📚 A-Level Physics: Key Formula Derivations from the June 2018 Insert Sheet | A-Level 物理：2018年6月插入表关键公式推导
In A-Level Physics, the insert sheet handed out with the exam paper is a valuable reference: it lists the equations you are not required to memorise, covering mechanics, materials, waves, fields, electricity, magnetism and nuclear physics. The June 2018 insert (for example AQA 7408/1, and the equivalent sheets for OCR A and Edexcel) is typical of this genre. Although the equations are printed for you, working through their derivations transforms them from symbols you copy into tools you actually understand, and it is exactly this understanding that examiners reward in the "show that" and multi-step calculation questions.
在 A-Level 物理考试中，随试卷一起发放的插入表是宝贵的参考资料：它列出了你不必死记硬背的方程，涵盖力学、材料、波动、场、电学、磁学和核物理。2018年6月的插入表（例如 AQA 7408/1，以及 OCR A 与 Edexcel 的对应版本）就是这类资料的典型代表。虽然这些方程会直接印给你，但亲手推导能把它们从"照抄的符号"变成"真正理解的工具"，而这正是考官在"证明题"和多步计算题中要奖励的能力。
This article rebuilds the key derivations from that insert, step by step, in the order the topics usually appear on a specification. Each derivation states the starting assumption, shows the algebra, and explains what the result means physically. Where the insert lists several equivalent forms, we show how one form turns into another, because fluency in switching between forms is what saves time under exam pressure. All formulae are written with standard symbols so they match the insert sheet exactly.
本文将按考试大纲通常出现的顺序，逐步重建该插入表中的关键推导。每一段推导都会先说明起始假设，再展示代数过程，最后解释结果在物理上意味着什么。当插入表列出多个等价形式时，我们会展示它们之间如何互相转换，因为在考场压力下能够自如互换形式，正是节省时间的关键。所有公式都使用与插入表完全一致的标准符号书写。
1. How to Use the Insert Sheet Effectively | 如何高效使用插入表
The insert sheet is a formula list, not a solution manual. The first practical habit is to match every printed equation to its conditions of validity. For instance, v = u + a t is only valid for constant acceleration in a straight line, and E = ½ C V² assumes the capacitor started uncharged. Marking these conditions next to each formula when you revise prevents the most common source of lost marks: applying a correct equation to an incorrect situation.
插入表是公式清单，不是解题手册。第一个实用习惯是把每一条印刷方程与它的适用条件对应起来。例如 v = u + a t 只适用于直线上的匀加速运动，而 E = ½ C V² 假设电容器最初不带电。复习时在每个公式旁标注这些条件，就能避免最常见的失分原因：把正确的方程用在错误的情境上。
The second habit is dimensional checking. If a derivation produces an expression whose units do not match the target quantity, an algebraic slip has occurred. This is especially useful for long derivations such as the discharge equation or the resonance condition for a mass-spring system, where a missing factor of 2π is easy to introduce and hard to notice.
第二个习惯是量纲检验。如果推导得到的表达式单位与目标物理量不符，说明代数出了错。这对较长的推导尤其有用，例如放电方程或弹簧振子的共振条件，因为漏掉一个 2π 很容易发生，却很难察觉。
The third habit is to record what each symbol means. On the insert, the same letter can carry different meanings across topics: k is a spring constant in materials but the Coulomb constant in electrostatics; E is energy, electric field strength and the Young modulus depending on context. A short glossary written by hand alongside the sheet removes this ambiguity.
