📚 Year 13 CIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 13 CIE 体育：公式定理速查手册
This handbook brings together the essential formulas, laws, and theorems that underpin the Year 13 CIE Physical Education syllabus. From biomechanical equations that explain human movement to physiological indices that measure fitness, each entry is paired with clear English and Chinese explanations to support revision and exam preparation.
本手册汇集了支撑 Year 13 CIE 体育课程的核心公式、定律与定理。从解释人体运动的生物力学方程，到衡量体能的生理指标，每个条目均配有清晰的中英文解释，助力复习与备考。
1. Linear Kinematic Equations | 线性运动学方程
Linear kinematics describes motion along a straight line without reference to its causes. The four SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) when acceleration is constant.
线性运动学描述直线运动而不涉及成因。四个 SUVAT 方程在加速度恒定时将位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)联系起来。
v = u + at
Link between velocity, acceleration and time. Useful for finding final speed after a given period of constant acceleration.
速度、加速度与时间的关系。用于求恒定加速度下经过一定时间的末速度。
s = ut + ½ at²
Displacement expressed in terms of initial velocity, time and acceleration. Commonly applied when time is known but final velocity is not.
用初速度、时间和加速度表示位移。通常在已知时间而未知末速度时使用。
v² = u² + 2as
Relates velocity and displacement without involving time. Ideal for braking distances or sprint finishes.
关联速度与位移，不含时间变量。适用于刹车距离或冲刺终点分析。
average velocity = (u + v) / 2
Simple mean of initial and final velocities when acceleration is uniform.
当加速度均匀时，初末速度的简单平均值。
2. Newton’s Laws of Motion & Momentum | 牛顿运动定律与动量
Newton’s three laws form the foundation of linear kinetics. They explain how forces change motion and are vital for analysing sporting actions such as jumping, throwing and tackling.
牛顿三定律是线性动力学的基础。它们解释力如何改变运动，对于分析跳跃、投掷和抢断等运动动作至关重要。
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by an external force.
第一定律（惯性）： 物体保持静止或匀速直线运动，除非受外力作用。
Second Law (Acceleration): F = ma — the acceleration of an object is proportional to the net force and inversely proportional to its mass.
第二定律（加速度）： F = ma —— 物体加速度与合外力成正比，与质量成反比。
Third Law (Reaction): For every action there is an equal and opposite reaction.
第三定律（反作用）： 作用力与反作用力大小相等、方向相反。
Momentum, p, is the product of mass and velocity. The principle of conservation of momentum states that in a closed system the total momentum before a collision equals the total momentum after, provided no external forces act.
动量 p 是质量与速度的乘积。动量守恒定律指出，在无外力作用的封闭系统中，碰撞前后的总动量保持不变。
p = m × v
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
3. Impulse and Momentum Theorem | 冲量与动量定理
Impulse is the product of force and the time for which it acts. It equals the change in momentum of an object. This theorem explains how extending the time of force application — such as following through in a golf swing — can increase the impulse and thus the resulting change in momentum.
冲量是力与其作用时间的乘积，等于物体动量的变化。该定理解释了延长力的作用时间（如高尔夫挥杆中的随挥动作）如何增大冲量，进而加大动量的变化。
Impulse = F × Δt = Δp = m(v − u)
In sports, increasing the contact time between the foot and a ball can generate greater impulse, leading to a faster ball speed. Conversely, reducing contact time with rigid surfaces can amplify the force experienced during landing, increasing injury risk.
在运动中，增大脚与球的接触时间可产生更大的冲量，获得更快的球速。反之，减小与硬地面的接触时间会放大落地时所受的力，增加受伤风险。
4. Centre of Mass and Stability | 重心与稳度
The centre of mass (CoM) is the point where the mass of a body is concentrated. Stability is governed by the height of the CoM, the size of the base of support, and the line of gravity’s position relative to the base.
重心是物体质量集中的点。稳度取决于重心高度、支撑面大小以及重力线相对于支撑面的位置。
A body is more stable when the CoM is low, the base of support is wide, and the line of gravity falls centrally within the base. For an object to topple, the line of gravity must move outside the base of support, creating a turning moment.
当重心较低、支撑面较宽且重力线落在支撑面中心时，物体稳度更大。物体倾倒的条件是重力线移出支撑面，产生转动力矩。
Moment of a force = F × perpendicular distance from pivot
In static equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. This principle underpins safe lifting techniques and balance control in gymnastics.
在静态平衡中，顺时针力矩之和等于逆时针力矩之和。这一原理是安全举重技术和体操平衡控制的基础。
5. Angular Kinematics | 角运动学
Angular kinematics describes circular motion. Quantities include angular displacement (θ), angular velocity (ω), and angular acceleration (α). They are linked to linear variables through the radius of rotation.
