Year 11 Cambridge Science: Formula & Theorem Quick Reference Handbook | 剑桥11年级科学：公式定理速查手册

📚 Year 11 Cambridge Science: Formula & Theorem Quick Reference Handbook | 剑桥11年级科学：公式定理速查手册

This handbook compiles the essential formulas, equations, and key theorems you need to master for Year 11 Cambridge IGCSE Science (Co-ordinated Sciences or separate Biology, Chemistry, Physics). It serves as a quick revision tool organised by topic, with each entry presented in English and Chinese to reinforce bilingual understanding. Use it to test your recall of definitions, symbol meanings, and SI units.

本手册汇编了剑桥11年级IGCSE科学（协调科学或单独的生物学、化学、物理学）必须掌握的核心公式、方程式和关键定理。它按主题编排，是快速复习的工具，每个条目均以中英文呈现，以强化双语理解。用它来检验你对定义、符号含义和国际单位制的记忆。

1. Kinematics – Motion in a Straight Line | 运动学 – 直线运动

The equations of motion apply when acceleration is constant. Always define a positive direction before substituting values.

当加速度恒定时，运动学方程适用。代入数值前务必先规定正方向。

Average speed = total distance ÷ total time
平均速率 = 总路程 ÷ 总时间
v = u + a t
最终速度 = 初始速度 + 加速度 × 时间
Symbols: v = final velocity (m/s), u = initial velocity (m/s), a = acceleration (m/s²), t = time (s)
s = u t + ½ a t²
位移 = 初始速度 × 时间 + ½ × 加速度 × 时间²
s = displacement (m)
v² = u² + 2 a s
最终速度² = 初始速度² + 2 × 加速度 × 位移
s = (u + v) / 2 × t
位移 = (初始速度 + 最终速度) / 2 × 时间

Remember: in free fall near Earth’s surface, a = g ≈ 9.8 m/s² downward (use ± appropriately).

记住：在地球表面附近自由下落时，加速度 a = g ≈ 9.8 m/s² 方向向下（注意正负号的选用）。

2. Forces and Newton’s Laws | 力与牛顿定律

Newton’s three laws form the backbone of mechanics. Always draw a free-body diagram to identify forces.

牛顿三定律是力学的基石。务必画出受力分析图以明确各个力。

First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a resultant force.
第一定律（惯性）：物体保持静止或匀速直线运动状态，除非受到合外力的作用。
Second Law: F = m a
合力 = 质量 × 加速度
F = resultant force (N), m = mass (kg), a = acceleration (m/s²)
Third Law: Forces occur in action–reaction pairs; equal in magnitude, opposite in direction, acting on different bodies.
第三定律：力以作用力–反作用力成对出现；大小相等、方向相反、作用在不同物体上。
Weight: W = m g
重力 = 质量 × 重力加速度
Hooke’s Law: F = k x (within the limit of proportionality)
胡克定律：F = k x（在弹性限度内）
k = spring constant (N/m), x = extension (m)
Moment of a force: moment = force × perpendicular distance from pivot
力矩 = 力 × 力臂（支点到力作用线的垂直距离）
Pressure: P = F / A
压强 = 压力 / 受力面积
P = pressure (Pa or N/m²), F = force (N), A = area (m²)

3. Energy, Work and Power | 能量、功与功率

Energy is a conserved quantity; it can be stored, transferred, but never destroyed. Use energy transformations to solve problems.

能量是守恒量，可以被储存、转移，但不会消失。运用能量转化来解题。

Kinetic energy: Eₖ = ½ m v²
动能 = ½ × 质量 × 速度²
Gravitational potential energy: Eₚ = m g h
重力势能 = 质量 × 重力加速度 × 高度
h = height (m)
Work done: W = F d (when force is parallel to displacement)
做功 = 力 × 在力的方向上移动的距离
W = work (J), d = distance (m)
Power: P = W / t or P = E / t
功率 = 做功 / 时间 = 能量转化 / 时间
P = power (W), t = time (s)
Efficiency = (useful energy output / total energy input) × 100%
效率 = (有用能量输出 / 总能量输入) × 100%

4. Thermal Physics – Heat and Temperature | 热物理 – 热量与温度

Specific heat capacity and latent heat govern energy changes during heating and phase transitions.