第三个习惯是记录每个符号的含义。在插入表中，同一个字母在不同主题下含义不同：k 在材料学中是劲度系数，在静电学中却是库仑常量；E 视语境可表示能量、电场强度或杨氏模量。手写一份简短的符号表贴在插入表旁，就能消除这种歧义。
2. Uniform Acceleration Equations (SUVAT) | 匀加速运动方程 (SUVAT)
Under constant acceleration a, the change in velocity changes at a uniform rate. Acceleration is defined as the rate of change of velocity, so we start from the definition itself:
在恒定加速度 a 下，速度以均匀速率变化。加速度的定义是速度的变化率，因此我们从定义本身出发：
a = (v − u) ÷ t
Rearranging immediately gives the first SUVAT equation, v = u + a t. Because acceleration is constant, the velocity-time graph is a straight line, so , and the average velocity over the interval is simply (u + v) ÷ 2. Displacement equals average velocity multiplied by time:
整理后立刻得到第一个 SUVAT 方程 v = u + a t。由于加速度恒定，速度-时间图是一条直线，因此该区间内的平均速度就是 (u + v) ÷ 2。位移等于平均速度乘以时间：
s = ((u + v) ÷ 2) × t
Substituting v = u + a t into this expression gives the third equation:
将 v = u + a t 代入上式，得到第三个方程：
s = u t + ½ a t²
To eliminate time, substitute t = (v − u) ÷ a into the displacement equation and simplify. The cross terms cancel and the result is the time-independent equation:
若要消去时间，把 t = (v − u) ÷ a 代入位移方程并化简。交叉项相消，得到与时间无关的方程：
v² = u² + 2 a s
A fifth form, s = v t − ½ a t², follows by the same substitution applied to u = v − a t. Together the five equations form a closed toolkit: each one omits exactly one of the five quantities s, u, v, a, t, so choosing the equation that omits the quantity you neither know nor need is the fastest route to the answer. In graph terms, the gradient of a velocity-time graph gives acceleration and the area beneath it gives displacement, which is precisely where equations 2 and 3 come from.
第五种形式 s = v t − ½ a t² 由同一代换以 u = v − a t 得到。这五个方程构成一个封闭工具箱：每一个都恰好缺少 s、u、v、a、t 中的一个量，因此选择那个"缺少你既不知道也不需要求的量"的方程，是通向答案最快的路径。从图像上看，速度-时间图的斜率给出加速度，图线下的面积给出位移，这恰恰就是方程2和方程3的来源。
3. Newton's Second Law and the Momentum Form | 牛顿第二定律与动量形式
Newton's second law in its most general form states that the resultant force equals the rate of change of momentum:
牛顿第二定律最普遍的形式是：合力等于动量的变化率：
F = Δp ÷ Δt
Since momentum p = m v, for a body of constant mass the change in momentum is Δp = m Δv, so the law reduces to the familiar form F = m a. The general form is essential when mass changes during the interaction, as with a rocket expelling fuel or a conveyor belt being loaded, where F = m a alone gives the wrong answer.
由于动量 p = m v，对于质量不变的物体，动量变化为 Δp = m Δv，于是该定律简化为熟悉的形式 F = m a。当相互作用过程中质量发生变化时，一般式就不可或缺，例如火箭喷射燃料或传送带被不断加料的情形，此时单用 F = m a 会得到错误答案。
Multiplying both sides by the contact time gives the impulse-momentum equation:
两边同乘接触时间，得到冲量-动量方程：
F Δt = Δp = m v − m u
This is the physics behind crumple zones and airbags: extending Δt for a fixed change in momentum reduces the peak force. Where no external resultant force acts, Δp = 0 for the system, giving conservation of linear momentum, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. In collisions you must also check whether kinetic energy is conserved: elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum only.
