📚 Year 12 CAIE Physical Education: Formulas & Theorems Quick Reference Handbook | Year 12 CAIE 体育：公式定理速查手册
This quick reference handbook compiles essential formulas and theorems for Year 12 CAIE Physical Education. Covering biomechanics, exercise physiology, and sports psychology, it provides a bilingual resource to support revision and application in exam scenarios.
本速查手册汇集了 Year 12 CAIE 体育的核心公式与定理，涵盖生物力学、运动生理学和运动心理学等模块，以双语形式帮助复习和应对考试中的实际应用。
1. Linear Kinematics | 直线运动学
Linear kinematics describes motion along a straight line without reference to its cause. The SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), constant acceleration (a), and time (t).
直线运动学研究沿直线的运动而不涉及运动的起因。SUVAT 方程建立了位移 (s)、初速度 (u)、末速度 (v)、恒加速度 (a) 和时间 (t) 之间的关系。
v = u + a t
Final velocity is determined by the initial velocity plus the product of acceleration and time. This is used when distance is not needed.
末速度等于初速度加上加速度与时间的乘积，适用于不计位移的情景。
s = u t + ½ a t²
Displacement is calculated from initial velocity, time, and half the acceleration multiplied by time squared. It assumes constant acceleration.
位移由初速度、时间以及加速度与时间平方乘积的一半求得，前提是加速度恒定。
v² = u² + 2 a s
This equation links velocities and displacement without requiring time, useful for analysing motion over a known distance.
该式避开时间，直接勾连初速、末速、加速度与位移，常用于分析已知距离的运动。
average velocity = (u + v) / 2
When acceleration is uniform, the average velocity equals the arithmetic mean of initial and final velocities.
匀加速运动中，平均速度等于初速度与末速度的算术平均值。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s three laws form the foundation of linear kinetics, explaining how forces affect motion. They are essential for analysing sprint starts, collisions, and projectile motion in sport.
牛顿三大定律构成线性动力学的基础，解释了力如何影响运动，是分析短跑起跑、碰撞和抛射体运动的关键。
First Law (Inertia): F = 0 → v constant
A body remains at rest or in uniform motion unless acted upon by a net external force. This explains why a football stays still until kicked.
物体保持静止或匀速直线运动状态，除非有净外力作用于其上。这解释了足球为何在踢动前保持静止。
Second Law (Acceleration): F = m a
The net force on an object equals its mass times the acceleration it experiences. This explains how a heavier shot put requires more force to achieve the same acceleration.
物体的净力等于其质量与加速度的乘积，诠释了推铅球时更大质量需要更大的力才能获得同样的加速度。
Third Law (Reaction): F₁₂ = -F₂₁
For every action force, there is an equal and opposite reaction force. This principle underlies ground reaction force in running.
每一作用力都存在大小相等、方向相反的反作用力，跑步中的地面反作用力即基于此原理。
3. Momentum and Impulse | 动量与冲量
Momentum is the product of mass and velocity, while impulse is the change in momentum. These concepts are crucial for understanding collisions and protective equipment in sport.
动量是质量与速度的乘积，冲量则是动量的变化量，这两者对于理解运动中的碰撞与护具设计至关重要。
momentum p = m v
Momentum is a vector quantity measured in kg·m·s⁻¹. An object’s momentum increases with mass and velocity.
动量是矢量，单位为 kg·m·s⁻¹，物体的质量越大、速度越快，动量越大。
impulse J = F Δt = Δp = m v – m u
Impulse equals the average force multiplied by the time interval and equals the change in momentum. Increasing the time of contact, as in a crumple zone or boxing glove, reduces the peak force.
冲量等于平均力与时间间隔的乘积，也等于动量的变化量。增加接触时间（如汽车的缓冲区和拳击手套）可以降低峰值力。
conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
In a closed system with no external forces, total momentum before a collision equals total momentum after. Used to analyse tackles and impacts.
在无外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量，用于分析擒抱和撞击等动作。
4. Work, Energy, and Power | 功、能与功率
Mechanical work, energy, and power quantify the effort and efficiency of human movement. Understanding these helps optimise performance and reduce fatigue.
机械功、能量和功率量化了人体运动的用力程度与效率，理解它们有助于优化运动表现并减少疲劳。
work done W = F d cos θ
Work is the product of force and displacement in the direction of the force. Only the component of force parallel to the movement does work.
功是力与沿力方向位移的乘积，只有与位移平行的分力才做功。
kinetic energy KE = ½ m v²
Kinetic energy depends on mass and the square of velocity. Sprinters generate enormous kinetic energy, which then must be absorbed when stopping.
