Pre-U Science Formula & Theorem Quick Reference Handbook | Pre-U 科学公式定理速查手册

📚 Pre-U Science Formula & Theorem Quick Reference Handbook | Pre-U 科学公式定理速查手册

This handbook brings together the most essential formulas, equations and theorems you will encounter across Physics, Chemistry and Biology in the Pre-U Science syllabus. Use it as a quick revision companion to reinforce your quantitative problem-solving skills.

本手册汇集了 Pre-U 科学课程中物理、化学和生物学最核心的公式、方程与定理，可作为快速复习伴侣，帮助你强化定量解题能力。

1. Equations of Motion | 运动学方程

For constant acceleration along a straight line, the four SUVAT equations connect displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).

在恒定加速度的直线运动中，以下四个运动学方程将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。

v = u + a t

The velocity–time relation: final velocity equals initial velocity plus acceleration multiplied by time.

速度–时间关系：末速度等于初速度加加速度乘以时间。

s = u t + ½a t²

Displacement when initial velocity is known; the term ½ a t² accounts for the distance gained from acceleration.

已知初速度时的位移公式；½ a t² 项反映由加速度产生的额外距离。

v² = u² + 2 a s

This form eliminates time, making it useful when acceleration and distance are given but time is not.

该方程消去了时间，当已知加速度和位移而时间未知时非常实用。

s = (u + v) t / 2

Displacement as the average velocity multiplied by time – a direct consequence of uniform acceleration.

位移等于平均速度乘以时间——这是匀加速运动的直接结果。

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2. Newton’s Laws and Momentum | 牛顿定律与动量

Newton’s laws govern the relationship between force, mass and motion. Momentum, defined as mass times velocity, is conserved in isolated systems.

牛顿定律描述了力、质量与运动的关系。动量定义为质量乘以速度，在孤立系统中守恒。

F = m a

The net force acting on a body equals its mass multiplied by its acceleration – Newton’s second law in its simplest form.

作用在物体上的合外力等于其质量与加速度的乘积——牛顿第二定律的最简形式。

p = m v

Linear momentum (p) is the product of mass and velocity; its unit is kg m s⁻¹.

线动量 p 是质量与速度的乘积；单位为千克·米/秒 (kg·m·s⁻¹)。

Σ p_before = Σ p_after

In the absence of external forces, total momentum before a collision equals total momentum after the collision – the principle of conservation of momentum.

在没有外力的条件下，碰撞前的总动量等于碰撞后的总动量——动量守恒原理。

F = Δp / Δt

Newton’s second law can also be expressed as force equals the rate of change of momentum; this links impulse (F Δt) to the change in momentum.

牛顿第二定律也可表示为力等于动量的变化率；这建立了冲量 (F Δt) 与动量变化之间的联系。

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3. Work, Energy and Power | 功、能与功率

Energy is the capacity to do work. The work–energy theorem and the principle of conservation of energy are fundamental to analysing mechanical systems.

能量是做功的能力。功能定理与能量守恒原理是分析力学系统的基础。

W = F s cos θ

Work done by a constant force equals the product of force, displacement and the cosine of the angle between them. When force and displacement are parallel, W = F s.

恒力所做的功等于力、位移及两者夹角余弦的乘积。当力与位移平行时，W = F s。

KE = ½ m v²

Kinetic energy of a body of mass m moving with speed v. It is always positive and scalar.

质量为 m、速度为 v 的物体的动能。动能恒为正标量。

PE = m g h

Gravitational potential energy near the Earth’s surface, where g is the gravitational field strength (≈ 9.8 N kg⁻¹).

近地表的重力势能，其中 g 为重力场强度（约 9.8 N kg⁻¹）。

P = W / t = F v

Power is the rate of doing work; for constant force and velocity in the same direction, power equals force times velocity.

功率是做功的速率；当力与速度方向相同且恒定时，功率等于力乘以速度。

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4. Waves and Optics | 波与光学

Wave phenomena are described by quantities such as frequency, wavelength and speed. Interference and diffraction provide evidence for the wave nature of light and sound.

波的特性由频率、波长和波速等物理量描述。干涉与衍射为光和声音的波动性提供了证据。

v = f λ

The wave speed (v) equals frequency (f) multiplied by wavelength (λ). This applies to all periodic waves.

波速 v 等于频率 f 与波长 λ 的乘积。这适用于所有周期性波。

n = sin i / sin r

Snell’s law of refraction, where n is the relative refractive index, i is the angle of incidence and r is the angle of refraction.

斯涅尔折射定律，其中 n 为相对折射率，i 为入射角，r 为折射角。

n₁ sin θ₁ = n₂ sin θ₂

General form of Snell’s law using absolute refractive indices n₁ and n₂ of two media.

使用两种介质的绝对折射率 n₁ 和 n₂ 的斯涅尔定律通式。

d sin θ = n λ

For a diffraction grating, the n-th order maximum occurs at angle θ when the grating spacing is d and the wavelength is λ.

