📚 A-Level Physics Unit 2 Jan21 Formula Derivation | A-Level 物理 Unit 2 Jan21 公式推导
In the January 2021 A-Level Physics Unit 2 question paper, candidates are often required to derive and apply fundamental formulas from mechanics, materials, waves, and quantum physics. This article revisits the essential derivations that underpin the syllabus, providing step‑by‑step reasoning in both English and Chinese to strengthen your conceptual understanding.
在2021年1月的A-Level物理单元2考试中，常需要推导并应用力学、材料、波和量子物理的基本公式。本文以中英文对照的方式，重新梳理这些关键公式的推导过程，逐步展示其推理方法，帮助你深化对概念的理解。
1. Acceleration and the First Equation of Motion | 加速度与第一个运动方程
Acceleration is defined as the rate of change of velocity. For constant acceleration a, initial velocity u, final velocity v, and time t, we write:
加速度定义为速度的变化率。对于匀加速度 a、初速度 u、末速度 v 和时间 t，我们有：
a = (v − u) / t
Rearranging this definition gives the first equation of motion, which shows how velocity varies linearly with time.
移项后得到第一个运动方程，它描述了速度随时间线性变化的关系。
v = u + at
2. Displacement and the Second Equation of Motion | 位移与第二个运动方程
Displacement s equals the area under a velocity–time graph. For constant acceleration, the graph is a trapezium, so the area is the average velocity multiplied by time:
位移 s 等于速度–时间图像下的面积。对于匀加速运动，图像是一个梯形，因此面积为平均速度乘以时间：
s = ½ (u + v) t
Substituting v = u + at into this average‑velocity expression yields the second equation of motion.
将 v = u + at 代入平均速度表达式中，可导出第二个运动方程。
s = ut + ½ at²
3. Relationship Between Velocity and Displacement | 速度与位移的关系式
By eliminating time t from v = u + at and s = ½ (u + v) t, we obtain a formula that directly links velocity, acceleration, and displacement.
通过从 v = u + at 和 s = ½ (u + v) t 中消去时间 t，我们得到一个直接关联速度、加速度和位移的公式。
From the first equation, t = (v − u) / a. Substituting into the displacement equation and simplifying:
由第一个方程得 t = (v − u) / a。代入位移方程并化简：
v² = u² + 2as
This equation is extremely useful when time is not given in a problem.
当题目未给出时间时，该方程非常实用。
4. Conservation of Momentum | 动量守恒
For two objects with masses m₁ and m₂ undergoing an interaction where no external force acts, the total momentum before collision equals the total momentum after collision.
对于质量分别为 m₁ 和 m₂ 的两个物体，在没有外力作用的相互作用中，碰撞前的总动量等于碰撞后的总动量。
Let initial velocities be u₁, u₂ and final velocities v₁, v₂. Newton’s third law ensures that the impulses on each object are equal and opposite, leading to:
设初速度为 u₁、u₂，末速度为 v₁、v₂。根据牛顿第三定律，两物体受到的冲量大小相等、方向相反，因此：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This principle is fundamental for analysing collisions and explosions.
这条原理是分析碰撞与爆炸问题的基础。
5. Work–Energy Theorem and Kinetic Energy | 动能定理与动能
When a constant net force F acts on a mass m over a displacement s, the work done changes the object’s kinetic energy. Using F = ma and v² = u² + 2as:
当一个恒定的净力 F 作用在质量 m 上并移动位移 s 时，所做的功会改变物体的动能。利用 F = ma 和 v² = u² + 2as：
Work done = F s = m a s = ½ m (v² − u²)
Hence, kinetic energy KE is defined as KE = ½ m v², and the work–energy theorem states that the net work done equals the change in kinetic energy.
因此，动能 KE 被定义为 KE = ½ m v²，而动能定理表明净功等于动能的变化量。
6. Elastic Potential Energy and Hooke’s Law | 弹性势能与胡克定律
For a spring obeying Hooke’s law, the force is proportional to the extension x: F = k x, where k is the spring constant.
对于遵循胡克定律的弹簧，力与伸长量 x 成正比：F = k x，其中 k 是弹簧常量。
The area under the force–extension graph is a triangle, so the work done to stretch the spring from 0 to x equals the elastic potential energy stored:
力–伸长图像下的面积是一个三角形，因此将弹簧从0拉伸到 x 所做的功等于储存的弹性势能：
Eₑₗ = ½ F x = ½ k x²
This derivation appears regularly in Unit 2 questions about energy stored in springs.
在有关弹簧储能问题的单元2考题中，经常用到这个推导。
7. Young’s Modulus | 杨氏模量
Young’s modulus E is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain.
