📚 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。
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°。
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。
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: Ff = μR.
两表面间的摩擦力通常与法向反作用力成正比：Ff = μR。
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）：
