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

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

This handbook offers a compact, structured collection of essential formulae and theorems across Physics, Chemistry and Biology for the Cambridge Pre-U programme. It serves as a quick-lookup companion for consolidating knowledge during revision and tackling problem-solving tasks with confidence.

本手册为剑桥Pre-U课程（涵盖物理、化学与生物）提供结构化、精要的公式与定理汇编，帮助你在复习中巩固知识，在解题时快速查阅、从容应对。

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1. Mathematical Toolkit | 数学工具包

Solid algebraic and trigonometric skills underpin quantitative reasoning in all three sciences.

扎实的代数与三角功底是所有科学定量推理的基础。

The roots of the quadratic equation ax² + bx + c = 0 are given by:

x = [–b ± √(b² – 4ac)] / (2a)

二次方程 ax² + bx + c = 0 的求根公式为：

x = [–b ± √(b² – 4ac)] / (2a)

Pythagoras’ theorem for a right‑angled triangle links side lengths:

a² + b² = c²

勾股定理将直角三角形边长关联起来：

a² + b² = c²

Key trigonometric identities include sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ.

关键三角恒等式有 sin²θ + cos²θ = 1 以及 tanθ = sinθ / cosθ。

Laws of logarithms (any base) simplify products and powers:

logₐ(xy) = logₐx + logₐy, logₐ(xⁿ) = n logₐx

对数的运算性质（任意底数）可简化乘积与幂：

logₐ(xy) = logₐx + logₐy, logₐ(xⁿ) = n logₐx

Exponential rules are equally important: aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.

指数法则同样重要：aᵐ × aⁿ = aᵐ⁺ⁿ 以及 (aᵐ)ⁿ = aᵐⁿ。

Radian measure and arc length are linked by s = rθ, with π rad equivalent to 180°.

弧度制与弧长满足 s = rθ，π 弧度等于 180°。

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2. Kinematics Equations | 运动学方程

Uniformly accelerated motion in one dimension is described by the SUVAT set.

一维匀加速运动由下列SUVAT方程组描述。

First equation (no displacement):

v = u + at

第一方程（不含位移）：

v = u + at

Second equation (no final velocity):

s = ut + ½ at²

第二方程（不含末速度）：

s = ut + ½ at²

Third equation (no time):

v² = u² + 2as

第三方程（不含时间）：

v² = u² + 2as

Fourth equation (no acceleration):

s = ½ (u + v) t

第四方程（不含加速度）：

s = ½ (u + v) t

For free fall near Earth’s surface, replace a with g ≈ 9.81 m s⁻².

地表附近的自由落体运动，将 a 替换为 g ≈ 9.81 m s⁻² 即可。

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3. Newton’s Laws & Momentum | 牛顿定律与动量

Newton’s three laws govern force, motion and interaction.

牛顿三定律支配着力、运动与相互作用。

The second law defines net force:

F = ma

第二定律定义了合外力：

F = ma

Weight near a planet’s surface is W = mg.

行星表面附近的物体重力为 W = mg。

Linear momentum is the product of mass and velocity:

p = mv

线动量是质量与速度的乘积：

p = mv

Impulse equals change in momentum: FΔt = Δp = mv – mu.

冲量等于动量的变化：FΔt = Δp = mv – mu。

In a closed system, total momentum is conserved: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.

封闭系统的总动量守恒：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。

The frictional force between surfaces is often proportional to the normal reaction: F_f = μR.

两表面间的摩擦力通常与法向反作用力成正比：F_f = μR。

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4. Work, Energy & Power | 功、能与功率

Work done by a constant force at an angle θ to displacement:

W = Fd cosθ

恒力作功（力与位移夹角为θ）：

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