对于衍射光栅，第 n 级极大出现在角度 θ 处，光栅间距为 d，波长为 λ。

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5. Electricity: Ohm’s Law and Circuits | 电学：欧姆定律与电路

Electric circuits are analysed using Ohm’s law, Kirchhoff’s rules and relationships for power and resistance. Charge and energy are conserved in every loop and junction.

利用欧姆定律、基尔霍夫定律以及功率与电阻关系来分析电路。每个回路和节点都遵守电荷与能量守恒。

V = I R

Ohm’s law: the potential difference (V) across a conductor is proportional to the current (I) through it, with resistance R as the constant of proportionality.

欧姆定律：导体两端的电压 V 与通过它的电流 I 成正比，比例常数为电阻 R。

P = I V = I² R = V² / R

Electrical power dissipated in a resistor; all three forms are equivalent when using Ohm’s law.

电阻消耗的电功率；结合欧姆定律，这三种形式是等价的。

R_total = R₁ + R₂ + … (series)

Resistances in series add directly; the same current flows through each component.

串联电阻直接相加；每个元件中流过相同的电流。

1/R_total = 1/R₁ + 1/R₂ + … (parallel)

For parallel resistors, the reciprocal of the total resistance equals the sum of reciprocals of individual resistances. The total resistance is always less than the smallest branch resistance.

并联电阻的总电阻倒数等于各支路电阻倒数之和；总电阻总是小于最小支路电阻。

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6. Thermal Physics and Gas Laws | 热物理与气体定律

The behaviour of ideal gases and the transfer of thermal energy are described by the kinetic theory and the laws of thermodynamics. Temperature, pressure and volume are linked by simple proportionalities under fixed conditions.

理想气体的行为及热能的传递由分子动理论和热力学定律描述。在固定条件下，温度、压力和体积之间具有简单的正比关系。

p V = n R T

The ideal gas equation, where p is pressure, V is volume, n is the amount of substance (mol), R is the molar gas constant (8.31 J mol⁻¹ K⁻¹) and T is the absolute temperature in kelvin.

理想气体状态方程，其中 p 为压强，V 为体积，n 为物质的量 (mol)，R 为摩尔气体常数 (8.31 J·mol⁻¹·K⁻¹)，T 为绝对温度 (开尔文)。

p ∝ T (V constant), V ∝ T (p constant)

Gay-Lussac’s law and Charles’s law – for a fixed mass of ideal gas, pressure varies directly with temperature at constant volume, and volume varies directly with temperature at constant pressure.

盖-吕萨克定律与查理定律——对于一定质量的理想气体，体积不变时压强与温度成正比；压强不变时体积与温度成正比。

p₁ V₁ / T₁ = p₂ V₂ / T₂

Combined gas law for a fixed mass of ideal gas undergoing a change in conditions; it unifies Boyle’s, Charles’s and the pressure law.

一定质量理想气体状态变化时的联合气体定律；统一了玻意耳定律、查理定律和压力定律。

ΔU = Q – W

The first law of thermodynamics: the change in internal energy of a system equals the heat added to the system minus the work done by the system on its surroundings.

热力学第一定律：系统内能的变化等于系统吸收的热量减去系统对外做的功。

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7. Quantum and Nuclear Physics | 量子与核物理

Photons, energy levels and radioactive decay are central to modern physics. The discrete nature of energy and the probabilistic character of decay are expressed by a few key relations.

光子、能级和放射性衰变是现代物理的核心。能量的分立性及衰变的概率特性由以下几个关键关系式表达。

E = h f = h c / λ

The energy of a photon: Planck’s constant h (6.63 × 10⁻³⁴ J s), frequency f, speed of light c and wavelength λ.

光子能量：E = h f = h c / λ，其中 h 为普朗克常数 (6.63 × 10⁻³⁴ J·s)，c 为光速，λ 为波长。

E_k_max = h f – Φ

Einstein’s photoelectric equation: the maximum kinetic energy of an emitted photoelectron equals the photon energy minus the work function Φ of the metal.

爱因斯坦光电方程：逸出光电子的最大动能等于光子能量减去金属的逸出功 Φ。

N = N₀ e⁻ᵏᵗ

Exponential decay law, where N is the number of undecayed nuclei at time t, N₀ is the initial number, and k is the decay constant. (Alternative form: N = N₀ exp(-λ t) with decay constant λ.)

指数衰变定律，其中 N 为时刻 t 尚未衰变的原子核数，N₀ 为初始数目，k 为衰变常数。（也可用 λ 表示，写作 N = N₀ exp(−λ t)。）

T_{1/2} = ln 2 / λ

Half-life T_{1/2} is related to the decay constant; it is the time taken for half of the radioactive nuclei to decay.

半衰期 T₁/₂ 与衰变常数 λ 的关系：T₁/₂ = ln 2 / λ；它是半数放射性核发生衰变所需的时间。

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8. Chemical Calculations: Moles, Concentration and Ideal Gas | 化学计算：摩尔、浓度与理想气体

Quantitative chemistry relies on the mole concept, stoichiometric ratios and the behaviour of gases under standard conditions. These formulas are used daily in titrations, yield calculations and gas volume determinations.