杨氏模量 E 是衡量材料刚度的量，定义为拉伸应力与拉伸应变之比。
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Stress σ = F / A, where A is the cross‑sectional area.
应力 σ = F / A，其中 A 为横截面积。
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Strain ε = ΔL / L, where L is the original length and ΔL the extension.
应变 ε = ΔL / L，L 为原长，ΔL 为伸长量。
Combining these gives the formula often used in the Jan21 paper:
将以上结合，得到Jan21试卷中常用的公式：
E = (F/A) / (ΔL/L) = F L / (A ΔL)
The derivation assumes the material remains within its elastic limit.
该推导假设材料仍处于弹性限度内。
8. The Wave Equation | 波动方程
A wave travels one wavelength λ in one period T. Since frequency f = 1/T, the wave speed v is given by:
一个周期 T 内，波传播一个波长 λ。由于频率 f = 1/T，波速 v 可表示为：
v = λ / T = f λ
This derivation is straightforward but must be understood as the link between frequency, wavelength, and speed for all harmonic waves.
这个推导虽然简单，但必须理解它是所有谐波中频率、波长和速度之间联系的纽带。
9. Snell’s Law and Refractive Index | 斯涅耳定律与折射率
When a wave passes from one medium to another, its speed changes. For light, the refractive index n is defined by the wave speeds v₁ and v₂:
当波从一种介质进入另一种介质时，其速度改变。对于光，折射率 n 由波速 v₁ 和 v₂ 定义：
n₁ = c / v₁, n₂ = c / v₂
Because the frequency remains constant, the change in speed is accompanied by a change in wavelength, which bends the wavefront. Applying geometry to the wavefronts yields Snell’s law:
由于频率保持不变，速度的改变伴随着波长的变化，导致波阵面弯曲。对波阵面应用几何分析可得出斯涅耳定律：
n₁ sin θ₁ = n₂ sin θ₂
Here θ₁ is the angle of incidence in medium 1 and θ₂ the angle of refraction in medium 2.
这里 θ₁ 是介质1中的入射角，θ₂ 是介质2中的折射角。
10. Critical Angle and Total Internal Reflection | 临界角与全内反射
When light travels from a denser medium (refractive index n₁) to a less dense medium (n₂, with n₁ > n₂), the critical angle θc occurs when the angle of refraction is 90°.
当光从光密介质（折射率 n₁）射向光疏介质（n₂，且 n₁ > n₂）时，临界角 θc 发生在折射角为90°的情况下。
Substituting θ₂ = 90° and θ₁ = θc into Snell’s law gives:
将 θ₂ = 90° 和 θ₁ = θc 代入斯涅耳定律，得到：
n₁ sin θc = n₂ sin 90° = n₂
sin θc = n₂ / n₁
This derivation is commonly required in fibre‑optic and prism questions on the Jan21 unit 2 paper.
在Jan21单元2考试中，光纤和棱镜类问题经常要求推导这个公式。
11. Einstein’s Photoelectric Equation | 爱因斯坦光电效应方程
Einstein proposed that light consists of photons, each with energy E = h f, where h is Planck’s constant and f is the frequency.
爱因斯坦提出光由光子组成，每个光子能量为 E = h f，其中 h 是普朗克常量，f 是频率。
When a photon strikes a metal surface, the energy is used to overcome the work function φ and to provide the emitted electron with kinetic energy. By conservation of energy:
当光子撞击金属表面时，能量一部分用于克服逸出功 φ，其余部分转化为发射电子的动能。根据能量守恒：
h f = φ + ½ m v²ₘₐₓ
If a stopping potential Vₛ is applied so that the fastest electrons are just repelled, then e Vₛ = ½ m v²ₘₐₓ. Substituting gives:
如果施加遏止电压 Vₛ 使最快的电子刚好被排斥，则 e Vₛ = ½ m v²ₘₐₓ。代入可得：
e Vₛ = h f − φ
This linear relationship between Vₛ and f was verified experimentally and features in the 2021 paper.
Vₛ 与 f 之间的线性关系已被实验证实，并在2021年试卷中出现。
12. Power and Efficiency | 功率与效率
Power is the rate of doing work. If a constant force F moves an object at constant speed v, the work done per second is P = F v.
功率是做功的速率。如果一个恒力 F 使物体以恒定速度 v 运动，每秒做的功为 P = F v。
Efficiency compares useful output power to total input power, often expressed as a percentage:
效率将有用输出功率与总输入功率进行比较，通常以百分比表示：
Efficiency = (useful power output / total power input) × 100%
These relationships, though simple in form, are derived from energy considerations and are applied to electrical and mechanical systems in the Unit 2 exam.
这些关系式形式简单，但都是从能量角度推导而来，在单元2考试中应用于电气和机械系统。
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