角运动学描述圆周运动，涉及角位移(θ)、角速度(ω)和角加速度(α)。它们通过旋转半径与线量相关联。
ω = Δθ / Δt
Angular velocity is the rate of change of angular displacement, measured in rad/s.
角速度是角位移的变化率，单位为弧度/秒。
v = r ω
Linear velocity of a point on a rotating body is the product of the radius and angular velocity. This explains why the tip of a golf club or the end of a lever system moves faster.
旋转体上一点的线速度等于半径与角速度的乘积。这解释了为何高尔夫球杆杆头或杠杆末端运动更快。
tangential a = r α
centripetal a = v² / r = r ω²
Tangential acceleration changes the magnitude of linear speed; centripetal acceleration points toward the centre of rotation, maintaining circular motion.
切向加速度改变线速度的大小；向心加速度指向旋转中心，维持圆周运动。
6. Angular Dynamics and Moment of Inertia | 角动力学与转动惯量
Angular kinetics examines the torques that cause rotation. The moment of inertia (I) measures an object’s resistance to angular acceleration and depends on both mass and its distribution about the axis.
角动力学研究导致转动的力矩。转动惯量(I)衡量物体抵抗角加速度的能力，取决于质量和质量相对于轴的分布。
I = Σ m r²
This shows that moving mass further from the axis greatly increases moment of inertia. Athletes can manipulate their I to control spin speed — e.g., a diver tucking to reduce I and increase angular velocity.
此式表明，质量远离转轴会显著增大转动惯量。运动员可调节 I 来控制旋转速度——例如，跳水运动员通过团身减小 I，从而增大角速度。
torque τ = F × d (perpendicular distance)
τ = I α
Newton’s second law for rotation links net torque, moment of inertia and angular acceleration.
牛顿第二定律的转动形式将净力矩、转动惯量和角加速度联系起来。
Angular momentum L = I ω
In the absence of external torques, angular momentum is conserved: I₁ ω₁ = I₂ ω₂. This principle is used in figure skating spins and gymnastic somersaults.
在没有外力矩时，角动量守恒：I₁ ω₁ = I₂ ω₂。该原理应用于花样滑冰旋转和体操空翻。
7. Projectile Motion | 抛体运动
A projectile follows a parabolic path determined by its initial velocity, angle of release, and the acceleration due to gravity. Air resistance is often ignored in basic models but is important in real‑world sports.
抛体沿抛物线轨迹运动，由初速度、出手角度和重力加速度决定。基础模型通常忽略空气阻力，但在真实体育运动中空气阻力很重要。
vₓ = u cos θ , vᵧ = u sin θ
Horizontal component remains constant (ignoring drag); vertical component changes due to gravity.
水平分量保持不变（忽略阻力）；垂直分量受重力影响而变化。
Time of flight = 2 u sin θ / g
Maximum height = u² sin² θ / (2g)
Range = u² sin 2θ / g
The optimum release angle for maximum range is 45° in a vacuum, but lower angles are often used in sports like shot put or javelin to balance height and air resistance.
真空中达到最大射程的最佳出手角为45°，但在铅球或标枪等项目中常采用更低的角度以平衡高度和空气阻力。
8. Bernoulli’s Principle and Magnus Effect | 伯努利原理与马格努斯效应
Bernoulli’s principle states that for an ideal fluid, an increase in velocity occurs simultaneously with a decrease in pressure. This principle explains lift on aerofoils and the behaviour of spinning balls.
伯努利原理指出，对于理想流体，流速增加时压力会同时降低。这一原理可以解释机翼升力以及旋转球体的运动行为。
P + ½ ρ v² + ρ g h = constant
When air flows faster over one side of a spinning ball, a pressure difference is created. The Magnus effect is the curved flight path that results: a spinning ball is deflected towards the low‑pressure side. Topspin, backspin and sidespin all produce predictable deviations that are crucial in tennis, football and golf.
当空气流过旋转球体的一侧较快时，便产生压力差。马格努斯效应就是由此产生的弯曲飞行路径：旋转的球会偏向低压侧。上旋、下旋和侧旋都会产生可预测的偏差，这在网球、足球和高尔夫中至关重要。
9. Aerobic Capacity and Fitness Indices | 有氧能力与体能指数公式
Several field tests estimate maximal oxygen uptake (VO₂ max) without laboratory equipment. These formulae allow coaches to monitor cardiovascular fitness economically.
