📚 A-Level CIE Physical Education: Formula & Theorem Quick Reference Handbook | A-Level CIE 体育：公式定理速查手册
This handbook provides a concise summary of key formulas and principles required for the CIE A-Level Physical Education (9396) syllabus, covering biomechanics, exercise physiology, and skill acquisition. Mastering these equations is essential for tackling numerical problems and applying theoretical concepts in exam scenarios.
本手册简要总结了 CIE A-Level 体育 (9396) 大纲要求的关键公式和原理，涵盖生物力学、运动生理学和技能习得等领域。掌握这些方程对于应对考试中的计算题和应用理论概念至关重要。
1. Units and Conversions | 单位与换算
Before applying any formula, ensure all quantities use SI units: displacement (m), time (s), velocity (m/s), acceleration (m/s²), mass (kg), force (N), work and energy (J), power (W), and angle (rad). Use conversions: 1 mph = 0.447 m/s, 1 km/h = 0.278 m/s, 1 kg = 9.81 N (weight on Earth).
运用任何公式前，确保所有量都使用国际单位制：位移（米/m）、时间（秒/s）、速度（米每秒/m/s）、加速度（米每二次方秒/m/s²）、质量（千克/kg）、力（牛顿/N）、功和能量（焦耳/J）、功率（瓦特/W）以及角度（弧度/rad）。常用换算：1 英里/时 = 0.447 m/s，1 公里/时 = 0.278 m/s，1 kg 物体的重量约为 9.81 N。
2. Linear Kinematics | 线性运动学
Uniform acceleration equations apply when acceleration is constant. The four SUVAT equations are fundamental:
匀加速运动方程在加速度恒定时适用。以下四个 SUVAT 方程是基础：
v = u + at
Final velocity = initial velocity + acceleration × time.
末速度 = 初速度 + 加速度 × 时间。
s = ut + ½at²
Displacement = initial velocity × time + half × acceleration × time squared.
位移 = 初速度 × 时间 + ½ × 加速度 × 时间²。
v² = u² + 2as
Relates final velocity, initial velocity, acceleration, and displacement.
关联末速度、初速度、加速度和位移。
s = ½(u + v)t
Displacement equals average velocity multiplied by time.
位移 = 平均速度 × 时间。
Velocity is the rate of change of displacement: v = Δs / Δt. Acceleration: a = Δv / Δt. Vector direction must be indicated by sign.
速度是位移变化率：v = Δs / Δt。加速度：a = Δv / Δt。矢量方向必须用正负号表示。
3. Newton’s Laws and Forces | 牛顿定律与力
Newton’s First Law: An object remains at rest or uniform motion unless a net external force acts. Inertia is proportional to mass.
牛顿第一定律：除非受到净外力作用，物体保持静止或匀速直线运动。惯性大小与质量成正比。
Newton’s Second Law: ΣF = ma. Net force equals mass times acceleration. This is used to analyse sprint starts, tackles, and any change in motion.
牛顿第二定律：ΣF = ma。净外力等于质量乘以加速度。用于分析起跑、冲撞等任何运动变化。
Newton’s Third Law: For every action force there is an equal and opposite reaction force. Relevant to ground reaction force and swimming propulsion.
牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力。适用于地面反作用力和游泳推进。
Weight: W = mg, where g = 9.81 m/s². Friction: F = μR, where μ is the coefficient of friction and R is the normal reaction force. On an incline, components of weight are mg sinθ along the slope and mg cosθ perpendicular.
重力：W = mg，g = 9.81 m/s²。摩擦力：F = μR，μ 是摩擦系数，R 是法向反作用力。在斜面上，重力的分量是沿斜面 mg sinθ，垂直于斜面 mg cosθ。
4. Momentum and Impulse | 动量与冲量
Linear momentum: p = mv. It is a vector quantity. Impulse = force × time = FΔt = Δp = mv – mu. A large impulse is produced by a large force over a short time (striking a ball) or a moderate force over a longer time (following through).
线性动量：p = mv，为矢量。冲量 = 力 × 时间 = FΔt = Δp = mv – mu。在极短时间内施加大力（如踢球）可产生大冲量，或通过延长力作用时间（如随挥）来增大冲量。
Conservation of momentum: In a closed system, total momentum before collision equals total momentum after collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This principle explains recoil in shooting and rebound in racket sports.
动量守恒：在封闭系统中，碰撞前总动量等于碰撞后总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。这一原理解释了射击时的后坐力和球拍运动的反弹。
5. Work, Energy and Power | 功、能量与功率
Work done: W = Fd cosθ, where θ is the angle between force and displacement. If force is parallel to displacement, W = Fd. Kinetic energy: KE = ½mv². Gravitational potential energy: PE = mgh.