这正是溃缩区和安全气囊背后的物理原理：在动量变化量固定时延长 Δt，就能降低峰值力。当没有外力合力作用时，系统 Δp = 0，即得到线动量守恒：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。在碰撞问题中还必须检验动能是否守恒：弹性碰撞同时守恒动量和动能，而非弹性碰撞只守恒动量。
4. Work, Energy, Power and Efficiency | 功、能、功率与效率
Work is done when a force moves its point of application through a displacement. If the force acts at an angle θ to the displacement, only the component along the displacement does work:
当力的作用点沿力的方向发生位移时，就说该力做了功。若力与位移成 θ 角，只有沿位移方向的分量做功：
W = F s cos θ
The kinetic energy formula can be derived directly from the SUVAT toolkit. A resultant force F accelerates a body of mass m from rest over distance s. The work done is W = F s = m a s, and from v² = u² + 2 a s with u = 0 we have a s = v² ÷ 2. Substituting gives:
动能公式可以直接由 SUVAT 工具箱推出。合力 F 使质量 m 的物体从静止加速经过距离 s。所做的功为 W = F s = m a s；而由 v² = u² + 2 a s 取 u = 0 得 a s = v² ÷ 2。代入即得：
Eₖ = ½ m v²
For a body raised through a vertical height Δh near the Earth's surface, the work done against gravity is W = m g Δh, which defines the change in gravitational potential energy ΔEₚ = m g Δh. The linear form is only valid when g is effectively constant, that is over heights small compared with the Earth's radius; over larger distances the inverse-square form of potential energy must be used instead.
在地球表面附近把物体竖直提升高度 Δh 时，克服重力做的功为 W = m g Δh，这就是重力势能变化 ΔEₚ = m g Δh 的定义。该线性形式只在 g 可视为常量时成立，即高度远小于地球半径时；在更大距离上必须改用势能的平方反比形式。
Power is the rate of energy transfer, P = ΔW ÷ Δt = ΔE ÷ Δt, and for a force moving at constant velocity, P = F v. Efficiency compares useful output to total input, usually expressed as a percentage:
功率是能量转移的速率，P = ΔW ÷ Δt = ΔE ÷ Δt；对于以恒定速度运动的力，P = F v。效率比较有用输出与总输入，通常用百分比表示：
efficiency = (useful energy output ÷ total energy input) × 100%
5. Hooke's Law, the Young Modulus and Elastic Energy | 胡克定律、杨氏模量与弹性势能
For a wire or spring within its limit of proportionality, extension is directly proportional to the applied force:
对于处于比例极限内的金属丝或弹簧，伸长量与施加的力成正比：
F = k x
Here k is the stiffness constant, equal to the gradient of the force-extension graph in its linear region. Beyond the limit of proportionality the graph curves, and beyond the elastic limit the material deforms plastically so that removing the load does not restore the original length.
这里的 k 是劲度系数，等于力-伸长量图在线性区的斜率。超过比例极限后图线弯曲；超过弹性极限后材料发生塑性形变，卸载后无法恢复到原长。
The energy stored in a stretched spring is the area under the force-extension graph. Since the graph is a triangle of base x and height F = k x, the area is half the base times the height:
拉伸弹簧中储存的能量就是力-伸长量图下的面积。由于图线是底为 x、高为 F = k x 的三角形，其面积为底乘高的一半：
Eₚ = ½ F x = ½ k x²
For a wire of original length L, cross-sectional area A and extension ΔL, the stress is F ÷ A and the strain is ΔL ÷ L. The Young modulus is defined as the ratio of stress to strain:
对于原长 L、横截面积 A、伸长 ΔL 的金属丝，应力为 F ÷ A，应变为 ΔL ÷ L。杨氏模量定义为应力与应变之比：
E = (F ÷ A) ÷ (ΔL ÷ L) = F L ÷ (A ΔL)
Because stress has units N m⁻² and strain is dimensionless, the Young modulus also has units N m⁻² (or Pa). On a stress-strain graph the modulus is the gradient, and the area under the graph up to a given strain is the energy stored per unit volume.
