📚 A-Level Physics: Common Mathematical Equations and Their Physical Applications | A-Level 物理：常用数学方程及其物理应用
Mathematics is the language of physics. In the CIE A-Level Physics syllabus, students are expected not only to recall key equations but also to understand the physical principles behind them, apply them to unfamiliar contexts, and manipulate them algebraically with confidence. This article presents a structured review of the most frequently tested mathematical equations across the core topics, with explanations of their physical meaning and typical exam applications.
数学是物理的语言。在 CIE A-Level 物理考纲中，学生不仅要熟记关键方程，更要理解这些方程背后的物理原理，能够在新情境中加以运用，并熟练进行代数变换。本文按核心专题系统梳理了最常考到的数学方程，解释其物理意义，并指出典型的考试应用场景。
1. Kinematics Equations of Motion | 运动学运动方程
The equations of motion, often called the ‘suvat’ equations, describe the motion of an object moving with constant acceleration in a straight line. The five variables are: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). Each equation omits exactly one variable, so the choice of equation depends on which quantities are given and which is required.
运动方程（常称为 suvat 方程）描述物体在直线上做匀加速运动的情况。五个变量分别为：s（位移）、u（初速度）、v（末速度）、a（加速度）和 t（时间）。每个方程恰好省略一个变量，因此选择哪个方程取决于已知量和待求量。
For constant acceleration, the four key equations are:
对于匀加速运动，四个关键方程为：
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
The first equation relates velocity and time; the second gives displacement from initial velocity and acceleration; the third uses average velocity; the fourth is useful when time is not known. In projectile motion problems, these equations are applied separately to the horizontal (constant velocity) and vertical (constant acceleration due to gravity) components.
第一个方程联系速度与时间；第二个方程由初速度和加速度求位移；第三个方程利用平均速度；第四个方程在时间未知时最为方便。在抛体运动问题中，这些方程分别应用于水平方向（匀速）和竖直方向（重力引起的匀加速）的分运动。
2. Newton’s Laws and Dynamics | 牛顿定律与动力学
Newton’s second law is the cornerstone of classical mechanics. It states that the resultant force acting on an object equals the rate of change of its momentum, and for constant mass, it simplifies to the familiar form:
牛顿第二定律是经典力学的基石。它指出物体所受合外力等于其动量的变化率；当质量恒定时，可简化为如下熟悉的形式：
F = ma
Here F is the resultant force in newtons, m is the mass in kilograms, and a is the acceleration in m s⁻². A common exam scenario involves a box on a rough inclined plane, where the net force is found by resolving weight into components: mg sin θ down the slope and mg cos θ perpendicular to the slope. The frictional force is then μR, where R = mg cos θ is the normal reaction.
其中 F 为合外力（单位牛顿），m 为质量（单位千克），a 为加速度（单位米每二次方秒）。常见的考试情境包括粗糙斜面上的物体：将重力分解为沿斜面方向 mg sin θ 和垂直斜面方向 mg cos θ，摩擦力为 μR，其中 R = mg cos θ 为正压力。
Momentum is defined as the product of mass and velocity:
动量定义为质量与速度的乘积：
p = mv
The principle of conservation of linear momentum states that in an isolated system, the total momentum before a collision equals the total momentum after. For two objects of masses m₁ and m₂ with initial velocities u₁ and u₂, and final velocities v₁ and v₂:
动量守恒定律指出：在孤立系统中，碰撞前后的总动量相等。对于质量分别为 m₁ 和 m₂、初速度为 u₁ 和 u₂、末速度为 v₁ 和 v₂ 的两个物体：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
When solving collision problems, always check whether the collision is elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved) before applying the appropriate equations.
在求解碰撞问题时，务必先判断碰撞是弹性碰撞（动能守恒）还是非弹性碰撞（动能不守恒），再选用相应的方程。
3. Work, Energy and Power | 功、能量与功率
Work is done when a force causes displacement. The general equation is:
当力使物体发生位移时，力就做了功。一般方程为：
W = Fd cos θ
where d is the displacement and θ is the angle between the force and the displacement direction. When θ = 0°, W = Fd; when θ = 90°, W = 0, which explains why no work is done by a centripetal force.
其中 d 为位移，θ 为力与位移方向之间的夹角。当 θ = 0° 时，W = Fd；当 θ = 90° 时，W = 0，这也解释了向心力为什么不做功。
Kinetic energy is the energy an object possesses due to its motion, and gravitational potential energy is the energy stored due to height in a gravitational field:
动能是物体因运动而具有的能量，重力势能是物体在重力场中因高度而储存的能量：
KE = ½mv²
PE = mgh
The work-energy theorem states that the net work done on an object equals its change in kinetic energy. This is extremely useful in problems involving variable forces, where direct use of F = ma is difficult. Power is the rate of doing work:
动能定理指出：合外力对物体所做的功等于物体动能的变化量。这在涉及变力的问题中非常有用，因为此时直接使用 F = ma 较为困难。功率是做功的快慢：
P = W/t = Fv
The form P = Fv is particularly important for vehicles: at constant power, as speed increases, the driving force decreases. This is why lorries climb hills more slowly when heavily loaded.
