📚 Year 12 CIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 12 CIE 体育公式定理速查手册
This quick reference guide compiles all essential formulas, equations and key theorems required for the Year 12 CIE Physical Education syllabus. It covers biomechanics, exercise physiology and sport psychology, presented in clear English–Chinese paired explanations to support revision and application.
本速查手册汇总了 Year 12 CIE 体育课程所需的全部核心公式、方程和关键定理，涵盖生物力学、运动生理学和运动心理学等内容，以清晰的中英对照形式呈现，便于复习和应用。
1. Linear Kinematic Equations | 匀变速直线运动方程
The first equation of uniform acceleration: v = u + at, where v = final velocity, u = initial velocity, a = acceleration, t = time.
第一个匀加速运动方程：v = u + at，其中 v = 末速度，u = 初速度，a = 加速度，t = 时间。
Displacement with uniform acceleration: s = ut + ½ at², where s = displacement.
匀加速运动的位移公式：s = ut + ½ at²，其中 s = 位移。
Velocity–displacement relation: v² = u² + 2as. Useful when time is unknown.
速度–位移关系：v² = u² + 2as。当时间未知时非常实用。
2. Newton’s Laws and Momentum | 牛顿定律与动量
Newton’s Second Law: F = ma, where force (N) equals mass (kg) times acceleration (m/s²). This explains how net force changes motion.
牛顿第二定律：F = ma，力（N）等于质量（kg）乘以加速度（m/s²），解释了合外力如何改变运动状态。
Momentum is defined as p = mv, where p = momentum (kg m/s). Momentum is a vector quantity.
动量定义为 p = mv，其中 p = 动量（kg·m/s）。动量是矢量。
Conservation of linear momentum: in a closed system, total momentum before collision equals total momentum after collision: Σp_initial = Σp_final.
线动量守恒：在封闭系统中，碰撞前总动量等于碰撞后总动量：Σp_初始 = Σp_最终。
3. Impulse and Momentum Relationship | 冲量与动量关系
Impulse is the product of force and the time for which it acts: J = FΔt. The impulse–momentum theorem states FΔt = Δp = mv – mu.
冲量是力与其作用时间的乘积：J = FΔt。冲量–动量定理指出 FΔt = Δp = mv – mu。
This relationship explains how extending impact time reduces peak force, crucial in sports like landing from a jump.
这一关系解释了为何延长作用时间可减小峰值力，在跳远落地等运动动作中至关重要。
4. Work, Energy and Power | 功、能与功率
Work done: W = Fd cos θ, where θ is the angle between force and displacement vectors. Measured in joules (J).
功：W = Fd cos θ，其中 θ 是力与位移的夹角，单位为焦耳（J）。
Kinetic energy: KE = ½ mv². Gravitational potential energy: PE = mgh, where h is height above a reference level.
动能：KE = ½ mv²。重力势能：PE = mgh，其中 h 是相对参考面的高度。
Power is the rate of doing work: P = W/t or P = Fv for constant velocity. Measured in watts (W).
功率是做功的快慢：P = W/t，或当速度恒定时 P = Fv，单位为瓦特（W）。
5. Projectile Motion | 抛体运动
Horizontal component of velocity is constant: v_x = u cos θ. Horizontal range: R = (u² sin 2θ)/g (launch and landing at same height).
水平分速度恒定：v_x = u cos θ。水平射程：R = (u² sin 2θ)/g（起抛点与落点等高时）。
Maximum height: H = (u² sin² θ) / (2g). Flight time: t = (2u sin θ)/g. The optimal angle for maximum range is 45° in a vacuum.
最大高度：H = (u² sin² θ) / (2g)。飞行时间：t = (2u sin θ)/g。真空中最大射程的最优角度为 45°。
6. Angular Motion | 角运动
Angular velocity: ω = Δθ/Δt (rad/s). Angular acceleration: α = Δω/Δt (rad/s²).
角速度：ω = Δθ/Δt（rad/s）。角加速度：α = Δω/Δt（rad/s²）。
Linear–angular conversion: v = rω, tangential acceleration a = rα. Centripetal acceleration: a_c = rω² = v²/r.
线量与角量转换：v = rω，切向加速度 a = rα。向心加速度：a_c = rω² = v²/r。
Moment of inertia for a point mass: I = mr². Rotational kinetic energy: KE_rot = ½ Iω².
点质量的转动惯量：I = mr²。转动动能：KE_rot = ½ Iω²。
Angular momentum: L = Iω. Conservation of angular momentum explains spins in diving and ice skating: I₁ω₁ = I₂ω₂.
