📚 Year 12 CIE PE: Formula & Theorem Quick Reference Handbook | Y12 CIE 体育公式定理速查手册
This handbook compiles the essential formulas, equations, and fundamental theorems required for the Cambridge International AS Level Physical Education syllabus. Each entry is presented with clear notation, application context in sports, and a paired Chinese translation to reinforce understanding. Use this quick reference to review key quantitative concepts across exercise physiology, biomechanics, and skill acquisition.
本手册汇编了剑桥国际 AS Level 体育课程所必需的核心公式、方程和基本定理。每一个条目都以清晰的符号表示、在体育运动中的应用背景以及配对的中文翻译呈现，以加深理解。使用本快速手册复习运动生理学、生物力学和技能学习领域的关键量化概念。
1. Maximum Heart Rate & Target Zones | 最大心率与目标区间
The simplest estimation of maximum heart rate (HRmax) is derived from age. It forms the basis for prescribing aerobic training intensities.
最大心率（HRmax）的最简单估计来自于年龄。它是制定有氧训练强度的基础。
HRmax = 220 − Age (years)
Target heart rate zones are typically calculated as a percentage of HRmax. For moderate-intensity aerobic work, 60–70% HRmax is common; for vigorous intensity, 70–85% HRmax is recommended.
目标心率区间通常以最大心率的百分比计算。对于中等强度有氧运动，通常采用 60–70% HRmax；对于高强度运动，推荐 70–85% HRmax。
2. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen method uses the heart rate reserve (HRR) to determine target heart rate more accurately by factoring in resting heart rate (HRrest).
卡沃宁方法利用心率储备（HRR）来更准确地确定目标心率，因为它考虑了安静心率（HRrest）。
Target HR = (HRmax − HRrest) × Intensity (%) + HRrest
For instance, an athlete aged 20 with HRrest = 60 bpm training at 70% intensity would have a target HR of (200 − 60) × 0.7 + 60 = 158 bpm. This method aligns training zones more closely with individual fitness levels.
例如，一位 20 岁、安静心率为 60 bpm 的运动员以 70% 强度训练，其目标心率为 (200 − 60) × 0.7 + 60 = 158 bpm。该方法使训练区间与个人体能水平更紧密地匹配。
3. Body Mass Index (BMI) | 身体质量指数
BMI is a simple anthropometric measure used to classify underweight, normal weight, overweight, and obesity in populations.
BMI 是一项简单的人体测量指标，用于在人群中对体重过轻、正常体重、超重和肥胖进行分类。
BMI = Weight (kg) ÷ Height2 (m2)
While BMI does not distinguish between fat mass and lean mass, it remains a widely used screening tool. For athletes with high muscle mass, BMI may overestimate adiposity.
虽然 BMI 不能区分脂肪质量与瘦体质量，但它仍是一种广泛使用的筛查工具。对于肌肉质量较高的运动员，BMI 可能会高估体脂程度。
4. Basal Metabolic Rate (BMR) Estimation | 基础代谢率估算
BMR represents the energy expended at rest to maintain vital functions. The Mifflin-St Jeor equation is a validated estimation method.
基础代谢率（BMR）代表在休息状态下维持生命机能所消耗的能量。Mifflin-St Jeor 方程是一种经过验证的估算方法。
Male: BMR = (10 × weight kg) + (6.25 × height cm) − (5 × age) + 5
Female: BMR = (10 × weight kg) + (6.25 × height cm) − (5 × age) − 161
These values can then be multiplied by a Physical Activity Level (PAL) factor to estimate total daily energy expenditure (TDEE), ranging from 1.2 (sedentary) to 2.5 (elite endurance training).
这些数值可进一步乘以体力活动水平（PAL）系数以估算每日总能量消耗（TDEE），系数范围从 1.2（久坐）到 2.5（精英耐力训练）。
5. Energy Expenditure & Metabolic Equivalents (METs) | 能量消耗与代谢当量
Energy cost of physical activities is often expressed in METs, where 1 MET equals the resting oxygen consumption of approximately 3.5 mL O2/kg/min.
体力活动的能量消耗常用代谢当量（MET）表示，1 MET 约等于 3.5 mL O2/kg/min 的安静耗氧量。
Energy expenditure (kcal/min) ≈ METs × 3.5 × Body mass (kg) / 200
The following table illustrates typical MET values for selected activities:
下表展示了选定活动的典型 MET 值：
| Activity 活动 | METs |
|---|---|
| Walking (5 km/h) 步行 | 3.5 |
| Jogging (8 km/h) 慢跑 | 8.0 |
| Cycling (moderate, 20 km/h) 自行车中速 | 8.0 |
| Swimming (freestyle, moderate) 自由泳中等 | 8.0 |
| Basketball game 篮球比赛 | 8.5 |
| Running (12 km/h) 跑步 | 12.5 |
This approach allows coaches to estimate caloric demands and plan nutritional strategies.
这种方法使教练能够估算热量需求并制定营养策略。
6. Newton’s Laws of Motion | 牛顿运动定律
Newton’s three laws govern the motion of bodies and are fundamental in biomechanical analysis of sports techniques.
牛顿三大定律支配物体的运动，是运动技术生物力学分析的基础。
First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by an external net force.
第一定律（惯性）： 物体将保持静止或匀速直线运动状态，除非受到外净力的作用。
Second Law (Acceleration): The acceleration of an object is directly proportional to the net force and inversely proportional to its mass.