P = Fv 的形式对车辆问题尤为重要：在恒定功率下，速度增大时牵引力减小。这就是重型卡车满载爬坡时速度变慢的原因。
4. Circular Motion | 圆周运动
Uniform circular motion involves an object moving at constant speed along a circular path. Although the speed is constant, the velocity changes continuously because its direction changes; hence there is an acceleration directed towards the centre of the circle.
匀速圆周运动指物体沿圆形路径以恒定速率运动。虽然速率不变，但由于方向不断改变，速度持续变化，因此存在一个指向圆心的加速度。
The angular displacement θ is related to the arc length s by s = rθ. Angular velocity ω is defined as the rate of change of angular displacement:
角位移 θ 与弧长 s 的关系为 s = rθ。角速度 ω 定义为角位移的变化率：
ω = Δθ/Δt = 2π/T = 2πf
The linear speed v is related to angular velocity by:
线速度 v 与角速度的关系为：
v = ωr
The centripetal acceleration and the centripetal force are given by:
向心加速度与向心力分别为：
a = v²/r = ω²r
F = mv²/r = mω²r
In a vertical circle, the tension in a string is greatest at the bottom and least at the top. At the top, mg provides part of the centripetal force; at the bottom, tension must overcome both gravity and provide the centripetal force.
在竖直圆周运动中，绳子张力在最低点最大、在最高点最小。在最高点，重力提供部分向心力；在最低点，张力既要克服重力，又要提供向心力。
5. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) occurs when the restoring force is proportional to the displacement from equilibrium and acts in the opposite direction. The defining equation is:
简谐运动发生在回复力与偏离平衡位置的位移成正比且方向相反时。其定义方程为：
a = -ω²x
The negative sign indicates that acceleration is always directed towards the equilibrium position. The displacement of an object in SHM as a function of time is:
负号表示加速度始终指向平衡位置。物体做简谐运动时，位移随时间的变化为：
x = A cos(ωt)
where A is the amplitude and ω is the angular frequency. The velocity and acceleration are obtained by differentiation:
其中 A 为振幅，ω 为角频率。速度和加速度通过对位移求导得到：
v = -Aω sin(ωt)
a = -Aω² cos(ωt) = -ω²x
The period of a mass-spring system and a simple pendulum are:
弹簧振子和单摆的周期分别为：
T = 2π√(m/k)
T = 2π√(L/g)
For SHM, the maximum speed is v_max = Aω at the equilibrium position, and the maximum acceleration is a_max = Aω² at the extreme positions. Energy is exchanged cyclically between kinetic and potential forms.
对于简谐运动，最大速度 v_max = Aω 出现在平衡位置，最大加速度 a_max = Aω² 出现在极端位置。动能与势能周期性相互转化。
6. Wave Properties | 波动性质
The wave equation relates the speed of a wave to its frequency and wavelength:
波动方程将波的传播速度与频率和波长联系起来：
v = fλ
Here v is the wave speed in m s⁻¹, f is the frequency in hertz, and λ is the wavelength in metres. For electromagnetic waves in a vacuum, v = c = 3.00 × 10⁸ m s⁻¹.
其中 v 为波速（单位米每秒），f 为频率（单位赫兹），λ 为波长（单位米）。对于真空中的电磁波，v = c = 3.00 × 10⁸ 米每秒。
The period T is the reciprocal of frequency:
周期 T 是频率的倒数：
T = 1/f
For stationary waves on a string fixed at both ends, the wavelength of the nth harmonic is λₙ = 2L/n, where L is the string length. The frequency of the fundamental (first harmonic) is therefore:
对于两端固定的弦上的驻波，第 n 次谐波的波长为 λₙ = 2L/n，其中 L 为弦长。基频（第一谐波）的频率为：
f₁ = v/2L = (1/2L)√(T/μ)
where T is the tension in the string and μ is the mass per unit length. This equation explains how string instruments are tuned: increasing tension raises the pitch. In interference and diffraction problems, the path difference Δx determines whether constructive interference (Δx = nλ) or destructive interference (Δx = (n + ½)λ) occurs.