角动量：L = Iω。角动量守恒解释了跳水和花样滑冰中的旋转动作：I₁ω₁ = I₂ω₂。
7. Levers and Torque | 杠杆与力矩
Torque (moment of force): τ = F × d, where d is the perpendicular distance from the pivot. It causes angular acceleration: τ = Iα.
力矩（力偶矩）：τ = F × d，d 是力到支点的垂直距离。它产生角加速度：τ = Iα。
Mechanical advantage (MA) of a lever = effort arm / resistance arm. Three classes of levers are classified by relative positions of pivot, effort and load.
杠杆的机械利益（MA）= 力臂 / 阻力臂。三类杠杆根据支点、动力和阻力的相对位置划分。
8. Fluid Mechanics and Drag | 流体力学与阻力
Drag force acting on an object moving through a fluid: F_D = ½ C_D ρ A v², where C_D is drag coefficient, ρ is fluid density, A is cross-sectional area, v is velocity.
物体在流体中运动所受的阻力：F_D = ½ C_D ρ A v²，其中 C_D 为阻力系数，ρ 为流体密度，A 为横截面积，v 为速度。
Lift force generated by an airfoil or spinning ball (Magnus effect): F_L = ½ C_L ρ A v², where C_L is the lift coefficient affected by spin.
翼型或旋转球体产生的升力（马格努斯效应）：F_L = ½ C_L ρ A v²，其中 C_L 为升力系数，受旋转影响。
9. Cardiovascular Equations | 心血管系统公式
Cardiac output: Q = HR × SV, where Q is cardiac output (L/min), HR is heart rate (bpm), SV is stroke volume (mL/beat).
心输出量：Q = HR × SV，其中 Q 为心输出量（L/min），HR 为心率（次/分），SV 为每搏输出量（mL/次）。
Estimated maximum heart rate: HR_max ≈ 220 – age. Used to prescribe training intensities.
估算最大心率：HR_max ≈ 220 – 年龄。常用于制定训练强度。
Oxygen consumption (Fick principle): VO₂ = Q × (a-v O₂ diff), where a-v O₂ diff is the arteriovenous oxygen difference.
摄氧量（菲克原理）：VO₂ = Q × (a-v O₂ diff)，a-v O₂ diff 为动静脉氧差。
10. Respiratory Equations | 呼吸系统公式
Minute ventilation: VE = TV × f, where VE is minute ventilation (L/min), TV is tidal volume (L/breath), f is breathing frequency (breaths/min).
每分通气量：VE = TV × f，VE 为每分通气量（L/min），TV 为潮气量（L/次），f 为呼吸频率（次/分）。
Alveolar ventilation takes dead space into account: VA = (TV – dead space) × f. This represents fresh air reaching the alveoli.
肺泡通气量考虑了无效腔：VA = (TV – 无效腔量) × f，代表实际进入肺泡的新鲜空气量。
11. Body Composition and Efficiency | 身体成分与效率
Body Mass Index: BMI = weight (kg) / (height (m))². Simple tool to categorise underweight, normal, overweight and obese.
身体质量指数：BMI = 体重 (kg) / (身高 (m))²，是一种划分偏瘦、正常、超重和肥胖的简单工具。
Mechanical efficiency: Efficiency (%) = (Work output / Energy expended) × 100. Typically around 20–25% for cycling and other gross movements.
机械效率：效率 (%) = （输出功 / 能量消耗）× 100。骑自行车等整体运动效率通常在 20–25% 左右。
12. Arousal and Performance Models | 唤醒与表现模型
Drive theory (Spence): Performance = habit strength × drive, written as P = H × D. It predicts a linear relationship between arousal and performance for well-learned skills.
驱力理论（斯彭斯）：表现 = 习惯强度 × 驱力，即 P = H × D。该理论预测对于熟练技能，唤醒水平与表现呈线性关系。
Inverted‑U hypothesis: performance improves with increased arousal up to an optimal point, then deteriorates. The optimum varies with skill complexity and individual differences. There is no single equation, but the relationship can be described as an inverted‑U curve.
倒U假说：表现随唤醒增加而提高，直至最佳点，随后下降。最佳点因技能复杂程度和个体差异而异。该关系没有单一公式，可用倒U形曲线描述。
Catastrophe model: when cognitive anxiety is high, performance can drop dramatically after passing the optimum, not gradually. This is represented graphically but not by a mathematical theorem.
突变模型：当认知焦虑较高时，一旦超过最佳唤醒水平，表现会急剧下降而非渐进下降。该模型用图形表示，并非数学定理。
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