第二定律（加速度）： 物体的加速度与净力成正比，与质量成反比。
F = m × a
Third Law (Action-Reaction): For every action force, there is an equal and opposite reaction force.
第三定律（作用力与反作用力）： 每个作用力都有一个大小相等、方向相反的反作用力。
In sprinting, the athlete pushes backward against the ground (action), and the ground pushes the athlete forward (reaction).
在短跑中，运动员向后蹬地（作用力），地面将运动员向前推进（反作用力）。
7. Momentum & Impulse | 动量与冲量
Momentum quantifies the quantity of motion an object possesses. Impulse is the product of force and the time over which it acts, causing a change in momentum.
动量定量描述物体所具有的运动量。冲量是力与其作用时间的乘积，导致动量的变化。
Linear momentum: p = m × v (kg·m/s)
Impulse: J = F × Δt = Δp = m × Δv
Increasing the time of contact (e.g., catching a ball by moving the hands backward) reduces the impact force, a principle used to prevent injury.
增加接触时间（例如接球时向后移动双手）会减小冲击力，这是预防受伤的原理。
8. Linear Kinematics | 线性运动学
Linear kinematics describes motion without considering forces. Key quantities include displacement, velocity, and acceleration.
线性运动学在不考虑力的情况下描述运动。关键量包括位移、速度和加速度。
Average velocity: v = Δd / Δt (m/s)
Average acceleration: a = Δv / Δt (m/s2)
Equations of uniform acceleration (SUVAT) are often applied to projectile motion in sport, such as a long jumper’s take-off:
匀加速方程（SUVAT）常应用于体育中的抛射体运动，例如跳远运动员的起跳：
- v = u + a t
- v2 = u2 + 2 a s
- s = u t + ½ a t2
where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.
其中 u = 初速度，v = 末速度，a = 加速度，t = 时间，s = 位移。
9. Work & Power | 功与功率
Mechanical work is done when a force moves an object over a distance. Power represents the rate at which work is performed, a key determinant of athletic performance.
当力使物体移动一段距离时，就做了机械功。功率表示做功的速率，是运动表现的关键决定因素。
Work: W = F × d × cosθ (Joules)
Power: P = W / t or P = F × v (Watts)
In cycling, power output is measured directly. A higher power-to-weight ratio is advantageous in climbing events. Work done against gravity: W = m × g × h, where g = 9.81 m/s2.
在自行车运动中，功率输出可直接测量。较高的功率重量比在爬坡赛事中具有优势。克服重力所做的功：W = m × g × h，其中 g = 9.81 m/s2。
10. Levers & Torque | 杠杆与力矩
The human musculoskeletal system operates as a series of levers. Torque (moment of force) is the rotational effect of a force applied at a distance from an axis.
人体肌肉骨骼系统作为一系列杠杆运作。力矩是力在离轴一定距离处产生的转动效应。
Torque: τ = F × d × sinθ (N·m)
where d is the moment arm (perpendicular distance from the axis to the line of force). In the body, muscles produce torque to overcome resistance. Three classes of levers exist, with third-class levers being most common (e.g., elbow flexion: effort between axis and resistance). Mechanical advantage = effort arm / resistance arm.
其中 d 是力臂（从轴到力线的垂直距离）。在人体中，肌肉产生力矩以克服阻力。杠杆分三类，其中第三类杠杆最为常见（如肘屈：动力点在轴与阻力点之间）。机械利益 = 动力臂 / 阻力臂。
11. Angular Motion | 角运动
Rotational movements in sport obey principles analogous to linear motion. Key descriptors include angular displacement, angular velocity, and angular momentum.
体育运动中的旋转运动遵循与线性运动相似的原理。关键描述量包括角位移、角速度和角动量。
Angular velocity: ω = Δθ / Δt (rad/s)
Angular momentum: L = I × ω (kg·m2/s)
where I is the moment of inertia (resistance to angular acceleration), dependent on mass distribution about the axis. Figure skaters spin faster by pulling arms in to reduce I, conserving angular momentum.
其中 I 是转动惯量（对角加速度的阻力），取决于质量绕轴的分布。花样滑冰运动员通过收臂减小 I，从而在角动量守恒下旋转更快。
12. Fluid Dynamics: Bernoulli’s Principle & Magnus Effect | 流体动力学：伯努利原理与马格努斯效应
Bernoulli’s principle explains lift forces on objects moving through a fluid (air or water). It states that faster fluid flow creates lower pressure.
伯努利原理解释了穿过流体（空气或水）运动的物体所受的升力。它指出流体流速越快，压力越低。
Bernoulli’s equation: P + ½ ρ v2 + ρ g h = constant
The Magnus effect is a consequence of pressure differences created by a spinning object: a ball spinning clockwise experiences a lift force perpendicular to the flow direction. This effect is exploited in tennis topspin, football swerve, and golf shots.
马格努斯效应是由旋转物体产生的压力差所致：顺时针旋转的球会受到垂直于来流方向的升力。该效应在网球上旋球、足球弧线球及高尔夫击球中被利用。
Drag force also acts opposite to motion, given by:
阻力还作用于运动的反方向，公式为：
Fd = ½ ρ v2 Cd A
where ρ = fluid density, v = velocity, Cd = drag coefficient, A = cross-sectional area. Streamlining reduces Cd and A, improving performance in cycling and swimming.
其中 ρ = 流体密度，v = 速度，Cd = 阻力系数，A = 横截面积。流线型减少 Cd 和 A，从而提升自行车和游泳成绩。
Published by TutorHao | Physical Education Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导