其中 T 为弦的张力，μ 为单位长度的质量。这个方程解释了弦乐器如何调音：增大张力使音调升高。在干涉和衍射问题中，光程差 Δx 决定了是发生相长干涉（Δx = nλ）还是相消干涉（Δx = (n + ½)λ）。
7. Electric Current and Resistance | 电流与电阻
Ohm’s law states that the current through a metallic conductor is directly proportional to the potential difference across it, provided the temperature remains constant:
欧姆定律指出：在温度保持不变的条件下，通过金属导体的电流与其两端的电势差成正比：
V = IR
The resistance of a conductor depends on its dimensions and material:
导体的电阻取决于其尺寸和材料：
R = ρL/A
where ρ is the resistivity of the material in ohm-metres, L is the length, and A is the cross-sectional area. This equation is frequently tested in questions about wire stretching: when a wire is stretched to double its length, its cross-sectional area halves (volume conserved), so the resistance increases by a factor of four.
其中 ρ 为材料的电阻率（单位欧姆·米），L 为长度，A 为横截面积。这个方程常出现在金属丝拉伸的问题中：当金属丝被拉伸为原来两倍长时，其横截面积减半（体积保持不变），因此电阻增大为原来的四倍。
Electrical power can be expressed in three equivalent forms:
电功率可以用三种等价形式表示：
P = VI = I²R = V²/R
When analysing circuits, use P = I²R for resistors in series (same current) and P = V²/R for resistors in parallel (same voltage). The maximum power transfer theorem states that maximum power is delivered to a load when the load resistance equals the internal resistance of the source.
在分析电路时，串联电阻（电流相同）适用 P = I²R，并联电阻（电压相同）适用 P = V²/R。最大功率传输定理指出：当负载电阻等于电源内阻时，负载获得最大功率。
8. Capacitance and Exponential Decay | 电容与指数衰减
A capacitor stores charge Q when connected to a potential difference V:
电容器在连接电势差 V 时储存电荷 Q：
Q = CV
The energy stored in a capacitor is:
电容器储存的能量为：
E = ½CV² = ½QV
When a capacitor discharges through a resistor, the charge, voltage, and current all decay exponentially with time:
当电容器通过电阻放电时，电荷、电压和电流均随时间指数衰减：
Q = Q₀e^(-t/RC)
The product RC is called the time constant τ, which has units of seconds. After one time constant (t = RC), the charge has fallen to 1/e ≈ 37% of its initial value. The time constant can also be determined graphically from the gradient of the ln Q against t graph, which is a straight line with gradient -1/RC.
乘积 RC 称为时间常数 τ，单位为秒。经过一个时间常数（t = RC）后，电荷下降到初始值的 1/e ≈ 37%。时间常数也可以通过 ln Q 对 t 图像求得：该图像为直线，斜率为 -1/RC。
A similar exponential law applies to radioactive decay:
类似的指数规律也适用于放射性衰变：
A = A₀e^(-λt)
where A is the activity, A₀ is the initial activity, and λ is the decay constant in s⁻¹. The half-life T½ is related to the decay constant by:
其中 A 为放射性活度，A₀ 为初始活度，λ 为衰变常数（单位 s⁻¹）。半衰期 T½ 与衰变常数的关系为：
T½ = ln 2 / λ = 0.693/λ
9. Gravitational Fields | 引力场
Newton’s law of universal gravitation states that the gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation:
牛顿万有引力定律指出：两个质点之间的引力与它们质量的乘积成正比，与它们之间距离的平方成反比：
F = GMm/r²
The gravitational field strength g at a distance r from the centre of a planet of mass M is:
距质量为 M 的行星中心距离 r 处的引力场强度 g 为：
g = GM/r²
For an object in a circular orbit of radius r around a planet, the gravitational force provides the centripetal force:
对于绕行星做半径为 r 的圆周运动的物体，引力提供向心力：
GMm/r² = mv²/r
Simplifying gives the orbital speed v = √(GM/r). The orbital period T satisfies Kepler’s third law:
化简得到轨道速度 v = √(GM/r)。轨道周期 T 满足开普勒第三定律：
T² = (4π²/GM)r³
This equation is used to determine the mass of planets or stars from the orbital period and radius of a satellite or moon. In geostationary orbit, T = 24 hours, giving r ≈ 42,300 km from the Earth’s centre.
该方程可用于根据卫星或月球的轨道周期和轨道半径来测定行星或恒星的质量。对于地球同步轨道，T = 24 小时，求得 r ≈ 42,300 千米（距地心）。
10. Ideal Gas Equation | 理想气体方程
The ideal gas equation links pressure, volume, temperature, and the amount of gas:
理想气体方程将压强、体积、温度和气体的量联系起来：
pV = nRT
where p is the pressure in pascals, V is the volume in cubic metres, n is the number of moles, R is the molar gas constant (8.31 J mol⁻¹ K⁻¹), and T is the absolute temperature in kelvin.
