📚 Essential Physics for Cambridge IGCSE: Key Formula Derivations | 剑桥IGCSE物理核心公式推导
Mastering IGCSE Physics is not just about memorising equations — it’s about understanding where they come from. In this article, we systematically derive the most essential formulas you will encounter in the Cambridge IGCSE syllabus, using clear logical steps and fundamental principles. By seeing how each relationship is built, you’ll gain deeper insight and the confidence to tackle both calculation and explanation questions in your exams.
掌握IGCSE物理不仅仅是要记住公式——更重要的是理解它们的来源。本文将系统地推导剑桥IGCSE课程中最核心的公式，运用清晰的逻辑步骤和基本原理。通过了解每个关系的由来，你将获得更深刻的理解，并有信心应对考试中的计算题和解释题。
1. Deriving the SUVAT Equations from a Velocity–Time Graph | 从速度-时间图推导运动学方程
Consider an object moving with uniform acceleration. Its velocity–time graph is a straight line with initial velocity u, final velocity v, and time taken t. The gradient of the line is acceleration a, so we can immediately write a = (v – u) / t, which rearranges to:
考虑一个匀加速运动的物体。其速度-时间图像是一条直线，初速度为u，末速度为v，用时为t。该直线的斜率即为加速度a，因此可直接写出a = (v – u) / t，整理得：
v = u + at
The total displacement s is the area under the graph. This area can be split into a rectangle of area ut and a triangle of area ½ (v – u)t. Using v – u = at, the total area becomes s = ut + ½ at².
总位移s等于图像下的面积。该面积可分为一个面积为ut的矩形和一个面积为½ (v – u)t的三角形。将v – u = at代入，总面积变为s = ut + ½ at²。
s = ut + ½ at²
We can also express displacement using the average velocity. For uniform acceleration, average velocity = (u+v)/2, so s = (u+v)t/2. Combining this with v = u + at to eliminate t yields the third equation:
我们还可以用平均速度表示位移。对于匀加速运动，平均速度 = (u+v)/2，因此s = (u+v)t/2。将此式与v = u + at联立消去t，即得到第三个方程：
v² = u² + 2as
2. Newton’s Second Law as F = ma | 牛顿第二定律 F = ma 的推导
Newton’s second law states that the net force on an object is proportional to the rate of change of its momentum. Momentum p = mv, so for constant mass, force F = (mv – mu) / t = m(v – u) / t. Since (v – u)/t = a, we obtain the familiar form:
牛顿第二定律指出，作用在物体上的合力与其动量的变化率成正比。动量p = mv，因此质量恒定时，力F = (mv – mu) / t = m(v – u) / t。由于(v – u)/t = a，即得到常见形式：
F = ma
This derivation emphasises that force is directly linked to acceleration, not just motion, and it is the foundation for almost all mechanics calculations in the IGCSE syllabus.
这一推导强调了力与加速度直接相关，而不仅仅是运动，它是IGCSE课程中几乎所有力学计算的基础。
3. Work Done and the Kinetic Energy Formula | 功与动能公式的推导
Work done by a constant force over a displacement s is W = F s. If this force accelerates a mass from rest, we can substitute F = ma and v² = 2as (since u = 0), giving a = v²/(2s). Thus W = m × v²/(2s) × s = ½ m v². This energy transferred is stored as kinetic energy:
恒力作用一段位移s所做的功为W = F s。如果这个力使一个物体从静止开始加速，代入F = ma以及v² = 2as（因为u=0），可得a = v²/(2s)。因此W = m × v²/(2s) × s = ½ m v²。这部分转移的能量即储存为动能：
KE = ½ m v²
If the object already had initial speed u, the same method using v² = u² + 2as gives the net work done as the change in kinetic energy: W = ½ m (v² – u²).
若物体已有初速度u，使用v² = u² + 2as同样可推导出净功等于动能的变化量：W = ½ m (v² – u²)。
4. Gravitational Potential Energy near Earth’s Surface | 地球表面附近的重力势能
When an object of weight mg is lifted through a vertical height Δh, the work done against gravity is W = force × distance = mg Δh. This work is stored as gravitational potential energy. Assuming the field is uniform, we define:
当一个重量为mg的物体被提升垂直高度Δh时，克服重力所做的功为W = 力 × 距离 = mg Δh。这部分功储存为重力势能。假设重力场均匀，定义：
GPE = mgΔh
IGCSE questions often ask you to equate GPE gained to kinetic energy lost in a falling object, leading to mgh = ½ m v², which allows calculation of impact speed.
IGCSE题目经常要求你将物体下落时获得的重力势能与失去的动能等同起来，即mgh = ½ m v²，从而计算撞击速度。
5. Density and Pressure in Fluids | 密度与液体压强
Density ρ is defined as mass per unit volume: ρ = m / V. To derive fluid pressure at depth h, consider a column of liquid of cross-sectional area A. The weight of the column is mg = (ρ × A × h) g. This weight acts on area A, producing pressure:
密度ρ定义为单位体积的质量：ρ = m / V。为推导深度h处的液体压强，考虑一截面积为A的液柱。液柱的重量为mg = (ρ × A × h) g。该重量作用在面积A上，产生的压强为：
P = F / A = ρgh
The total pressure at a point also includes atmospheric pressure on the surface, but the formula above gives the pressure due to the liquid alone.