比热容和潜热决定了加热和相变过程中的能量变化。

Heat energy: Q = m c Δθ
热量 = 质量 × 比热容 × 温度变化
Q = heat energy (J), c = specific heat capacity (J/(kg·°C)), Δθ = change in temperature (°C)
Latent heat: Q = m L
潜热 = 质量 × 比潜热
L = specific latent heat (J/kg) – fusion or vaporisation
Gas laws (ideal gas): p₁V₁ / T₁ = p₂V₂ / T₂ (for a fixed mass, T in kelvin)
理想气体定律：p₁V₁ / T₁ = p₂V₂ / T₂（质量固定，温度用开尔文）
Absolute zero: 0 K = –273 °C
绝对零度：0 K = –273 °C

5. Waves – Light and Sound | 波动 – 光与声

The wave equation links speed, frequency and wavelength. Reflection, refraction and total internal reflection rely on the laws of optics.

波速公式将速度、频率和波长联系起来。反射、折射和全反射依赖于光学定律。

Wave speed: v = f λ
波速 = 频率 × 波长
v = wave speed (m/s), f = frequency (Hz), λ = wavelength (m)
Refractive index: n = sin i / sin r (when light enters from vacuum/air)
折射率：n = sin i / sin r（光从真空/空气射入介质）
Critical angle: sin c = 1 / n
临界角：sin c = 1 / n
Total internal reflection occurs when angle of incidence > c and light travels from denser to less dense medium.
Lens formula (not always required but useful): 1/f = 1/u + 1/v
透镜公式：1/f = 1/u + 1/v（f = 焦距，u = 物距，v = 像距）

6. Electricity – Circuits and Components | 电学 – 电路与元件

Ohm’s law and the relationships for series and parallel circuits are essential for circuit analysis.

欧姆定律以及串并联电路的关系是分析电路的基础。

Ohm’s law: V = I R (at constant temperature)
欧姆定律：电压 = 电流 × 电阻（温度恒定）
V = potential difference (V), I = current (A), R = resistance (Ω)
Series circuits: I same everywhere, V_total = V₁ + V₂ + …, R_total = R₁ + R₂ + …
串联电路：电流处处相等，总电压 = 各电压之和，总电阻 = 各电阻之和。
Parallel circuits: V same across branches, I_total = I₁ + I₂ + …, 1/R_total = 1/R₁ + 1/R₂ + …
并联电路：各支路电压相等，总电流 = 各支路电流之和，总电阻倒数 = 各电阻倒数之和。
Electrical power: P = I V = I² R = V² / R
电功率：P = I V = I² R = V² / R
Energy transferred: E = I V t
电能转移：E = I V t
E = energy (J), t = time (s)
Charge: Q = I t
电荷量 = 电流 × 时间 (C = A·s)

7. Magnetism and Electromagnetism | 磁学与电磁学

Moving charges produce magnetic fields, and changing magnetic fields induce voltage.

运动的电荷产生磁场，变化的磁场产生感应电压。

Transformer equation: Vₚ / Vₛ = Nₚ / Nₛ (for an ideal transformer)
变压器公式：初级电压 / 次级电压 = 初级匝数 / 次级匝数
p = primary, s = secondary, N = number of turns
Power input ≈ Power output: Vₚ Iₚ = Vₛ Iₛ (ideal case)
输入功率 ≈ 输出功率：Vₚ Iₚ = Vₛ Iₛ（理想情况）
Fleming’s left-hand rule: used for motor effect (force on current-carrying conductor in magnetic field).
弗莱明左手定则：用于电动机效应（磁场对载流导体的力）。
Fleming’s right-hand rule: used for generator effect (induced current direction).
弗莱明右手定则：用于发电机效应（感应电流方向）。

8. Chemical Formulae and Moles | 化学式与摩尔

The mole concept is central to quantitative chemistry. Always work in moles to compare quantities.

摩尔概念是定量化学的核心。始终用摩尔来比较各物质的数量。

Number of moles: n = m / M
摩尔数 = 质量 / 摩尔质量
n = moles (mol), m = mass (g), M = molar mass (g/mol)
Concentration: c = n / V
浓度 = 摩尔数 / 体积
c = concentration (mol/dm³), V = volume (dm³); also c₁V₁ = c₂V₂ for dilutions.
Volume of gas at RTP (room temperature and pressure): 1 mole = 24 dm³
常温常压下气体摩尔体积：1 摩尔 = 24 dm³
Mole ratio: use balanced equation coefficients to relate moles of reactants and products.
摩尔比：利用配平方程式的系数关联反应物和产物的摩尔数。
Percentage yield: (actual yield / theoretical yield) × 100%
产率：(实际产量 / 理论产量) × 100%
Empirical formula: simplest whole-number ratio of atoms; molecular formula = n × empirical formula.
实验式：最简单的整数原子个数比；分子式 = n × 实验式。

9. Acids, Bases and Salts | 酸、碱和盐

Acid–base reactions can be described in terms of proton transfer. Neutralisation always forms a salt and water.