其中 p 为压强（单位帕斯卡），V 为体积（单位立方米），n 为物质的量（单位摩尔），R 为摩尔气体常数（8.31 焦耳每摩尔每开尔文），T 为热力学温度（单位开尔文）。
In terms of the number of molecules N, the equation becomes:
用分子数 N 表示时，方程为：
pV = NkT
where k is the Boltzmann constant (1.38 × 10⁻²³ J K⁻¹). Combining the ideal gas equation with the kinetic theory of gases gives the average translational kinetic energy of a molecule:
其中 k 为玻尔兹曼常数（1.38 × 10⁻²³ 焦耳每开尔文）。将理想气体方程与气体动理论结合，得到分子的平均平动动能：
½m⟨c²⟩ = (3/2)kT
A common exam question involves using the ideal gas equation to calculate the number of moles and then converting to the number of molecules using Avogadro’s constant N_A = 6.02 × 10²³ mol⁻¹. Remember that all temperatures must be converted to kelvin (T = θ + 273.15) before substitution.
常见考题要求用理想气体方程计算物质的量，然后通过阿伏伽德罗常数 N_A = 6.02 × 10²³ mol⁻¹ 换算为分子数。切记所有温度必须先换算为开尔文（T = θ + 273.15）再代入计算。
11. Summary and Exam Strategy | 总结与应试策略
Mastery of these equations requires more than memorisation; you must understand the conditions under which each equation applies. The table below summarises the key equations and their applicability.
掌握这些方程不能仅靠死记硬背，还必须理解每个方程的适用条件。下表总结了关键方程及其适用范围。
| Topic | 专题 | Key Equation | 关键方程 | Condition | 适用条件 |
| Kinematics 运动学 | v = u + at; s = ut + ½at² | Constant acceleration 匀加速 |
| Dynamics 动力学 | F = ma; p = mv | Constant mass 质量恒定 |
| Energy 能量 | KE = ½mv²; PE = mgh | Non-relativistic speeds 非相对论速度 |
| Circular motion 圆周运动 | F = mv²/r; v = ωr | Uniform circular motion 匀速圆周运动 |
| SHM 简谐运动 | a = -ω²x; T = 2π√(m/k) | Small oscillations 小幅度振动 |
| Waves 波动 | v = fλ | All waves 所有波 |
| Electricity 电学 | V = IR; P = VI | Constant temperature 温度恒定 |
| Capacitors 电容器 | Q = CV; Q = Q₀e^(-t/RC) | RC circuit 电阻电容电路 |
| Gravitation 万有引力 | F = GMm/r²; g = GM/r² | Point masses / spherical bodies 质点或球体 |
| Ideal gas 理想气体 | pV = nRT | Low pressure, high temperature 低压高温 |
In the exam, always write down the equation before substituting numbers. Check units carefully: convert centimetres to metres, grams to kilograms, and degrees Celsius to kelvin. For graphs, identify whether the relationship is linear, inverse, or exponential, and use suitable graph transformations such as plotting ln A against t to obtain a straight line.
考试中务必先写出方程再代入数值。仔细检查单位：将厘米换算为米，克换算为千克，摄氏度换算为开尔文。对于图像问题，判断关系是线性、反比还是指数，并使用合适的图像变换（如绘制 ln A 对 t 的图像）以获得直线。
12. Final Advice | 最后建议
Physics equations are tools for reasoning, not just formulas to quote. When tackling a problem, first identify the physical situation, then select the relevant equations, and finally solve step by step. Practise deriving one equation from another; for example, derive v² = u² + 2as from v = u + at and s = ut + ½at² by eliminating t. This deepens your understanding and prepares you for the ‘show that’ type questions that CIE exams frequently include.
物理方程是推理的工具，而不仅仅是可以引用的公式。解题时，先判断物理情境，再选择相关方程，最后逐步求解。练习方程之间的相互推导，例如从 v = u + at 和 s = ut + ½at² 中消去 t，推导出 v² = u² + 2as。这会加深你的理解，并为 CIE 考试中常见的”证明”类题目做好准备。
Build a formula sheet in your revision notes, organised by topic, and revisit it regularly. Use flashcards for equations that you frequently confuse, such as those for capacitor discharge and radioactive decay. Most importantly, apply these equations to past paper questions — this is the most effective way to internalise them and to recognise the patterns that examiners repeat year after year.
在复习笔记中建立按专题组织的公式表，并定期复习。对于容易混淆的方程（如电容器放电与放射性衰变的公式），使用闪卡加强记忆。最重要的是，用真题来练习这些方程——这是内化知识、识别考官年复一年重复考查模式的最有效途径。
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