📚 A-Level Physics Unit 2 Formula Derivation (Jan21) | A-Level物理单元2公式推导（2021年1月）
Welcome to the ultimate guide for deriving the key formulas found in the A-Level Physics Unit 2 Insert for January 2021. Mastering these derivations not only deepens your understanding but also equips you to tackle any calculation problem with confidence. This article walks you through each equation step by step, from fundamental principles to the final expression, ensuring you see the logical flow behind every symbol.
欢迎阅读A-Level物理单元2（2021年1月）公式推导完整指南。掌握这些推导不仅能加深理解，也能让你充满信心地应对任何计算题。本文将逐步带你推导每一个关键公式，从基本原理到最终表达式，让你看清每个符号背后的逻辑脉络。
1. Introduction to the Unit 2 Insert | 单元2公式表简介
The A-Level Physics Unit 2 Insert (January 2021) provides a collection of essential formulas for mechanics, materials, and waves. While you can use the sheet in the exam, understanding where these formulas come from is crucial for applying them correctly. We will cover derivations for SUVAT equations, force and momentum, energy, elasticity, wave optics, and quantum phenomena.
A-Level物理单元2（2021年1月）的公式表提供了力学、材料和波动的核心公式。虽然考试时可以查阅，但理解这些公式的来源对于正确运用至关重要。我们将涵盖匀加速运动方程、力与动量、功与能、弹性、波动光学以及量子现象的推导。
2. Deriving the SUVAT Equations | 推导匀加速运动方程
The SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) for uniformly accelerated motion. Start with the definition of acceleration: a = (v – u) / t. Rearranging gives the first equation:
SUVAT方程描述了匀加速运动中位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）的关系。从加速度的定义出发：a = (v – u) / t，整理即得第一式：
v = u + at
Since acceleration is constant, average velocity = (u + v)/2. Displacement is average velocity multiplied by time: s = ((u + v)/2) t. Substitute v = u + at into this expression:
由于加速度恒定，平均速度 = (u + v)/2。位移等于平均速度乘以时间：s = ((u + v)/2) t。将 v = u + at 代入：
s = ½ (u + u + at) t = ut + ½ at²
To eliminate time, solve v = u + at for t: t = (v – u)/a. Substitute into s = ((u + v)/2) t:
为消去时间，由 v = u + at 解出 t = (v – u)/a，代入 s = ((u + v)/2) t：
s = (u + v)(v – u) / (2a) = (v² – u²) / (2a) ⇒ v² = u² + 2as
These four equations (including s = vt – ½ at²) form the SUVAT set. They are derived purely from definitions of velocity and acceleration under constant a.
这四个方程（包括 s = vt – ½ at²）构成了SUVAT集合，完全由恒定加速度下的速度和位移定义导出。
3. Newton’s Second Law and Impulse | 牛顿第二定律与冲量
Newton’s second law states that the net force acting on an object equals the rate of change of its momentum: F = dp/dt. For constant mass, p = mv, so F = m (dv/dt) = ma. The impulse J delivered by a constant force over time Δt is J = F Δt, which equals the change in momentum:
牛顿第二定律指出，作用于物体的净力等于其动量变化率：F = dp/dt。对于质量恒定的情况，p = mv，因此 F = m (dv/dt) = ma。恒力在时间 Δt 内产生的冲量 J = F Δt，等于动量的变化：
F = m a
J = F Δt = Δp = m(v – u)
In collisions where force varies, the area under a force–time graph gives the impulse, still equal to Δp. This link between force, time, and momentum is fundamental in mechanics.
在变力碰撞中，力-时间图像下的面积即为冲量，仍等于 Δp。力、时间与动量之间的这种联系是力学的基础。
4. Work, Energy, and Power | 功、能与功率
Work done by a constant force F acting over a displacement s in the direction of the force is W = F s. If the force is at an angle θ to the displacement, W = F s cos θ. The work–energy principle states that the net work done equals the change in kinetic energy. For an object starting at speed u and accelerating to v, using F = ma and v² = u² + 2as:
恒力 F 沿位移 s 方向做的功为 W = F s。若力与位移夹角为 θ，则 W = F s cos θ。功-能原理表明，净功等于动能的变化。对一个从速度 u 加速到 v 的物体，利用 F = ma 和 v² = u² + 2as：
W = F s = (ma) × ((v² – u²)/(2a)) = ½ mv² – ½ mu²
This shows kinetic energy Eₖ = ½ mv². Power P is the rate of doing work: P = W/t = F v (for constant force and velocity in the same direction). The unit of power is the watt (W) = J s⁻¹.