动能取决于质量与速度的平方。短跑运动员产生巨大的动能，停止时则需被吸收。
gravitational potential energy GPE = m g h
Potential energy stored due to height above a reference level. In high jump or pole vault, an athlete converts kinetic energy into gravitational potential energy.
重力势能因高于参考面而储存，跳高或撑竿跳中运动员将动能转化为重力势能。
power P = W / t = F v
Power is the rate of doing work, measured in watts. Power output is critical in explosive activities like weightlifting. P = F v relates power to force and instantaneous velocity.
功率是做功的速率，单位为瓦特，爆发性活动如举重依赖高功率输出，P = F v 将功率与力及瞬时速度联系起来。
5. Levers and Torque | 杠杆与力矩
Levers magnify force or speed in the human body. Torque quantifies the turning effect of a force around an axis, fundamental for joint biomechanics.
杠杆可放大人体中的力或速度，力矩则量化了力对某一轴的转动效应，是关节生物力学的基础。
torque τ = F × d (d is moment arm)
Torque is the product of force and the perpendicular distance from the axis of rotation to the line of action of the force. A larger moment arm increases torque for the same force.
力矩是力与转动轴到力作用线的垂直距离（力臂）的乘积。力臂越长，相同力产生的力矩越大。
principle of moments: Σ clockwise τ = Σ anticlockwise τ
For a body in rotational equilibrium, the sum of clockwise torques equals the sum of anticlockwise torques. This is the basis of balance and static holds.
处于旋转平衡的物体，顺时针力矩之和等于逆时针力矩之和，这是平衡和静态支撑动作的基础。
mechanical advantage MA = effort arm / resistance arm
Mechanical advantage indicates whether a lever amplifies force (MA > 1) or speed (MA < 1). The biceps curl is a third-class lever with MA < 1, favouring range and speed.
机械利益表明杠杆是增力（MA > 1）还是增速（MA < 1）。肱二头肌弯举属于第三类杠杆，MA < 1，偏向幅度与速度。
6. Angular Motion | 角运动
Angular kinematics and kinetics describe rotation, which dominates nearly all human movement. Key variables include angular displacement, angular velocity, and angular acceleration.
角运动学和角动力学描述旋转运动，几乎所有人体运动都涉及旋转，关键变量包括角位移、角速度和角加速度。
angular velocity ω = Δθ / Δt
Angular velocity is the rate of change of angular displacement, measured in rad/s. A gymnast spinning faster has a larger ω.
角速度是角位移的变化率，单位为弧度/秒，体操运动员旋转越快 ω 越大。
angular acceleration α = Δω / Δt
Angular acceleration describes how quickly angular velocity changes. Explosive turning actions require high α.
角加速度描述角速度变化的快慢，爆发性转身动作需要较大的 α。
centripetal acceleration a꜀ = v² / r = r ω²
An object in circular motion experiences centripetal acceleration directed towards the centre. This requires a centripetal force, F꜀ = m v² / r.
作圆周运动的物体具有指向圆心的向心加速度，由此需要向心力 F꜀ = m v² / r。
moment of inertia I = Σ m r²
Moment of inertia is the resistance to angular acceleration, depending on mass and its distribution about the axis. A tucked figure skater reduces I to spin faster.
转动惯量抵抗角加速度，取决于质量及其到转轴的分布；花样滑冰者收拢身体可减小 I 以加速旋转。
angular momentum L = I ω (conserved if τₑₓₙ = 0)
Angular momentum is conserved when no external torque acts. This explains the increase in ω when I decreases during a dive pike.
当无外力矩时角动量守恒，这解释了跳水屈体时 I 减小会导致 ω 增大的现象。
7. Fluid Mechanics – Drag and Lift | 流体力学——阻力与升力
Air and water resistance significantly affect sport performance. Drag opposes motion, while lift acts perpendicular to flow, both depending on velocity, fluid density, and shape.
空气和水的阻力对运动表现影响显著。阻力阻碍运动，升力则垂直于流向，两者均与速度、流体密度和物体形状有关。
drag force Fᑧ = ½ Cᑧ ρ A v²
Drag force depends on the drag coefficient (Cᑧ), fluid density (ρ), frontal cross-sectional area (A), and the square of velocity. Cyclists reduce A by crouching to minimise drag.
阻力大小取决于阻力系数 (Cᑧ)、流体密度 (ρ)、迎风横截面积 (A) 及速度的平方。自行车运动员通过俯身减小 A 来降低阻力。
lift force Fₗ = ½ Cₗ ρ A v²
Lift force arises from pressure differences, governed by the lift coefficient (Cₗ). Aerofoils on racing cars and discus throws use lift to extend flight.