功：W = Fd cosθ，θ 是力与位移的夹角。若力与位移平行，W = Fd。动能：KE = ½mv²。重力势能：PE = mgh。
Conservation of mechanical energy: KE + PE = constant if only conservative forces act. Used to calculate take-off velocity from jump height.
机械能守恒：如果只有保守力做功，动能与势能之和保持不变。可用于通过跳跃高度推算起跳速度。
Power: P = W / t = work done per unit time. Also P = Fv for an object moving at constant speed against a resistive force. Measured in watts (W).
功率：P = W / t = 单位时间做功。对于以恒定速度对抗阻力运动的物体，P = Fv。单位是瓦特（W）。
6. Angular Motion | 角运动
Angular displacement (θ) measured in radians. Angular velocity: ω = Δθ / Δt (rad/s). Angular acceleration: α = Δω / Δt (rad/s²).
角位移（θ）以弧度计。角速度：ω = Δθ / Δt（弧度/秒）。角加速度：α = Δω / Δt（弧度/秒²）。
Relationship between linear and angular motion for a rotating body: v = rω, a_tangential = rα, where r is the radius of rotation. Centripetal acceleration: a_c = v²/r = rω². Centripetal force: F_c = mv²/r = mrω².
旋转物体线性运动与角运动的关系：v = rω，切向加速度 a = rα，r 为旋转半径。向心加速度：a_c = v²/r = rω²。向心力：F_c = mv²/r = mrω²。
Moment of inertia (I) depends on mass and its distribution from the axis. Angular momentum: L = Iω. Torque: T = Iα. Conservation of angular momentum (I₁ω₁ = I₂ω₂) explains how a diver or skater changes spin rate by altering body shape.
转动惯量（I）取决于质量及其距轴分布。角动量：L = Iω。力矩：T = Iα。角动量守恒（I₁ω₁ = I₂ω₂）解释了跳水员或溜冰者通过改变身体姿态来改变旋转速度的原理。
7. Projectile Motion | 抛体运动
For a projectile launched from ground level with speed u at angle θ to horizontal, resolving into components: horizontal velocity u_x = u cosθ, vertical velocity u_y = u sinθ. Horizontal motion is constant, vertical motion under gravity a = –g.
对于从地面以速度 u、角度 θ 发射的抛体，分解速度：水平速度 u_x = u cosθ，垂直速度 u_y = u sinθ。水平运动匀速，垂直运动在重力作用下 a = –g。
Time of flight: T = 2u sinθ / g
飞行时间：T = 2u sinθ / g
Maximum height: H = u² sin²θ / (2g)
最大高度：H = u² sin²θ / (2g)
Range: R = u² sin2θ / g
水平射程：R = u² sin2θ / g
The optimal angle for maximum range is 45° in the absence of air resistance. In sports, release height above landing level modifies optimum angle (e.g., shot put typically below 40°).
在没有空气阻力的情况下，最大射程的理想角度为 45°。在体育运动中，出手点高于落点会改变最佳角度（如铅球通常低于 40°）。
8. Levers and Torque | 杠杆与力矩
A lever consists of an effort force, a resistance load, and a fulcrum. Torque (moment of force): τ = F × d, where d is the perpendicular distance from the fulcrum to the line of force action. Principle of moments: for equilibrium, sum of clockwise torques = sum of anticlockwise torques.
杠杆由动力、阻力和支点组成。力矩（力偶矩）：τ = F × d，d 是支点到力作用线的垂直距离。力矩原理：平衡时，顺时针力矩之和等于逆时针力矩之和。
Mechanical advantage (MA): MA = effort arm / resistance arm. MA > 1 means large load can be moved by small effort (e.g., second-class lever in ankle). MA < 1 favours speed and range of motion (e.g., third-class lever in biceps curl). Velocity ratio = effort arm length / resistance arm length.
机械优势：MA = 动力臂 / 阻力臂。MA > 1 表示可用较小动力移动较大负荷（如脚踝处的第二类杠杆）。MA < 1 有利于速度和活动范围（如肱二头肌弯举的第三类杠杆）。速度比 = 动力臂长 / 阻力臂长。
In human body, most muscles operate as third-class levers, prioritising speed over force.
人体内大多数肌肉作为第三类杠杆运作，以速度优先于力量。
9. Fluid Mechanics and Drag | 流体力学与阻力
Drag force acting on a body moving through air or water: 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 relative to fluid. Reducing A or C_d (streamlining) lowers drag.