酸碱反应可用质子转移来描述。中和反应总是生成盐和水。

pH scale: pH = –log₁₀[H⁺]; [H⁺] = 10⁻^pH (for strong acids, approximate)
pH 值：pH = –lg[H⁺]；[H⁺] = 10⁻^pH（对强酸近似成立）
Neutralisation: H⁺(aq) + OH⁻(aq) → H₂O(l)
中和反应离子方程式：H⁺(aq) + OH⁻(aq) → H₂O(l)
Acid + metal → salt + hydrogen
酸 + 活泼金属 → 盐 + 氢气
Acid + carbonate → salt + water + carbon dioxide
酸 + 碳酸盐 → 盐 + 水 + 二氧化碳
Titration formula: nₐ / n_b = (cₐ Vₐ) / (c_b V_b) based on mole ratio from the equation.
滴定公式：根据方程式摩尔比，n_酸 / n_碱 = (c_酸 V_酸) / (c_碱 V_碱)

10. Rates of Reaction and Energetics | 反应速率与能量变化

The rate of a reaction can be followed by measuring how quickly a reactant is consumed or a product is formed. Collision theory explains the effect of temperature, concentration and surface area.

反应速率可通过测量反应物消耗或产物生成的快慢来跟踪。碰撞理论解释了温度、浓度和表面积的影响。

Rate = change in concentration (or mass/volume) / time
速率 = 浓度（或质量/体积）的变化量 / 时间
Activation energy (Eₐ): minimum energy needed for a collision to result in a reaction.
活化能 (Eₐ)：发生有效碰撞所需的最低能量。
Exothermic reaction: ΔH negative; energy released to surroundings.
放热反应：ΔH 为负；向环境释放能量。
Endothermic reaction: ΔH positive; energy absorbed from surroundings.
吸热反应：ΔH 为正；从环境吸收能量。
Bond energy calculation: ΔH = Σ (bond energies of bonds broken) – Σ (bond energies of bonds formed).
键能计算：ΔH = 断裂化学键吸收的总能量 – 形成化学键释放的总能量。

11. Biology – Key Equations and Relationships | 生物学 – 核心方程与关系

In Biology, mathematical formulas often appear in topics such as photosynthesis, respiration, and population studies. Even though they are not as numerous as in Physics, they are crucial for data analysis.

在生物学中，数学公式常出现在光合作用、呼吸作用和种群研究等主题。虽然数量不如物理多，但对数据分析至关重要。

Magnification: Magnification = size of image / actual size of specimen
放大倍数 = 图像尺寸 / 标本实际尺寸
Always use the same units; rearrange as needed.
Rate of reaction (e.g., enzyme activity): Rate = 1 / time or change in product amount / time
反应速率（如酶活性）：速率 = 1 / 时间或产物变化量 / 时间
Respiratory quotient (RQ): RQ = CO₂ produced / O₂ consumed
呼吸商 (RQ)：RQ = 产生的CO₂ 体积 / 消耗的O₂ 体积
Population growth rate: (population change / original population) × 100%
种群增长率：(种群变化量 / 初始种群数量) × 100%
Photosynthesis word equation: carbon dioxide + water → glucose + oxygen (in the presence of light and chlorophyll)
光合作用文字方程式：二氧化碳 + 水 → 葡萄糖 + 氧气（在光和叶绿素存在下）
Aerobic respiration: glucose + oxygen → carbon dioxide + water (+ energy)
有氧呼吸：葡萄糖 + 氧气 → 二氧化碳 + 水（+ 能量）
Anaerobic respiration in yeast: glucose → ethanol + carbon dioxide (+ energy)
酵母无氧呼吸：葡萄糖 → 乙醇 + 二氧化碳（+ 能量）
C₆H₁₂O₆ → 2 C₂H₅OH + 2 CO₂ (balanced fermentation equation)
C₆H₁₂O₆ → 2 C₂H₅OH + 2 CO₂（发酵的配平方程式）

Understanding these simple quantitative relationships will help you interpret experiments and graphs accurately.

理解这些简单的定量关系，有助于你准确解读实验和图表。

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Year 11 Cambridge Science: Formula & Theorem Quick Reference Handbook | 剑桥11年级科学：公式定理速查手册