定量化学依赖于摩尔概念、化学计量比以及标准状况下的气体行为。这些公式在滴定、产率计算和气体体积测定中天天使用。

n = m / M

Amount of substance (mol): n equals the mass of the sample (m) divided by its molar mass (M, g mol⁻¹).

物质的量 n (mol)：等于样品的质量 m 除以摩尔质量 M (g·mol⁻¹)。

c = n / V

Concentration of a solution: amount of solute (n) per unit volume of solution (V), typically expressed in mol dm⁻³.

溶液浓度：溶质的物质的量 n 除以溶液体积 V，通常以 mol·dm⁻³ 表示。

n = p V / (R T)

Using the ideal gas law to find the amount of a gas from measured pressure, volume and temperature.

利用理想气体状态方程，从测得的压强、体积和温度计算气体的物质的量。

percentage yield = (actual yield / theoretical yield) × 100%

Yield compares the amount of product actually obtained to the maximum amount predicted by the stoichiometric equation.

产率将实际获得的产物量与化学计量方程式预测的理论最大量进行比较。

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9. Chemical Equilibria and Thermodynamics | 化学平衡与热力学

Reversible reactions reach a dynamic equilibrium described by an equilibrium constant. Thermochemical calculations link energy changes to bond making and breaking.

可逆反应达到动态平衡，由平衡常数描述。热化学计算将能量变化与化学键的生成和断裂联系起来。

a A + b B ⇌ c C + d D

For the general equilibrium reaction, the equilibrium constant K_c = ([C]ᶜ [D]ᵈ) / ([A]ᵃ [B]ᵇ), where square brackets denote equilibrium concentrations in mol dm⁻³.

对于一般可逆反应，浓度平衡常数 K_c = ([C]ᶜ [D]ᵈ) / ([A]ᵃ [B]ᵇ)，方括号表示各物质的平衡浓度 (mol·dm⁻³)。

K_p = (p_Cᶜ p_Dᵈ) / (p_Aᵃ p_Bᵇ)

For gas-phase equilibria, the equilibrium constant in terms of partial pressures (p in atm or Pa), each raised to the power of its stoichiometric coefficient.

对于气相平衡，以分压 (单位为 atm 或 Pa) 表示的平衡常数 K_p，各分压的指数等于其化学计量数。

ΔH = Σ E_bonds_broken – Σ E_bonds_formed

An approximate enthalpy change can be calculated using average bond energies; breaking bonds absorbs energy, forming bonds releases energy.

可以使用平均键能近似计算焓变；断裂化学键吸收能量，形成化学键释放能量。

q = m c ΔT

Heat energy transferred to a substance: mass m, specific heat capacity c, temperature change ΔT. This equation is used in calorimetry.

传递给物质的热量：质量 m，比热容 c，温度变化 ΔT。该方程用于量热计算。

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10. Biological Tools: Hardy–Weinberg, Chi-squared and Magnification | 生物工具：哈代–温伯格、卡方检验与放大率

Population genetics, statistical analysis of data and microscopy all require precise formulas. These quantitative tools allow biologists to test hypotheses and measure specimens accurately.

群体遗传学、数据统计分析及显微镜使用都需要精确的公式。这些定量工具使生物学家能够验证假设并准确测量样本。

p² + 2 p q + q² = 1

Hardy–Weinberg principle: for a gene with two alleles, the frequencies of homozygous dominant (p²), heterozygous (2pq) and homozygous recessive (q²) sum to 1, provided the population is large and mating is random.

哈代–温伯格原理：对于有两个等位基因的群体，在群体大且随机交配的条件下，显性纯合子 (p²)、杂合子 (2pq) 与隐性纯合子 (q²) 的频率之和为 1。

χ² = Σ (O – E)² / E

The chi-squared (χ²) statistic measures the difference between observed (O) and expected (E) frequencies; a high χ² value suggests the null hypothesis is unlikely.

卡方统计量 (χ²) 衡量观测频数 (O) 与期望频数 (E) 之间的差异；χ² 值越大，说明零假设成立的可能性越小。

M = I / A

Magnification M is the ratio of the image size (I) to the actual object size (A). Both must be in the same units.

放大率 M 等于图像尺寸 I 与实际物体尺寸 A 的比值；两者必须使用相同单位。

actual size = image size / magnification

When using a light or electron micrograph, the actual size of a specimen can be found by dividing the measured image length by the stated magnification.

在光镜或电子显微照片中，将测量的图像长度除以标明的放大倍数即可计算出样本的实际尺寸。

These formulas underpin data-handling problems in ecology, genetics and cell biology. Always check degrees of freedom when using chi-squared tables.

这些公式是生态学、遗传学和细胞生物学中数据处理问题的基础。使用卡方分布表时，务必检查自由度。

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