📚 AS Physics Unit 1 Formula Insert Jan20 Derivations | AS物理单元1公式推导
The January 2020 AS Physics Unit 1 data sheet provided a compact reference of equations spanning mechanics, materials, waves and electricity. Rather than simply memorising them, understanding how each formula is derived from fundamental definitions and principles builds deeper insight and equips you to handle unfamiliar contexts. This article walks through the derivations of the key relations found on that insert, establishing links between quantities so you can see the logical structure of physics at this level.
2020年1月AS物理单元1公式表涵盖力学、材料、波和电学中的关键方程。与其死记硬背，不如从基本定义与原理出发推导每一个公式，这不仅加深理解，还让你能应对陌生情境。本文带你逐一推导该试卷插入表中的核心关系式，理清各物理量间的逻辑脉络。
1. SUVAT Equations of Motion | 匀加速运动学方程组
The kinematic equations for constant acceleration in a straight line start from the definition of acceleration. Acceleration a is the rate of change of velocity, so a = (v – u) / t, where u is the initial velocity, v is the final velocity after time t. Rearranging gives the first equation.
匀加速直线运动的运动学方程源于加速度的定义。加速度 a 是速度的变化率，因此 a = (v – u) / t，其中 u 为初速度，v 为时刻 t 的末速度。整理即得第一个公式。
v = u + at
Displacement s is average velocity multiplied by time. For uniform acceleration the average velocity is (u + v) / 2, so s = ((u + v) / 2) × t. Substituting v = u + at into this expression gives the second equation.
位移 s 等于平均速度乘以时间。匀加速运动中平均速度为 (u + v) / 2，故 s = ((u + v) / 2) × t。将 v = u + at 代入，得到第二个方程。
s = ut + ½ at²
To eliminate t, rearrange the first equation to t = (v – u) / a and substitute it into s = ((u + v) / 2) t. This yields the time-independent relation.
消去时间 t，可从第一式解出 t = (v – u) / a，代入 s = ((u + v) / 2) t，得到不含时间的公式。
v² = u² + 2as
If the initial position is at s = 0, the displacement after time t can also be written with v as the subject, but the form s = vt – ½ at² is less common in the insert. The three equations above are sufficient to solve any constant-acceleration problem in a straight line.
若初位置 s = 0，位移也可表示为 s = vt – ½ at²，但公式表中常用的就是上述三式，足以解决所有直线匀加速问题。
2. Momentum and Newton’s Second Law | 动量与牛顿第二定律
Momentum p is defined as the product of mass and velocity: p = m v. The insert lists this definition, which is fundamental for collision and explosion analyses. Newton’s second law in its most general form states that the resultant force on an object equals the rate of change of its momentum.
动量 p 定义为质量与速度的乘积：p = m v。这是分析碰撞与反冲的基础。牛顿第二定律最普适的形式为：物体所受合力等于其动量变化率。
F = Δp / Δt
For a constant mass, Δp = m(v – u), so the force becomes F = m (v – u) / t = m a. This shows how the familiar F = m a arises as a special case of the momentum version when mass remains constant. Both forms appear on the insert.
当质量恒定时，Δp = m(v – u)，因此 F = m (v – u) / t = m a。这解释了常见的 F = m a 其实是质量不变时动量表达式的特殊情形。公式表同时包含了这两种形式。
Impulse is defined as the product of force and the time for which it acts, equal to the change in momentum: Impulse = F Δt = Δp. This is directly useful when analysing forces during collisions or in impact time graphs.
冲量定义为力与其作用时间的乘积，等于动量的变化：冲量 = F Δt = Δp。在分析碰撞力或力-时间图像时非常有用。
3. Work and Energy Transfers | 功与能量转移
Work done by a constant force is given by W = F s cos θ, where θ is the angle between the force and displacement vectors. When the force is parallel to displacement, cos θ = 1 and W = F s. This definition quantifies the energy transferred mechanically.
恒力做功公式为 W = F s cos θ，θ 是力与位移之间的夹角。当力与位移平行时，cos θ = 1，W = F s。这个定义量化了机械能转移的多少。
Kinetic energy is derived by considering the work done to accelerate an object from rest to speed v. Using F = m a and s = (v²) / (2a) from the kinematic equations, the work done W = F s = m a × (v² / (2a)) = ½ m v².
动能的推导：将物体从静止加速到速度 v，力做功 W = F s。利用运动学公式 v² = 2as 得 s = v²/(2a)，代入 F = m a，得到 W = m a × v²/(2a) = ½ m v²。
Eₖ = ½ m v²
Gravitational potential energy near Earth’s surface is the work done against gravity to lift a mass m through a height Δh: W = m g Δh. This explains why the insert lists ΔEₚ = m g Δh.
地表附近的重力势能等于克服重力将质量 m 提升高度 Δh 所做的功：W = m g Δh，因此公式表给出 ΔEₚ = m g Δh。
4. Power and Efficiency | 功率与效率
Power is the rate of doing work: P = ΔW / Δt. If work is done by a constant force moving at constant speed v in its direction, then ΔW = F Δs, so P = F Δs / Δt = F v. This relation is explicitly provided on the insert.
