- E = ½mv²: kinetic energy is proportional to the square of speed, so doubling speed quadruples energy. | E = ½mv²：动能与速度的平方成正比，所以速度翻倍，动能变四倍。
3. Dimensional Analysis as a Memory Check | 用量纲分析做记忆校验
Every physically valid equation must be homogeneous: the base units on the left must equal the base units on the right. This single rule catches most memory errors before you lose marks. Test v² = u² + 2as: the left side is (m s⁻¹)² = m² s⁻², and the right side is m² s⁻² + (m s⁻²)(m) = m² s⁻². It passes, so the structure is credible.
任何物理上成立的方程都必须”量纲齐次”：左边的基本单位必须等于右边的基本单位。这一条规则能在丢分之前拦住大部分记忆错误。检验 v² = u² + 2as：左边是 (m s⁻¹)² = m² s⁻²，右边是 m² s⁻² + (m s⁻²)(m) = m² s⁻²，通过，说明结构可信。
Now test a common slip: writing s = ut + ½at. The right side gives m s⁻¹ · s + m s⁻² · s = m + m s⁻¹, which is nonsense. The failed check instantly tells you a time factor is missing, and you recover s = ut + ½at². Do the same for R = ρL/A: Ω = (Ω m)(m)/(m²) = Ω, correct. For 1/f = 1/u + 1/v: m⁻¹ = m⁻¹ + m⁻¹, correct, which also explains why lens powers add in dioptres (m⁻¹).
再检验一个常见错误写法：s = ut + ½at。右边给出 m s⁻¹ · s + m s⁻² · s = m + m s⁻¹，明显不通。校验失败立刻告诉你少了一个时间因子，你就能恢复成 s = ut + ½at²。同样检验 R = ρL/A：Ω = (Ω m)(m)/(m²) = Ω，正确。检验 1/f = 1/u + 1/v：m⁻¹ = m⁻¹ + m⁻¹，正确，这也顺便解释了为什么透镜焦度用屈光度（m⁻¹）相加。
Be clear about the limits: dimensional analysis cannot detect a missing ½, a stray minus sign, or a factor of 2π. Those must come from meaning and from derivation, which is why sections 5 to 9 matter.
但要清楚它的边界：量纲分析查不出漏掉的 ½、多余的负号或 2π。这些必须靠意义和推导来解决，这正是第 5 至第 9 节存在的原因。
4. Build a Formula Map Instead of a Formula List | 搭建”公式地图”而不是”公式清单”
A flat list of forty equations is hard to store; a map of five families is easy. Group every formula you meet into one of these families: definitions (a quantity defined as a ratio), laws (relationships discovered experimentally), conservation statements (energy, momentum, charge), rate relationships (something per unit time), and geometry factors (2π, 4π, ½, squares and inverse squares).
四十个公式的平铺清单很难储存，五个家族构成的地图却很好记。把所有公式归入以下几类：定义（某量被定义为一个比值）、定律（实验发现的关系）、守恒陈述（能量、动量、电荷）、变化率关系（某量每单位时间的变化）、以及几何因子（2π、4π、½、平方与平方反比）。
The most powerful family in A-Level physics is the rate family, because one pattern generates dozens of formulas:
A-Level 物理中最强大的家族是”变化率家族”，因为一个模式能生成几十个公式：
| Rate pattern | Formula | Physical meaning |
| s per t | v = Δs/Δt | velocity is displacement per unit time |
| v per t | a = Δv/Δt | acceleration is velocity change per unit time |
| Q per t | I = ΔQ/Δt | current is charge flow per unit time |
| W per t | P = ΔW/Δt | power is energy transfer per unit time |
| Φ per t | ε = −N ΔΦ/Δt | induced emf is flux change per unit time |
| N per t | A = λN | activity is decays per unit time |
Learn the pattern once and you have effectively learned six formulas. The same trick works for graph skills: if gradient equals rate of change, then on a displacement–time graph the gradient is velocity, on a velocity–time graph it is acceleration, and on a charge–time graph it is current.
把这个模式学一次，等于学会了六个公式。同样的技巧也适用于图像题：如果”斜率 = 变化率”，那么位移–时间图的斜率是速度，速度–时间图的斜率是加速度，电荷–时间图的斜率是电流。
5. The Mechanics Core Set | 力学核心公式组
Mechanics is where most marks are won, and the equations form a tidy chain. Learn them in the order constant acceleration, force, energy, momentum, so that each group reminds you of the next.
力学是分值的主战场，而这些公式构成一条整齐的链条。按”匀变速—力—能量—动量”的顺序记忆，每一组都会提示下一组。
v = u + at s = ut + ½at² v² = u² + 2as s = ½(u + v)t
F = ma p = mv F = Δp/Δt W = Fs cos θ P = W/t = Fv
Eₖ = ½mv² ΔEₚ = mgΔh F = kx E = ½kx² a = v²/r = ω²r
The suvat set is best stored as a story: you always know three of the five quantities, and each equation omits exactly one. Omit s and you get v = u + at; omit v and you get s = ut + ½at²; omit t and you get v² = u² + 2as. Students who memorise “which letter is missing” almost never pick the wrong suvat equation.