这表明动能 Eₖ = ½ mv²。功率 P 是做功的速率：P = W/t = F v（力与速度同向且恒定时）。功率单位为瓦特（W）= J s⁻¹。
5. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能
For a spring obeying Hooke’s law, the extension x is directly proportional to the applied force: F = k x, where k is the spring constant. The elastic potential energy stored in a stretched or compressed spring is the area under the force–extension graph, which is a triangle for a linear spring:
对于满足胡克定律的弹簧，伸长量 x 与施加的力成正比：F = k x，其中 k 为劲度系数。拉伸或压缩弹簧所储存的弹性势能等于力-伸长图下的面积，对于线性弹簧为三角形面积：
Eₑₗ = ½ F x = ½ k x²
This derivation assumes the spring’s limit of proportionality is not exceeded. If a force–extension graph is non-linear, the energy stored is still the area under the curve but cannot be given by a simple ½ k x².
该推导假设弹簧未超过比例极限。若力-伸长图非线性，储存的能量仍为曲线下的面积，但不能用简单的 ½ k x² 表示。
6. Young’s Modulus and Stress-Strain | 杨氏模量与应力-应变
Stress σ is the force per unit cross-sectional area: σ = F / A. Strain ε is the extension per unit original length: ε = ΔL / L. Young’s modulus E quantifies the stiffness of a material, defined as the ratio of stress to strain in the linear region:
应力 σ 是单位横截面积上的力：σ = F / A。应变 ε 是单位原长的伸长量：ε = ΔL / L。杨氏模量 E 衡量材料的刚度，定义为线弹性区域内应力与应变之比：
E = σ / ε = (F/A) / (ΔL/L) = FL / (A ΔL)
Combining with Hooke’s law for a wire, the effective spring constant is k = EA / L. Substituting into F = k ΔL gives the same expression. The unit of Young’s modulus is N m⁻² or Pa.
结合金属丝的胡克定律，其等效劲度系数 k = EA / L。代入 F = k ΔL 可得一致表达式。杨氏模量的单位是 N m⁻² 或 Pa。
7. The Wave Equation: v = f λ | 波动方程：v = f λ
Waves transfer energy without transferring matter. The frequency f is the number of complete oscillations per second. The wavelength λ is the distance between consecutive points in phase. In one period T = 1/f, the wave advances by one wavelength. Therefore, the speed v is distance/time:
波传递能量而不传递物质。频率 f 是每秒完整振动的次数。波长 λ 是相邻同相位点之间的距离。在一个周期 T = 1/f 内，波前进一个波长。因此，波速 v = 距离/时间：
v = λ / T = f λ
This relationship holds for all types of waves: mechanical (sound, water) and electromagnetic. It is used extensively in analysing diffraction, interference, and standing waves.
此关系适用于所有类型的波：机械波（声波、水波）和电磁波。它广泛用于分析衍射、干涉和驻波。
8. Refractive Index and Snell’s Law | 折射率与斯涅尔定律
When light travels from medium 1 to medium 2, its speed changes, causing refraction. Snell’s law links the angles of incidence θ₁ and refraction θ₂ to the refractive indices. The refractive index n of a medium is n = c / v, where c is the speed of light in vacuum. Using wavefronts and the principle of least time or boundary conditions, we derive:
当光从介质1进入介质2时，其速度改变，从而发生折射。斯涅尔定律将入射角 θ₁ 和折射角 θ₂ 与折射率联系起来。介质的折射率 n = c / v，其中 c 是真空光速。利用波前和最短时间原理或边界条件，可以推导出：
n₁ sin θ₁ = n₂ sin θ₂
For light entering a block of refractive index n from air (n ≈ 1), the equation simplifies to sin θ₁ = n sin θ₂. The critical angle for total internal reflection occurs when θ₂ = 90°, giving sin θc = 1/n.
