📚 Deriving Key AS Physics Formulae (June 2022 Insert 2) | AS物理核心公式推导（2022年6月公式表第二页）
The AS Physics examination often provides a formula sheet, and the June 2022 Insert 2 contains many of the essential relationships that students need to apply. Understanding how these equations are derived not only helps with memorisation but also deepens your grasp of the underlying physical principles. In this article, we will work through the derivations of key formulae from mechanics, materials, electricity, and waves.
AS物理考试通常会提供公式表，2022年6月的Insert 2包含了学生需要运用的许多基本关系式。了解这些公式的推导过程不仅有助于记忆，还能加深对背后物理原理的理解。在本文中，我们将逐步推导力学、材料、电学和波等领域的关键公式。
1. Deriving s = ½(u+v)t from a Velocity-Time Graph | 从速度-时间图推导 s = ½(u+v)t
For an object moving with constant acceleration, the velocity-time graph is a straight line. The area under this graph between time 0 and t represents the displacement s. Since the line starts at initial velocity u and ends at final velocity v, the shape is a trapezium.
对于做匀加速运动的物体，速度-时间图是一条直线。0到t时刻之间，图形下方的面积表示位移s。由于直线从初速度u开始，到末速度v结束，该图形是一个梯形。
The area of a trapezium is the average of the parallel sides multiplied by the width. Thus, s = (u + v)/2 × t. This gives us the equation s = ½ (u + v) t.
梯形的面积等于平行边的平均值乘以宽度。因此，s = (u + v)/2 × t。这就得到了公式 s = ½ (u + v) t。
s = ½ (u + v) t
2. Deriving v = u + at and the Other SUVAT Equations | 推导 v = u + at 及其他匀加速运动方程
Acceleration is defined as the rate of change of velocity: a = (v − u)/t. Rearranging gives the familiar form v = u + at. This relation can be combined with the displacement equation to eliminate variables.
加速度定义为速度的变化率：a = (v − u)/t。整理后得到熟悉的形式 v = u + at。将这个关系与位移方程结合，可以消去变量。
Substituting v = u + at into s = ½(u+v)t yields s = ½ [u + (u + at)] t = ½ (2u + at) t = ut + ½at². Alternatively, expressing t as (v − u)/a and inserting into s = ½(u+v)t eliminates t, leading to v² = u² + 2as after cross-multiplication.
将 v = u + at 代入 s = ½(u+v)t 得到 s = ½ [u + (u + at)] t = ½ (2u + at) t = ut + ½at²。或者，将 t 表示为 (v − u)/a 并代入 s = ½(u+v)t 消去时间，经过交叉相乘得到 v² = u² + 2as。
s = ut + ½ at²
v² = u² + 2as
These equations hold only when acceleration a is constant. They form the basis of many mechanics problems in the AS syllabus.
这些方程仅在加速度a恒定时成立。它们构成了AS考纲中许多力学问题的基础。
3. Newton’s Second Law in Terms of Momentum | 动量形式的牛顿第二定律
Newton’s second law states that the resultant force on an object is equal to the rate of change of its momentum. Momentum p is defined as mv.
牛顿第二定律指出，作用在物体上的合外力等于其动量的变化率。动量 p 定义为 mv。
The change in momentum over a time interval Δt is Δp = mΔv if mass is constant. Therefore, the force can be written as F = Δp/Δt. For constant mass, this reduces to F = m (Δv/Δt) = ma.
如果质量不变，在时间间隔 Δt 内动量的变化量为 Δp = mΔv。因此，力可以写成 F = Δp/Δt。对于恒定的质量，这简化为 F = m (Δv/Δt) = ma。
F = Δp / Δt
This formulation is particularly useful when mass is not constant, such as in rocket propulsion, but in AS Physics it reinforces the concept of impulse.
这个公式在质量不恒定时（如火箭推进）特别有用，但在AS物理中它强化了冲量的概念。
4. Derivation of Kinetic Energy, Eₖ = ½ mv² | 动能 Eₖ = ½ mv² 的推导
Work done by a constant force is W = Fs cosθ. When the force is parallel to the displacement, W = Fs. Using F = ma and the equation v² = u² + 2as, we can find the work needed to accelerate a mass from rest.
恒力做功为 W = Fs cosθ。当力与位移平行时，W = Fs。利用 F = ma 和方程 v² = u² + 2as，我们可以求出将质量从静止加速所需的功。
For an object starting at rest (u = 0) and reaching speed v, v² = 2as gives as = v²/2. Multiply both sides by mass m: Fs = m a s = m (v²/2) = ½ mv². This work is stored as kinetic energy, so Eₖ = ½ mv².
对于从静止（u = 0）开始达到速度 v 的物体，由 v² = 2as 得到 as = v²/2。两边乘以质量 m：Fs = m a s = m (v²/2) = ½ mv²。该功储存为动能，因此 Eₖ = ½ mv²。
Eₖ = ½ mv²
5. Gravitational Potential Energy, Eₚ = mgΔh | 重力势能 Eₚ = mgΔh
To lift an object of weight mg through a vertical height Δh at constant speed, the applied force must equal the weight. The work done is W = force × distance = mg × Δh.