升力源于压力差，由升力系数 (Cₗ) 主导，赛车翼板和铁饼投掷均利用升力延长飞行。
Magnus effect: spinning ball curves
A spinning ball creates a pressure differential, causing it to deviate from its straight path. This is evident in tennis topspin, football free kicks, and golf ball dimples.
旋转的球体产生压力差，使其偏离直线轨迹，网球上旋、足球任意球和高尔夫球凹坑均体现马格努斯效应。
8. Cardiovascular System Formulas | 心血管系统公式
Cardiovascular efficiency is measured through heart rate, stroke volume, and cardiac output. These formulas quantify the oxygen delivery capacity of an athlete.
心血管效率通过心率、每搏输出量和心输出量来衡量，这些公式量化了运动员的氧运输能力。
cardiac output Q = HR × SV
Cardiac output (Q or CO) is the volume of blood pumped by the heart per minute, calculated as heart rate times stroke volume. Endurance training increases both SV and maximal Q.
心输出量 Q 是心脏每分钟泵出的血量，等于心率乘以每搏输出量。耐力训练可同时提高每搏输出量和最大心输出量。
stroke volume SV = EDV – ESV
Stroke volume is the difference between end-diastolic volume and end-systolic volume, representing blood ejected per beat. Enhanced venous return raises EDV, boosting SV via the Frank-Starling mechanism.
每搏输出量等于舒张末期容积减去收缩末期容积，代表每次心跳射血量。增加的静脉回流量提高舒张末期容积，通过弗兰克-斯塔林机制增大每搏输出量。
heart rate max estimate: HRmax = 220 – age
A simple population-based prediction for maximal heart rate. Individual variation is notable; lab testing yields more accurate values for training zones.
基于群体的最大心率简单预测公式，个体差异显著，实验室测试可提供更精确的训练区间数据。
9. Respiratory System Formulas | 呼吸系统公式
Pulmonary ventilation and alveolar ventilation equations describe how effectively air is moved into the lungs and reaches the gas-exchange surfaces.
肺通气量和肺泡通气量方程描述了空气进入肺部并到达气体交换表面的有效程度。
minute ventilation V̇E = VT × f
Minute ventilation (V̇E) is the total volume of air entering the lungs per minute, calculated as tidal volume (VT) multiplied by breathing frequency (f). At high intensities, both VT and f rise markedly.
每分通气量 V̇E 是每分钟进入肺部的气体总量，等于潮气量 (VT) 乘以呼吸频率 (f)。在高强度运动中，潮气量和呼吸频率均显著上升。
alveolar ventilation V̇A = (VT – VD) × f
Alveolar ventilation accounts for dead space (VD), the portion of air that does not participate in gas exchange. Only the air reaching the alveoli contributes to oxygen uptake.
肺泡通气量扣除了解剖无效腔 (VD)，后者是不参与气体交换的通气部分，只有抵达肺泡的空气才参与摄氧。
VO₂ max relative = absolute VO₂ max / body mass
Maximal oxygen uptake is often expressed in ml·kg⁻¹·min⁻¹ to enable fair comparison across athletes of different body sizes. Relative VO₂ max is a strong predictor of endurance performance.
最大摄氧量常以 ml·kg⁻¹·min⁻¹ 表示，便于不同体型的运动员公平比较，相对 VO₂ max 是耐力表现的强预测指标。
10. Training Heart Rate Formulae | 训练心率公式
Heart rate-based training zones use the Karvonen formula to prescribe exercise intensity. This method accounts for resting heart rate, improving individualisation.
基于心率的训练区间使用卡沃宁公式确定运动强度，该方法纳入安静心率，大幅提升了个体化程度。
heart rate reserve HRR = HRmax – HRrest
Heart rate reserve is the difference between maximum and resting heart rate. It reflects the physiological range available for exercise.
心率储备是最大心率与安静心率之差，反映了可调动用于运动的生理范围。
target heart rate THR = (HRR × % intensity) + HRrest
Your target heart rate for a given intensity zone is found by multiplying HRR by the desired percentage and adding resting heart rate. For a 17-year-old with HRrest of 60 bpm aiming at 70% intensity: HRmax = 203, HRR = 143, THR = (143 × 0.7) + 60 = 160 bpm.
某强度区间的靶心率等于 HRR 乘以目标百分比再加上安静心率。例如，一名 17 岁、安静心率 60 bpm 的运动员以 70% 强度训练：HRmax = 203, HRR = 143, THR = (143 × 0.7) + 60 = 160 bpm。
aerobic training zone ≈ 60–80% HRR
Sustained aerobic adaptations are typically achieved by exercising within 60–80% of heart rate reserve, although zone boundaries vary by fitness level.
持续有氧适应通常通过 60–80% 心率储备的运动实现，但区间边界因体能水平而异。
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