当光从空气（n ≈ 1）射入折射率为 n 的介质时，方程简化为 sin θ₁ = n sin θ₂。发生全内反射的临界角满足 θ₂ = 90°，得到 sin θc = 1/n。
9. Superposition and Standing Waves | 叠加与驻波
The principle of superposition states that when two or more waves meet, the resultant displacement is the vector sum of individual displacements. Standing waves are formed when two identical progressive waves travel in opposite directions. For a string fixed at both ends, the condition for a standing wave is that the length L equals an integer multiple of half-wavelengths:
叠加原理指出，当两个或更多波相遇时，合位移等于各波位移的矢量和。驻波由两列完全相同但反向传播的行波叠加形成。对于两端固定的弦，驻波条件为长度 L 等于半波长的整数倍：
L = n λ/2, where n = 1, 2, 3, …
The frequency of the nth harmonic is fₙ = n (v/2L), with v being the wave speed on the string. Similar conditions apply to pipes (open or closed), leading to different harmonic series.
第 n 次谐波的频率为 fₙ = n (v/2L)，v 为弦上的波速。类似的条件适用于管乐器（开管或闭管），得到不同的谐波序列。
10. Two-Source Interference and Young’s Double Slit | 双源干涉与杨氏双缝
Young’s double-slit experiment demonstrates the wave nature of light. Two coherent sources of light create an interference pattern. Constructive interference occurs when the path difference Δ between the two waves is an integer multiple of λ: Δ = nλ. Destructive interference occurs when Δ = (n + ½)λ. For small angles, the fringe spacing Δy on a screen at distance D is given by:
杨氏双缝实验证明了光的波动性。两个相干光源产生干涉图样。当两束光的路程差 Δ 为 λ 的整数倍时发生相长干涉：Δ = nλ。当 Δ = (n + ½)λ 时发生相消干涉。对于小角度，屏幕上距离 D 处的条纹间距 Δy 由下式给出：
Δy = λ D / d
This is derived by approximating sin θ ≈ tan θ ≈ θ for small θ, where d is the slit separation. The same formula applies to sound waves and microwaves in demonstration experiments.
其推导基于小角度近似 sin θ ≈ tan θ ≈ θ，d 为双缝间距。该公式同样适用于声波和微波的演示实验。
11. Photon Energy and the Photoelectric Effect | 光子能量与光电效应
The photoelectric effect provides evidence for the particle model of light. A single photon of frequency f carries energy E = h f, where h is Planck’s constant. When a photon hits a metal surface, its energy can be used to overcome the work function φ (the minimum energy needed to release an electron) and give the electron kinetic energy. Einstein’s photoelectric equation is:
光电效应为光的粒子模型提供了证据。一个频率为 f 的光子携带能量 E = h f，其中 h 为普朗克常数。当光子撞击金属表面时，其能量可用于克服功函数 φ（释放电子所需的最小能量）并赋予电子动能。爱因斯坦光电方程如下：
h f = φ + ½ m v²ₘₐₓ
The maximum kinetic energy Eₖ₍ₘₐₓ₎ can be measured by applying a stopping potential Vₛ: Eₖ₍ₘₐₓ₎ = e Vₛ, where e is the elementary charge. The threshold frequency f₀ is given by h f₀ = φ. This derivation shows that kinetic energy increases linearly with frequency, not intensity.
最大动能 Eₖ₍ₘₐₓ₎ 可通过施加遏止电压 Vₛ 测量：Eₖ₍ₘₐₓ₎ = e Vₛ，e 为元电荷。截止频率 f₀ 满足 h f₀ = φ。该推导表明，最大动能随频率线性增加，而与光强无关。
12. Key Takeaways and Exam Tips | 关键要点与应试技巧
When tackling Unit 2 questions, always start by identifying the fundamental principles at play. Write down the relevant derived equation from the insert, but be prepared to show how it follows from basic definitions if asked. Practice drawing labelled diagrams (e.g., force–extension, wavefronts) to support your derivations. Remember to check unit consistency and use standard form for very large or small numbers.
解答单元2题目时，首先确定所用的基本原理。写下公式表中的相关推导方程，但如果题目要求展示推导过程，务必能从基本定义开始展示。练习绘制标注清晰的示意图（如力-伸长图、波前图）来辅助推导。记得检查单位一致性，并对非常大或非常小的数字使用科学记数法。
The formulas derived here are interconnected. For instance, the conservation of energy ties together kinetic energy, potential energy, and work done. Mastering derivations gives you a safety net: even if you forget a formula, you can reconstruct it.
本文推导的公式相互关联。例如，能量守恒将动能、势能和做功联系起来。掌握推导能为你提供一张安全网：即使忘记了某个公式，你也能重新推导出来。
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