要以恒定速度将重量为 mg 的物体竖直举高 Δh，所施加的力必须等于物体的重量。所做的功为 W = 力 × 距离 = mg × Δh。
This work is transferred to the gravitational potential store, so the change in gravitational potential energy is ΔEₚ = mgΔh. The formula assumes a uniform gravitational field, as is the case near the Earth’s surface.
这些功转移到重力势能储存，因此重力势能的变化量为 ΔEₚ = mgΔh。该公式假设重力场均匀，正如地球表面附近的情况。
ΔEₚ = mgΔh
6. Stress, Strain and Young Modulus | 应力、应变与杨氏模量
In materials physics, stress (σ) is the force applied per unit cross-sectional area. Strain (ε) is the fractional change in length. Hooke’s law for a wire states that stress is proportional to strain up to the elastic limit.
在材料物理中，应力 (σ) 是单位截面积上所受的作用力。应变 (ε) 是长度的变化率。对于金属丝，胡克定律指出在弹性极限内应力与应变成正比。
Stress is defined as σ = F/A, where F is the tensile force and A is the original cross-sectional area. Strain is ε = ΔL/L, where ΔL is the extension and L is the original length.
应力定义为 σ = F/A，其中 F 是拉伸力，A 是原始截面积。应变定义为 ε = ΔL/L，其中 ΔL 是伸长量，L 是原始长度。
σ = F / A
ε = ΔL / L
The Young modulus (E) is the ratio of stress to strain when the material obeys Hooke’s law. Hence E = σ/ε = (F/A) / (ΔL/L) = FL / (AΔL). This is a constant for a given material, measuring its stiffness.
杨氏模量 (E) 是材料遵守胡克定律时应力与应变的比值。因此 E = σ/ε = (F/A) / (ΔL/L) = FL / (AΔL)。对于给定的材料这是一个常数，衡量其刚度。
E = FL / (A ΔL)
7. Resistors in Series and Parallel | 电阻的串联与并联
When resistors are connected in series, the same current I flows through each. The total potential difference across the combination is the sum of the individual p.d.s: Vₜₒₜₐₗ = V₁ + V₂.
当电阻串联时，相同的电流 I 流过每个电阻。组合两端的总电位差等于各个电阻两端电位差之和：Vₜₒₜₐₗ = V₁ + V₂。
Using Ohm’s law, V = IR, we have IRₛ = IR₁ + IR₂, where Rₛ is the total series resistance. Cancelling I gives Rₛ = R₁ + R₂.
应用欧姆定律 V = IR，得到 IRₛ = IR₁ + IR₂，其中 Rₛ 是总串联电阻。消去 I 得到 Rₛ = R₁ + R₂。
Rseries = R₁ + R₂
For parallel resistors, the potential difference across each is the same, V. The total current splits: Iₜₒₜₐₗ = I₁ + I₂. Applying Ohm’s law, V/Rₚ = V/R₁ + V/R₂, where Rₚ is the total parallel resistance. Dividing by V gives 1/Rₚ = 1/R₁ + 1/R₂.
对于并联电阻，每个电阻两端的电位差相同，均为 V。总电流分流：Iₜₒₜₐₗ = I₁ + I₂。应用欧姆定律得到 V/Rₚ = V/R₁ + V/R₂，其中 Rₚ 是总并联电阻。除以 V 得到 1/Rₚ = 1/R₁ + 1/R₂。
1 / Rparallel = 1 / R₁ + 1 / R₂
8. Deriving the Resistivity Formula ρ = RA / L | 电阻率公式 ρ = RA / L 的推导
Resistivity (ρ) is an intrinsic property of a material that quantifies how strongly it opposes electric current. The resistance of a uniform wire depends on its length L, cross-sectional area A, and the material’s resistivity.
电阻率 (ρ) 是材料的内禀性质，用于量化其对电流阻碍作用的强弱。一根均匀导线的电阻取决于其长度 L、截面积 A 以及材料的电阻率。
Experimentally, resistance is proportional to length and inversely proportional to area: R ∝ L/A. Introducing the constant of proportionality ρ gives R = ρL/A. Rearranging produces the formula found on the data sheet: ρ = RA/L.
实验表明，电阻与长度成正比，与面积成反比：R ∝ L/A。引入比例常数 ρ 得到 R = ρL/A。重新整理即得公式表中的公式：ρ = RA/L。
ρ = RA / L
The units of resistivity are ohm-metres (Ω m). This relationship is fundamental when analysing conducting materials.
电阻率的单位是欧姆·米 (Ω m)。在分析导体材料时，这个关系式是基础。
9. Potential Divider Equation | 分压器公式
A potential divider consists of two resistors, R₁ and R₂, connected in series across a supply voltage Vin. The output voltage Vout is taken across one of the resistors, typically R₂.