一些场地测试可以在没有实验室设备的情况下估算最大摄氧量(VO₂ max)。这些公式使教练能经济地监测心血管体适能。
Cooper 12‑min run: VO₂ max = (distance in metres − 504.9) / 44.73
For the Harvard Step Test, the fitness index reflects how quickly heart rate recovers after exercise.
哈佛台阶测试中，体能指数反映运动后心率恢复的快慢。
Fitness Index = (100 × test duration in seconds) / (2 × sum of heart beats in recovery)
The Multistage Fitness Test (bleep test) provides a VO₂ max prediction: VO₂ max = 3.46 × (shuttles level + 1.2) + 2.3 or using other validated equations. In all cases, higher values indicate superior aerobic endurance.
多阶段体能测试(bleep test)也提供VO₂ max预测公式，如 VO₂ max = 3.46 × (折返级别 + 1.2) + 2.3。通常数值越高代表有氧耐力越强。
10. Training Intensity and Load Formulae | 训练强度与负荷公式
Precision in prescribing exercise intensity is essential for effective training. The Karvonen formula uses heart rate reserve to set target zones.
精确设定运动强度对有效训练至关重要。卡沃宁公式利用心率储备来设定目标区间。
Target HR = ((HRmax − HRrest) × %intensity) + HRrest
HRmax is often estimated as 220 − age. For example, an 18‑year‑old athlete with a resting HR of 60 bpm aiming for 80% intensity would have a target HR of ((200 − 60) × 0.8) + 60 = 172 bpm.
最大心率通常估算为 220 − 年龄。例如一名18岁运动员安静心率60 bpm，目标强度80%，其目标心率为 ((200 − 60) × 0.8) + 60 = 172 bpm。
Training load can be monitored via session RPE: Load = RPE (1−10 scale) × duration (min). The acute:chronic workload ratio (ACWR) is then used to assess injury risk by dividing the current week’s load by the rolling 4‑week average load.
训练负荷可通过课次自觉用力程度(sRPE)监测：负荷 = RPE (1−10) × 时长(分钟)。急慢性工作负荷比(ACWR)用于评估受伤风险，用本周负荷除以滚动四周平均负荷。
11. Psychomotor Laws (Hick’s & Fitts’) | 心理运动定律（Hick与Fitts定律）
Hick’s Law describes the relationship between the number of choices and reaction time. As the number of stimulus‑response alternatives increases, reaction time grows logarithmically.
Hick定律描述选择数量与反应时间的关系。刺激‑反应选项越多，反应时间呈对数增长。
RT = a + b log₂(n)
In open skills such as catching in cricket, reducing the batter’s decision options (e.g., by varying only two types of delivery) can slow their response. Coaches use this to simplify practice or strategically overload opponents.
在板球接球等开放性技能中，减少击球员的决策选项（如只变化两种投球）可延缓其反应。教练利用此原理简化训练或策略性地增加对手负荷。
Fitts’ Law predicts the time required to move to a target area as a function of distance and target width. It applies to aiming tasks in sports.
Fitts定律预测移动到目标区域所需的时间，是距离与目标宽度的函数。适用于运动中的瞄准任务。
MT = a + b log₂(2D / W)
This explains why larger targets (e.g., a wider goal) or shorter distances allow faster and more accurate movements. It underpins equipment design, such as larger tennis racquet sweet spots.
这解释了为何更大的目标（如较宽的球门）或较短的距离能实现更快更准的动作。它指导装备设计，比如更大的网球拍甜区。
12. Body Composition and Energy Expenditure | 身体成分与能量消耗公式
Body Mass Index (BMI) provides a simple screening tool for weight categories, though it does not distinguish between muscle and fat.
体重指数(BMI)是简单的体重分级筛查工具，但不区分肌肉与脂肪。
BMI = weight (kg) / height² (m²)
More precise body composition can be estimated via skinfold measurements and prediction equations. Basal metabolic rate (BMR) can be estimated using the Harris‑Benedict formula, which accounts for gender, weight, height and age.
更精确的身体成分可通过皮褶测量和预测方程估算。基础代谢率(BMR)可用Harris‑Benedict公式估算，该公式考虑性别、体重、身高和年龄。
Energy expenditure in physical activity is often expressed in METs. One MET equals the resting metabolic rate of approximately 3.5 mL O₂ per kg per minute. Total daily energy expenditure (TDEE) is the sum of BMR, the thermic effect of food and physical activity energy expenditure.
身体活动的能量消耗常用MET表示。1 MET相当于约3.5毫升/公斤/分钟的静息摄氧量。每日总能耗(TDEE)是基础代谢率、食物热效应与身体活动能耗的总和。
Energy expended (kcal) ≅ METs × body mass (kg) × duration (h)
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