物体在空气或水中运动时所受阻力：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 为相对于流体的速度。减小 A 或 C_d（流线型）可降低阻力。
Lift force explained by Bernoulli’s principle: faster fluid flow over an object creates lower pressure, generating an upward lift (airfoil effect). Magnus effect: a spinning ball experiences a lateral force due to pressure differences (topspin dips, backspin floats).
升力由伯努利原理解释：物体上方流体流速较快，压力较低，产生向上的升力（翼型效应）。马格努斯效应：旋转的球因压力差而受到侧向力（上旋球下坠，下旋球飘浮）。
Surface drag and form drag are key considerations in swimming, cycling, and ski jumping.
表面阻力和形状阻力是游泳、自行车和跳台滑雪中的关键因素。
10. Exercise Physiology Formulas | 运动生理学公式
Maximum heart rate estimation: HRmax = 220 – age. (Alternate: 208 – 0.7 × age). This is used to set training zones.
最大心率估算：HRmax = 220 – 年龄（替代公式：208 – 0.7 × 年龄）。用于设定训练区间。
Karvonen formula for target heart rate: Target HR = HRrest + (HRmax – HRrest) × exercise intensity (%). Ensures training at desired percentage of heart rate reserve.
卡氏靶心率公式：靶心率 = 安静心率 + (最大心率 – 安静心率) × 运动强度(%)。确保以心搏储备的所需百分比进行训练。
Cardiac output: CO = SV × HR, where SV is stroke volume (ml/beat). At rest, CO ≈ 5 L/min; can exceed 20–40 L/min in elite endurance athletes.
心输出量：CO = 每搏输出量 × 心率，SV 为每搏输出量（毫升/搏）。安静时 CO 约 5 升/分；精英耐力运动员可超过 20–40 升/分。
VO₂max: expressed in ml/kg/min. Often estimated via submaximal tests (e.g., Astrand cycle test). 1 MET = 3.5 ml/kg/min. Energy expenditure: kcal = METs × body mass (kg) × time (hours).
最大摄氧量 VO₂max：单位为 ml/kg/min。常用亚极量测试估算（如 Astrand 自行车测试）。1 MET = 3.5 ml/kg/min。能量消耗：千卡 = METs × 体重(kg) × 时间(小时)。
Body Mass Index: BMI = mass (kg) / height² (m²). Used to classify underweight, normal, overweight, obese, though it does not distinguish fat from muscle.
身体质量指数：BMI = 体重(kg) / 身高²(m²)。用于分类体重不足、正常、超重和肥胖，但不区分脂肪和肌肉。
11. Skill Acquisition Laws | 技能习得定律
Fitts’ Law describes the speed–accuracy trade-off in aiming tasks. Movement time: MT = a + b log₂(2A / W), where A is movement amplitude, W is target width. Index of difficulty ID = log₂(2A/W). Larger ID requires longer movement time.
费茨定律描述了瞄准任务中速度与准确性的权衡。动作时间：MT = a + b log₂(2A / W)，A 为动作幅度，W 为目标宽度。难度指数 ID = log₂(2A/W)。ID 越大，所需动作时间越长。
Hick’s Law predicts reaction time increases with the number of stimulus–response choices: RT = a + b log₂(n), where n is the number of alternatives. This underlies the advantage of anticipation and ‘cueing’ in open skills.
希克定律预测反应时随刺激-反应选项数量的增加而增加：RT = a + b log₂(n)，n 为选项数。这是开放式技能中预判和“提示”优势的理论基础。
Schmidt’s schema theory and dynamical systems theory are not equation-based but are key in explaining motor programme development.
施密特的图式理论和动力系统理论并非以方程为基础，但在解释动作程序发展方面至关重要。
12. Efficiency and Miscellaneous | 效率及其他
Mechanical efficiency: Efficiency (%) = (mechanical work output / energy expenditure) × 100. Gross efficiency in cycling is typically 18–24%. The remaining energy is lost as heat.
机械效率：效率(%) = (机械功输出 / 能量消耗) × 100。自行车骑行总效率通常为 18–24%。其余能量以热量形式散失。
Coefficient of restitution (e) measures elasticity of collisions: e = relative speed after / relative speed before = (v₂ – v₁) / (u₁ – u₂). e = 1 for perfectly elastic, e = 0 for plastic. Used to assess ball and surface compliance.
恢复系数 (e) 衡量碰撞的弹性：e = 碰撞后相对速度 / 碰撞前相对速度 = (v₂ – v₁) / (u₁ – u₂)。e = 1 为完全弹性碰撞，e = 0 为完全非弹性碰撞。用于评估球与表面的顺应性。
Power–duration relationship: critical power CP represents the highest sustainable aerobic power, W’ is the finite anaerobic work capacity. Time to exhaustion: t = W’ / (P – CP)
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