分压器由两个串联的电阻 R₁ 和 R₂ 组成，供电电压为 Vin。输出电压 Vout 取自其中一个电阻两端，通常是 R₂。
The same current I flows through both resistors, so Vin = I (R₁ + R₂) and Vout = I R₂. Dividing the two equations gives Vout / Vin = R₂ / (R₁ + R₂). Thus, Vout = Vin × R₂ / (R₁ + R₂).
相同的电流 I 流过两个电阻，因此 Vin = I (R₁ + R₂) 且 Vout = I R₂。将两式相除得到 Vout / Vin = R₂ / (R₁ + R₂)。于是，Vout = Vin × R₂ / (R₁ + R₂)。
Vout = Vin × R₂ / (R₁ + R₂)
This equation is extremely useful in sensor circuits where one of the resistors changes in response to a physical quantity.
当一个电阻随物理量变化时，这个方程在传感器电路中极为有用。
10. Electrical Power Equations: P = VI, P = I²R, P = V²/R | 电功率公式：P = VI、P = I²R、P = V²/R
Power is the rate of energy transfer. In an electrical circuit, the energy transferred when a charge Q moves through a potential difference V is E = QV. Substituting current I = Q/t gives P = E/t = QV/t = (I t) V / t = IV.
功率是能量传递的速率。在电路中，电荷 Q 通过电位差 V 时转移的能量为 E = QV。代入电流 I = Q/t 得到 P = E/t = QV/t = (I t) V / t = IV。
P = IV
For a resistor, Ohm’s law V = IR can be substituted into P = IV to obtain two alternative forms. Replacing V: P = I × (IR) = I²R. Replacing I: P = (V/R) × V = V²/R.
对于电阻，可将欧姆定律 V = IR 代入 P = IV 获得另外两种形式。替换 V：P = I × (IR) = I²R。替换 I：P = (V/R) × V = V²/R。
P = I²R and P = V² / R
These equations allow us to calculate the heat dissipated in a resistor, and are essential when considering energy conservation in circuits.
这些方程使我们能够计算电阻上耗散的热量，在考虑电路能量守恒时必不可少。
11. Snell’s Law and Critical Angle Derivation | 斯涅尔定律与临界角推导
When light passes from one medium to another, it changes speed and direction. The refractive index n of a medium is defined as the ratio of the speed of light in vacuum c to the speed in the medium v: n = c/v.
当光从一种介质进入另一种介质时，其速度和方向会发生改变。介质的折射率 n 定义为光在真空中的速度 c 与在介质中的速度 v 之比：n = c/v。
n = c / v
Snell’s law relates the angles of incidence and refraction to the refractive indices: n₁ sinθ₁ = n₂ sinθ₂. This can be derived from the wavefront principle, but on the data sheet it is given for direct application.
斯涅尔定律将入射角和折射角与折射率联系起来：n₁ sinθ₁ = n₂ sinθ₂。这可由波前原理推导，但公式表中直接给出了这个关系以供应用。
n₁ sinθ₁ = n₂ sinθ₂
The critical angle θc occurs when the angle of refraction is 90° (for light going from a denser to a less dense medium, n₁ > n₂). Substituting θ₂ = 90° gives n₁ sinθc = n₂ sin90° = n₂. Therefore, sinθc = n₂ / n₁. This is the formula for total internal reflection.
临界角 θc 发生在折射角为 90°（光从光密介质射向光疏介质，n₁ > n₂）时。代入 θ₂ = 90° 得到 n₁ sinθc = n₂ sin90° = n₂。因此 sinθc = n₂ / n₁。这就是全内反射的公式。
sinθc = n₂ / n₁ (n₁ > n₂)
12. Deriving the Diffraction Grating Equation nλ = d sinθ | 衍射光栅方程 nλ = d sinθ 的推导
A diffraction grating consists of many equally spaced slits, with a slit separation d. When coherent light of wavelength λ passes through, constructive interference occurs when the path difference between adjacent slits is a whole number of wavelengths.
衍射光栅由许多等间距的狭缝组成，狭缝间距为 d。当波长为 λ 的相干光通过时，当相邻狭缝的光程差等于波长的整数倍时发生相长干涉。
For light incident normally on the grating, the extra distance travelled by a ray from one slit compared to its neighbour is d sinθ, where θ is the angle of the diffracted beam to the normal. Setting this path difference equal to nλ gives nλ = d sinθ.
对于垂直入射到光栅上的光，从一个狭缝发出的光线与其相邻狭缝相比多走的光程为 d sinθ，其中 θ 是衍射光束与法线的夹角。令此光程差等于 nλ 即得 nλ = d sinθ。
nλ = d sinθ
The integer n is called the order of the maximum. This simple derivation explains why a grating produces a series of bright spots at specific angles, making it a powerful tool for spectroscopy.
整数 n 称为级次。这一简单推导解释了为何光栅会在特定角度产生一系列亮斑，使其成为光谱学中的强有力工具。
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