匀变速五式最好按”缺谁”来记：你总是已知五个量中的三个，而每个方程恰好不含其中一个。不含 s 得到 v = u + at；不含 v 得到 s = ut + ½at²；不含 t 得到 v² = u² + 2as。记住”缺哪个字母”的学生，几乎不会选错方程。
For simple harmonic motion, remember one anchor equation and one proportionality:
简谐运动只需记住一个锚点方程和一条正比关系：
a = −ω²x ω = 2πf = 2π/T v = ω√(A² − x²) T = 2π√(m/k)
The minus sign in a = −ω²x is the definition of SHM: acceleration is always directed towards the equilibrium position and is proportional to displacement. If you remember why the minus is there, you will never drop it.
a = −ω²x 中的负号就是简谐运动的定义：加速度始终指向平衡位置，且与位移成正比。只要记住负号的来历，就永远不会漏掉它。
6. Electricity and Circuits | 电学与电路
Electricity formulas are best remembered in three tiers: definitions, the resistance family, and circuit rules. The definitions tier answers “what is this quantity?” The resistance tier answers “how do these quantities link?”
电学公式最好分三层记忆：定义层、电阻家族、电路规则。定义层回答”这个量是什么”，电阻层回答”这些量如何关联”。
Q = It V = W/Q V = IR P = VI = I²R = V²/R
Notice the symmetry in the power family: P = VI is the definition, and substituting V = IR gives P = I²R, while substituting I = V/R gives P = V²/R. Only one of the three needs to be memorised; the other two are one substitution away. This is the “derive, don’t drill” principle in action.
注意功率家族中的对称性：P = VI 是定义，代入 V = IR 得 P = I²R，代入 I = V/R 得 P = V²/R。三个只需记一个，另外两个只差一次代换。这就是”推导优于死背”的实例。
Circuit rules and internal resistance complete the picture:
电路规则与内阻补全整个图景：
- Series: R = R₁ + R₂ + … and the current is the same everywhere. | 串联：R = R₁ + R₂ + …，各处电流相同。
- Parallel: 1/R = 1/R₁ + 1/R₂ + …, so the combined resistance is smaller than the smallest branch. | 并联：1/R = 1/R₁ + 1/R₂ + …，所以总电阻小于最小的支路电阻。
- Internal resistance: ε = I(R + r), rearranged as ε = V + Ir, and V = ε − Ir on a terminal-potential graph. | 内阻：ε = I(R + r)，变形得 ε = V + Ir，在端电压图上 V = ε − Ir。
- Potential divider: Vout = Vin × R₂/(R₁ + R₂). | 分压器：Vout = Vin × R₂/(R₁ + R₂)。
- Resistivity: R = ρL/A, so resistance doubles if length doubles, and halves if area doubles. | 电阻率：R = ρL/A，所以长度翻倍电阻翻倍，横截面积翻倍电阻减半。
Capacitors add two more, which pair neatly with the nuclear decay equations later:
电容器再加两个，而且能与后面的核衰变方程配对记忆：
Q = CV W = ½QV = ½CV² τ = RC Q = Q₀ e^(−t/RC)
7. Waves, Optics and Thermal Physics | 波、光学与热学
Waves reduce to one core equation plus one reciprocal relationship. Everything else in the topic is either a definition of a ratio or a consequence of superposition.
波的部分归结为一个核心方程加一个倒数关系，该主题其余内容要么是比值的定义，要么是叠加原理的结果。
v = fλ f = 1/T n = c/v n₁ sin θ₁ = n₂ sin θ₂ n = 1/sin C
x = λD/d I = P/(4πr²) I ∝ A²
The double-slit fringe spacing x = λD/d is easy to garble, so check it dimensionally: (m)(m)/(m) = m, a length, correct. Then check it logically: a longer wavelength spreads fringes further apart, a larger slit separation packs them closer. Both checks together make the formula almost impossible to forget.
双缝条纹间距 x = λD/d 很容易记混，所以先做量纲检验：(m)(m)/(m) = m，量纲是长度，正确。再做逻辑检验：波长越长条纹越散，缝间距越大条纹越密。两个检验合在一起，这个公式几乎不可能记错。
Thermal physics is built on two energy equations and one gas equation:
热学建立在两个能量方程和一个气体方程之上：
E = mcΔθ E = mL pV = nRT pV = NkT p₁V₁/T₁ = p₂V₂/T₂
Keep the two E = mc forms apart by their symbols: c with a Δθ is specific heat capacity (J kg⁻¹ K⁻¹), L is specific latent heat (J kg⁻¹) and applies at constant temperature. The gas constant R and Boltzmann constant k differ only by Avogadro’s number: R = Nₐk. Remembering that single link halves the memory burden.
两个 E = mc 形式要靠符号区分：带 Δθ 的 c 是比热容（J kg⁻¹ K⁻¹），L 是比潜热（J kg⁻¹），只在恒温相变时使用。气体常量 R 与玻尔兹曼常量 k 只差一个阿伏伽德罗常量：R = Nₐk。记住这一条联系，记忆负担立刻减半。
For the first law of thermodynamics, fix your sign convention in writing before the exam and never switch: ΔU = Q + W where W is work done on the gas, or ΔU = Q − W where W is work done by the gas. Write the convention next to the formula in your notes.
关于热力学第一定律，考前就把符号约定写定，考试中绝不切换：ΔU = Q + W（W 为外界对气体做功）或 ΔU = Q − W（W 为气体对外做功）。把约定写在笔记里公式旁边。
8. Fields, Magnetism and Nuclear Physics | 场、磁学与核物理
Gravitational and electric fields are the same mathematics with different constants, so learn them as a mirrored pair rather than two separate lists. This mirroring is one of the biggest time-savers in the whole syllabus.
引力场与电场是同一套数学、不同的常量，所以要当作一对镜像来记，而不是两张独立的清单。这个镜像结构是整个考纲里最省时间的技巧之一。
| Concept | Gravitational | Electric |
| Force between two bodies | F = Gm₁m₂/r² | F = (1/4πε₀)Q₁Q₂/r² |
| Field strength | g = GM/r² | E = (1/4πε₀)Q/r² |
| Potential | V = −GM
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