📚 KS3 Cambridge PE Formula & Theorem Quick Reference Guide | KS3 剑桥体育：公式定理速查手册
This quick reference handbook gathers the most important formulas, theorems and numerical scales you will use throughout the KS3 Cambridge Physical Education course. Understanding these principles will help you train smarter, analyse performance and link science to sport. Keep this guide handy whenever you plan a personal fitness programme or revise for an assessment. Each section presents a key concept first in English, followed immediately by the Chinese explanation.
本速查手册汇集了 KS3 剑桥体育课程中最常用的公式、定理和数值量表。理解这些原理能帮助你更科学地训练、分析运动表现，并将科学与运动联系起来。在制定个人健身计划或备考复习时，可随时查阅本指南。每个部分先用英文介绍核心概念，紧接提供中文解释。
1. Maximum Heart Rate & Target Heart Rate | 最大心率与靶心率
Maximum heart rate (MHR) represents the highest number of beats per minute your heart can safely achieve during all-out exercise. The simplest estimation is: MHR = 220 – age (in years). For a 13-year-old student, this would be 220 – 13 = 207 bpm. Although individual variation exists, this formula is widely used in KS3 to set training intensities.
最大心率（MHR）是你在全力运动中每分钟心跳的最高安全次数。最简单的估算公式为：MHR = 220 – 年龄（岁）。例如一名13岁学生的最大心率估算值为 220 – 13 = 207 bpm。尽管存在个体差异，该公式在 KS3 阶段被广泛用于设定训练强度。
Once you know your MHR, you can calculate target heart rate (THR) zones. The Karvonen formula uses resting heart rate (RHR) to give personalised zones: THR = RHR + Intensity × (MHR – RHR). Intensity is expressed as a decimal, for example 0.6 for 60% effort. If RHR = 70 bpm and MHR = 207 bpm, training at 60% intensity gives THR = 70 + 0.6 × (207 – 70) = 70 + 82.2 ≈ 152 bpm. Using heart rate zones helps you work at the right effort for aerobic or anaerobic goals.
获知最大心率后，便可用卡沃南公式计算靶心率（THR）区间，该公式纳入了安静心率（RHR），区间更具个性化：THR = RHR + 强度 × (MHR – RHR)。强度以小数表示，如 60% 即 0.6。若 RHR = 70 bpm，MHR = 207 bpm，以 60% 强度训练时，THR = 70 + 0.6 × (207 – 70) = 70 + 82.2 ≈ 152 bpm。运用心率区间能帮助你在有氧或无氧训练中维持恰当的运动强度。
2. Body Mass Index (BMI) | 身体质量指数
Body Mass Index is a screening tool that estimates whether a person has a healthy body weight for their height. The formula is: BMI = weight (kg) ÷ height² (m²). Measure weight in kilograms and height in metres, then square the height. For example, a student weighing 50 kg with a height of 1.60 m has BMI = 50 ÷ (1.6 × 1.6) = 50 ÷ 2.56 = 19.5 kg/m². This falls within the healthy weight range (typically 18.5–24.9 for adults; KS3 students should refer to age-adjusted centile charts).
身体质量指数是评估体重相对于身高是否在健康范围内的筛查工具。计算公式为：BMI = 体重（kg）÷ 身高²（m²）。先以千克为单位测体重，以米为单位测身高，再将身高数值平方。例如一名体重50 kg、身高1.60 m的学生，BMI = 50 ÷ (1.6 × 1.6) = 50 ÷ 2.56 = 19.5 kg/m²。该数值落在健康体重范围内（成人标准通常为 18.5–24.9；KS3 学生应参照年龄百分位曲线图）。
BMI does not distinguish between muscle and fat, so a highly muscular athlete may register a higher BMI despite low body fat. Therefore, it is always used alongside other fitness indicators such as waist-to-hip ratio or skinfold measures.
BMI 无法区分肌肉与脂肪，因此肌肉发达的运动者可能体脂率低而 BMI 数值偏高。所以要结合其他体能指标使用，例如腰臀比或皮褶厚度测量。
3. Energy Expenditure and Metabolic Equivalents (METs) | 能量消耗与代谢当量
A MET (Metabolic Equivalent) represents the energy cost of resting quietly. By definition: 1 MET = 3.5 ml of oxygen consumed per kilogram of body weight per minute (ml O₂ / kg / min). Activities can be compared using MET values; for instance, walking at 5 km/h costs around 3.5 METs, while jogging may cost 7 METs or more. This allows you to estimate energy expenditure with the formula: Energy (kcal) = MET value × body weight (kg) × time (hours).