某点的总压强还需加上液面的大气压，但上述公式给出了仅由液体产生的压强。
6. Ohm’s Law and Defining Resistance | 欧姆定律与电阻定义
For many conductors at constant temperature, the current I is directly proportional to the potential difference V across it. The constant of proportionality is the resistance R, defined as R = V / I. While Ohm’s law is an experimental observation, rearranging gives the universally applicable definition:
对于许多恒温下的导体，电流I与其两端的电势差V成正比。比例常数即为电阻R，定义为R = V / I。尽管欧姆定律是实验观察结果，但整理后可得到普遍适用的定义：
V = I R
Resistance can also be expressed in terms of physical properties: R = ρ L / A, where ρ is resistivity, L length, and A cross-sectional area. This formula is derived from the idea that resistance increases with length and decreases with area.
电阻还可用物理特性表示为：R = ρ L / A，其中ρ为电阻率，L为长度，A为横截面积。该公式的思路是电阻随长度增加而增大，随截面积增大而减小。
7. Resistors in Series and Parallel | 串联和并联电阻的公式推导
For resistors in series, the same current flows through each. The total p.d. is the sum of individual p.d.s: V_total = V₁ + V₂ + V₃. Applying V = IR to each gives I R_total = I R₁ + I R₂ + I R₃, so:
串联电阻中流过相同的电流。总电压等于各电阻电压之和：V_total = V₁ + V₂ + V₃。对每个电阻应用V = IR得I R_total = I R₁ + I R₂ + I R₃，因此：
R_total = R₁ + R₂ + R₃
For parallel resistors, the p.d. across each branch is the same. The total current is the sum of branch currents: I_total = I₁ + I₂ + I₃. Using I = V/R, we have V / R_total = V/R₁ + V/R₂ + V/R₃, which simplifies to:
并联电阻各支路两端电压相同。总电流等于各支路电流之和：I_total = I₁ + I₂ + I₃。代入I = V/R得V / R_total = V/R₁ + V/R₂ + V/R₃，化简为：
1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃
These derivations are simple but powerful in circuit analysis.
这些推导虽简单，但在电路分析中十分有力。
8. Electrical Power and Energy Dissipated | 电功率与消耗的电能
Power is the rate of doing work: P = W / t. In an electrical component, the work done to move charge Q through a p.d. V is W = V Q. Current I is charge per unit time: I = Q / t, so Q = I t. Substituting gives:
功率是做功的速率：P = W / t。在电器元件中，移送电荷Q通过电势差V所做的功为W = V Q。电流I是单位时间流过的电荷：I = Q / t，故Q = I t。代入得：
P = V I
Using Ohm’s law, we can replace V or I to obtain two more useful forms:
利用欧姆定律，可替换V或I得到另两种有用形式：
P = I² R and P = V² / R
These are essential when analysing heating effects in resistors.
这些公式在分析电阻的热效应时至关重要。
9. The Transformer Equation | 变压器公式
An ideal transformer operates on the principle of electromagnetic induction. The alternating flux links both primary and secondary coils, inducing e.m.f.s proportional to their respective turns. For 100% efficiency, input power equals output power: V_P × I_P = V_S × I_S. Moreover, the voltage ratio equals the turns ratio:
理想变压器基于电磁感应原理工作。交变磁通同时穿过初级和次级线圈，感应出的电动势与各自匝数成正比。效率为100%时，输入功率等于输出功率：V_P × I_P = V_S × I_S。此外，电压比等于匝数比：
V_P / V_S = N_P / N_S
This is derived from Faraday’s law: the induced e.m.f. in a coil is proportional to the rate of change of flux linkage, and with the same flux linking both coils, the ratio holds.
该公式源自法拉第定律：线圈中的感应电动势与磁通量链的变化率成正比，由于同一磁通穿过两组线圈，此比值成立。
10. The Wave Equation v = f λ | 波速公式 v = f λ
A wave’s speed, frequency and wavelength are connected by a simple relationship. Frequency f is the number of complete waves passing a point per second; wavelength λ is the distance travelled by one complete wave. In one second, f waves pass a fixed point, so the distance covered by the wave front is f × λ. Therefore, the speed is:
波速、频率和波长之间存在简单关系。频率f是每秒通过某点的完整波数；波长λ是一个完整波行进的距离。在一秒内，有f个波通过固定点，因此波前移动的距离为f × λ。因此速度为：
v = f λ
This equation applies to all waves — sound, water, light — and is fundamental in understanding phenomena like refraction.
该方程适用于所有波——声波、水波、光波——是理解折射等现象的基础。
11. Efficiency of Energy Transfers | 能量传递的效率
No device is perfect; some energy is always wasted as heat or sound. Efficiency measures how much of the input energy is converted to useful output. Expressed either in terms of energy or power:
没有完美的装置；总会有部分能量以热或声的形式浪费。效率衡量输入能量中有多少转化为有用输出。可用能量或功率表示：
Efficiency = (Useful energy output / Total energy input) × 100%
Efficiency = (Useful power output / Total power input) × 100%
This simple ratio is derived directly from the principle of conservation of energy and is vital for evaluating real-world systems like motors and lamps.
这个简单的比值直接来自能量守恒原理，对于评估电动机和灯泡等实际系统至关重要。
12. Hooke’s Law and the Spring Constant | 胡克定律与弹簧常数
For many elastic materials, the extension x is directly proportional to the applied force F, provided the elastic limit is not exceeded. This proportionality is Hooke’s law, written as:
对于许多弹性材料，只要未超过弹性极限，伸长量x与施加的力F成正比。这一比例关系即为胡克定律，写作：
F = k x
The constant k is the spring constant (stiffness). While this is an experimental law, the energy stored in a stretched spring can be derived from the area under the force–extension graph (a triangle), giving Elastic potential energy = ½ F x = ½ k x². This derivation mirrors the kinetic energy approach.
常数k为弹簧常数（劲度系数）。虽然这是一条实验定律，但通过力-伸长图下方三角形的面积可推导出弹簧储存的能量：弹性势能 = ½ F x = ½ k x²。这一推导与动能公式的方法类似。
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