由于应力单位是 N m⁻² 而无量纲的应变无单位，杨氏模量的单位也是 N m⁻²（即 Pa）。在应力-应变图上，该模量是斜率，而图线在给定应变前下方的面积则是单位体积储存的能量。
6. Circular Motion and Angular Speed | 圆周运动与角速度
For an object moving in a circle of radius r with constant speed v, its velocity direction changes continuously even though its speed does not, so it accelerates towards the centre. In one complete revolution the object travels the circumference 2πr in a period T and sweeps an angle of 2π radians, giving the two links between linear and angular quantities:
对于以恒定速率 v 沿半径 r 的圆运动的物体，虽然速率不变，但速度方向不断改变，因此它存在指向圆心的加速度。在一整圈中，物体在周期 T 内走过周长 2πr，并扫过 2π 弧度的角，由此得到线量与角量之间的两个关系：
v = 2πr ÷ T = ω r and ω = 2π ÷ T
The centripetal acceleration can be derived by considering two nearby positions of the object and the small velocity vector triangle they produce; the vector triangle is similar to the position triangle, so Δv ÷ v = Δs ÷ r. Dividing by Δt and taking the limit as Δt → 0 gives:
向心加速度可通过考察物体两个相邻位置及其形成的微小速度矢量三角形推出；该速度矢量三角形与位置三角形相似，因此 Δv ÷ v = Δs ÷ r。两边除以 Δt，并取 Δt → 0 的极限，得到：
a = v² ÷ r = ω² r
By Newton's second law the resultant force must point towards the centre and have magnitude F = m v² ÷ r = m ω² r. This "centripetal force" is a role, not a new kind of force: it is provided by tension, gravity, friction, or the normal contact force depending on the situation, and it must always be identified explicitly in an exam answer.
由牛顿第二定律，合力必须指向圆心，大小为 F = m v² ÷ r = m ω² r。这个"向心力"是一种角色而非新的力：它由张力、重力、摩擦力或法向接触力提供，视情境而定；在考试作答中必须明确指出它由什么提供。
7. Simple Harmonic Motion | 简谐运动
Simple harmonic motion is defined by the condition that acceleration is directly proportional to displacement from equilibrium and directed towards it:
简谐运动的定义是：加速度与离开平衡位置的位移成正比，且方向指向平衡位置：
a = −ω² x
If the motion starts at maximum displacement A, the displacement-time relation is x = A cos(ω t). The velocity is found by differentiating this with respect to time, giving v = −A ω sin(ω t ty sin²θ + cos²θ = 1, eliminates time and produces the useful speed-displacement relation:
若运动从最大位移 A 开始，则位移-时间关系为 x = A cos(ω t)。对时间求导得到速度 v = −A ω sin(ω t)。将两者平方相加，利用恒等式 sin²θ + cos²θ = 1 消去时间，即得实用的速度-位移关系：
v = ±ω √A² − x²
This shows that speed is a maximum at the centre (x = 0, v = ω A) and zero at the extremes (x = ±A). For a mass on a spring, the restoring force F = −k x combined with F = m a gives a = −(k ÷ m)x, so comparing with the definition, ω² = k ÷ m and:
这表明速率在中心处最大（x = 0，v = ω A），在两极处为零（x = ±A）。对于弹簧上的质量块，恢复力 F = −k x 与 F = m a 结合给出 a = −(k ÷ m)x，与定义比较得 ω² = k ÷ m，于是：
T = 2π √m ÷ k
For a simple pendulum of length l swinging through a small angle, the restoring force component is −m g sinθ ≈ −m g (x ÷ l), which again has the form −(constant)x. Identifying the constant with ω² yields T = 2π √l ÷ g. Note that in both cases the period is independent of amplitude for small oscillations — that is the isochronous property that makes pendulums useful as timekeepers.
对于摆长为 l、摆角很小的单摆，恢复力分量为 −m g sinθ ≈ −m g (x ÷ l)，同样具有 −(常数)x 的形式。把该常数等同于 ω² 即得 T = 2π √l ÷ g。注意两种情况下周期都与振幅无关（小振幅时）——正是这一"等时性"使单摆可以用作计时
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