功率是做功的快慢：P = ΔW / Δt。若恒力沿运动方向做功且速度 v 恒定，则 ΔW = F Δs，因此 P = F Δs / Δt = F v。公式表直接给出了这一关系。
P = F v
Efficiency is the ratio of useful output power (or energy) to total input power (or energy). It can be expressed as a percentage: efficiency = (useful output / total input) × 100 %. The insert gives efficiency = useful power out / total power in.
效率定义为有用输出功率（或能量）与总输入功率（或能量）之比，可用百分数表示：效率 = (有用输出 / 总输入) × 100 %。公式表列出 efficiency = useful power out / total power in。
5. Density and Pressure in Materials | 密度与压强
Density ρ is mass per unit volume, a simple definition: ρ = m / V. It is required in many contexts, from buoyancy to quantifying inertia of fluids and solids.
密度 ρ 是单位体积的质量，定义简单：ρ = m / V。在浮力、流体与固体惯性的量化等许多场景中都需要用到。
Pressure exerted by a solid on a surface is force per unit area: p = F / A. In fluids, pressure at a depth h is derived from the weight of the fluid column above: force = weight = m g = (ρ V) g = ρ (A h) g, so pressure p = F / A = ρ g h. This explains the fluid pressure equation on the insert.
固体作用在表面上的压强是单位面积所受的力：p = F / A。液体中深度 h 处的压强由上方的液柱重量推导而来：力 = 重力 = m g = (ρ V) g = ρ (A h) g，因此压强 p = F / A = ρ g h。这便是公式表中液体压强公式的由来。
p = ρ g h
6. Stress, Strain and the Young Modulus | 应力、应变与杨氏模量
The tensile stress in a material is the force applied per unit cross-sectional area: stress = F / A. Tensile strain is the extension per unit original length: strain = ΔL / L. Both are dimensionless or have units of Pa for stress.
拉伸应力是单位横截面积上所受的力：stress = F / A。拉伸应变是单位原长上的伸长量：strain = ΔL / L。应力单位为帕斯卡，应变无量纲。
The Young modulus E characterises the stiffness of a material in the linear elastic region and is defined as the ratio of tensile stress to tensile strain: E = (F / A) / (ΔL / L). This formula appears on the insert because it links the load applied and the resulting extension for a given sample.
杨氏模量 E 描述材料在线弹性区的劲度，定义为拉伸应力与拉伸应变之比：E = (F / A) / (ΔL / L)。公式表包含此式，因为它关联了给定试样的载荷与伸长量。
E = (F L) / (A ΔL)
This rearranged form is particularly useful in experiments where force, original length, area and extension are measured to determine the Young modulus of a wire.
此整理形式在实验中特别实用，通过测量力、原长、截面积与伸长量来确定金属丝的杨氏模量。
7. The Wave Equation | 波动方程
All progressive waves obey the relationship linking speed v, frequency f and wavelength λ. In one period T, a wave advances by one wavelength λ, so speed v = distance / time = λ / T. Since frequency f = 1 / T, we obtain v = f λ. This fundamental equation is central to AS wave topics.
所有行波都满足波速 v、频率 f 和波长 λ 的关系。在一个周期 T 内，波向前传播一个波长 λ，因此波速 v = 距离 / 时间 = λ / T。又因频率 f = 1 / T，得到 v = f λ。这个核心方程在AS波动部分至关重要。
For electromagnetic waves in a vacuum, v = c, so c = f λ, enabling calculation of frequency or wavelength across the spectrum.
对于真空中的电磁波，v = c，故 c = f λ，可据此计算不同波段的频率或波长。
8. Refraction and Snell’s Law | 折射与斯涅尔定律
When a wave passes from one medium to another, its speed changes, causing a change in direction unless it enters along the normal. Snell’s law is derived from the constancy of frequency and the boundary condition that wavefronts must match. For a wave incident at angle θ₁ in medium 1 with speed v₁, and refracted at angle θ₂ in medium 2 with speed v₂, the ratio sin θ₁ / sin θ₂ = v₁ / v₂. Introducing the refractive index n = c / v gives n₁ sin θ₁ = n₂ sin θ₂.
波从一种介质进入另一种介质时波速改变，除沿法线入射外都会改变方向。斯涅尔定律源于频率不变性及波前匹配的边界条件。以角 θ₁ 入射到波速为 v₁ 的介质1，折射角 θ₂ 进入波速为 v₂ 的介质2，有 sin θ₁ / sin θ₂ = v₁ / v₂。引入折射率 n = c / v，得到 n₁ sin θ₁ = n₂ sin θ₂。
n₁ sin θ₁ = n₂ sin θ₂
The critical angle for total internal reflection occurs when the angle of refraction is 90°. Setting θ₂ = 90° for light going from a denser medium (n₁ = n) into a less dense medium (n₂ = 1 for air), we obtain sin C = 1 / n. This is another important relation on the insert.
全内反射的临界角出现在折射角为 90°时。光从光密介质 (n₁ = n) 射向光疏介质 (n₂ ≈ 1)，令 θ₂ = 90°，即得 sin C = 1 / n。这也是公式表中的重要关系。
sin C = 1 / n
9. Electrical Charge, Current and Potential Difference | 电荷、电流与电势差
Electric current I is the rate of flow of charge: I = ΔQ / Δt. The charge transferred by a steady current over time t is Q = I t. This definition appears on the insert as the connection between current and charge.
电流 I 是电荷流动的速率：I = ΔQ / Δt。恒定电流在时间 t 内传递的电荷量为 Q = I t。公式表中这一关系定义了电流与电荷之间的联系。
Potential difference V between two points is the energy transferred per unit charge: V = W / Q. Combined with the expression for power (P = W / t), we get P = V I. This is the key power equation for electrical circuits.
两点间的电势差 V 是单位电荷所转移的能量：V = W / Q。结合功率定义 P = W / t，即得 P = V I。这是电路功率的核心方程。
P = V I
Using Ohm’s law V = I R for an ohmic conductor, the power can also be written in forms that depend only on current and resistance, or voltage and resistance.
对于欧姆导体，利用欧姆定律 V = I R，功率还可写成仅依赖电流与电阻，或电压与电阻的形式。
P = I² R = V² / R
10. Resistance, Resistivity and Combinations | 电阻、电阻率与串并联
Resistance R of a wire is determined by its material, length L and cross-sectional area A through the resistivity ρ: R = ρ L / A. Resistivity is a property of the material and depends on temperature. The formula is derived from the factors that make charge flow more difficult (longer path) or easier (wider area).
导线的电阻 R 由材料、长度 L 与横截面积 A 决定，关系式为 R = ρ L / A。电阻率 ρ 是材料的固有属性且与温度相关。该公式源自使电荷流动更困难（更长路径）或更容易（更大截面积）的因素。
For resistors in series, the same current passes through each, and the total potential difference is the sum of individual p.d.s: Vₜₒₜ = V₁ + V₂ + … . Using V = I R, we get I Rₜₒₜ = I R₁ + I R₂ + … , so Rₜₒₜ = R₁ + R₂ + … .
串联电阻中电流相同，总电势差等于各电阻电势差之和：Vₜₒₜ = V₁ + V₂ + … 。利用 V = I R，得 I Rₜₒₜ = I R₁ + I R₂ + … ，故 Rₜₒₜ = R₁ + R₂ + … 。
For resistors in parallel, the p.d. across each branch is the same, and the total current is the sum of branch currents. Using I = V / R, the total current I = V / Rₜₒₜ = V / R₁ + V / R₂ + … , so 1 / Rₜₒₜ = 1 / R₁ + 1 / R₂ + … . These derivations show why series increases resistance while parallel decreases it.
并联电阻两端电势差相同，总电流为各支路电流之和。利用 I = V / R，总电流 I = V / Rₜₒₜ = V / R₁ + V / R₂ + … ，因此 1 / Rₜₒₜ = 1 / R₁ + 1 / R₂ + … 。这些推导解释了串联增大电阻而并联减小电阻的原因。
| Series | Rₜₒₜ = R₁ + R₂ + … |
| Parallel | 1 / Rₜₒₜ = 1 / R₁ + 1 / R₂ + … |
Note: for two resistors in parallel, the combined resistance simplifies to Rₜₒₜ = (R₁ R₂) / (R₁ + R₂).
注意：两个电阻并联时，总电阻公式可简化为 Rₜₒₜ = (R₁ R₂) / (R₁ + R₂)。
11. Systematic Derivations for the Insert Equations | 插入表方程的系统推导总结
Having traced each key relation back to its definition or experimental law, you can see how the insert is not a random collection but a logically connected set. The SUVAT equations emerge from the definitions of acceleration and average velocity; momentum and impulse link force and motion; work and energy tie mechanics together; material properties follow from stress–strain proportionality; wave equations come from basic kinematics of oscillation; and electrical rules flow from charge conservation and energy transfer.
追溯每个核心关系式至其定义或实验规律后，你会发现公式表并非零散的汇编，而是一个逻辑相连的整体。SUVAT方程组源自加速度和平均速度的定义；动量与冲量联系了力与运动；功和能量统一了力学；材料属性源于应力-应变的比例关系；波动方程来自基本振动运动学；电学规律则由电荷守恒与能量转移导出。
Mastering these derivations not only secures marks in explanation questions but also empowers you to adapt equations when solving unfamiliar problems. Use the insert as a prompt for revising the underlying physics, and you will find the subject far more coherent.
掌握这些推导不仅能确保在解释题中得分，还能让你在解决陌生问题时灵活运用公式。将插入表当作回顾底层物理的线索，你会发现学科内容远比你想象的更连贯。
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