MET（代谢当量）代表安静休息时身体消耗的能量。定义为：1 MET = 每分钟每千克体重消耗 3.5 毫升氧气（ml O₂ / kg / min）。不同运动可用 MET 值比较；例如，以 5 公里/小时走路约消耗 3.5 METs，而慢跑则需 7 METs 甚至更高。利用公式可估算能量消耗：能量（千卡）= MET 值 × 体重（kg） × 时间（小时）。
A 50 kg student who cycles at 6 METs for 0.5 hours would burn roughly 6 × 50 × 0.5 = 150 kcal. Understanding METs helps balance energy intake with physical activity, which is foundational for a healthy lifestyle.
一名50 kg的学生以 6 METs 的强度骑行0.5小时，大约消耗 6 × 50 × 0.5 = 150 千卡。理解 MET 概念有助于平衡能量摄入与体力活动，为健康生活方式奠定基础。
4. Speed, Distance and Time | 速度、距离和时间
The relationship between speed, distance and time is fundamental in sport. The three forms of the formula are: Speed = Distance ÷ Time, Distance = Speed × Time, and Time = Distance ÷ Speed. Units must be consistent; in PE we commonly use metres per second (m/s) or kilometres per hour (km/h). For example, if a sprinter covers 100 m in 12.5 s, speed = 100 ÷ 12.5 = 8 m/s.
速度、距离和时间之间的关系是运动学的基础。公式可写成三种形式：速度 = 距离 ÷ 时间，距离 = 速度 × 时间，时间 = 距离 ÷ 速度。单位须一致；体育课中常使用米/秒（m/s）或公里/小时（km/h）。例如，一名短跑选手用 12.5 秒跑完 100 米，其速度 = 100 ÷ 12.5 = 8 m/s。
To convert between m/s and km/h, multiply by 3.6 (since 1 m/s = 3.6 km/h). An 8 m/s sprint corresponds to 8 × 3.6 = 28.8 km/h. Practising these calculations helps you interpret fitness test results and match tactics to your pace.
在 m/s 和 km/h 之间转换时，乘以 3.6 即可（因为 1 m/s = 3.6 km/h）。8 m/s 的冲刺相当于 8 × 3.6 = 28.8 km/h。反复练习这类计算有助于解读体能测试结果，并根据自身速度制定战术。
5. Force, Mass and Acceleration | 力、质量和加速度
Newton’s second law of motion is expressed as: Force (N) = mass (kg) × acceleration (m/s²). In sport, understanding force helps explain how athletes start, stop and change direction. A football player with a mass of 60 kg who accelerates at 2 m/s² requires a force of 60 × 2 = 120 N. Greater force production can improve acceleration and agility.
牛顿第二运动定律表述为：力（N）= 质量（kg） × 加速度（m/s²）。在运动中，理解力的原理有助于解释运动员如何启动、制动和变向。一名质量为 60 kg 的足球运动员以 2 m/s² 加速时，需要产生 120 N 的力。更强的发力能提升加速度和灵敏性。
When muscles contract, they apply force to bones via tendons. This force is transferred to the ground or an object, generating movement. Training to increase strength (maximum force) and power (rate of force development) is crucial for many sports.
肌肉收缩时通过肌腱将力施加到骨骼上，此力再传递至地面或物体，从而产生运动。训练以增加力量（最大力）和爆发力（产生力的速率）对众多运动项目至关重要。
6. Work, Power and Mechanical Advantage | 功、功率和机械效益
Work (W) is done when a force moves an object over a distance. The formula is: W = force (N) × distance (m), measured in joules (J). Lifting a 20 kg weight (≈ 200 N) through 0.5 m does 200 × 0.5 = 100 J of work. Power (P) is the rate of doing work: P = Work ÷ time (s), measured in watts (W). If the lift takes 1 second, power = 100 W.
力推动物体移动一段距离时即做功（W）。公式为：W = 力（N）× 距离（m），单位为焦耳（J）。将 20 kg 的重物（约 200 N）抬高 0.5 m，做功 200 × 0.5 = 100 J。功率（P）是做功的速率：P = 做功 ÷ 时间（s），单位为瓦特（W）。若完成该上举耗时 1 秒，功率则为 100 W。
Mechanical advantage (MA) describes how a lever amplifies force. For a simple lever, MA = effort arm length ÷ resistance arm length. In the human body, bones act as levers, joints are pivots, and muscles provide the effort. A second-class lever, such as the ankle during plantar flexion (standing on tiptoes), has MA > 1 because the effort arm (Achilles tendon) is longer than the resistance arm. Understanding MA explains why some joint movements are stronger but slower, while others favour speed.
机械效益（MA）描述杠杆如何放大力量。对简单杠杆，MA = 动力臂长度 ÷ 阻力臂长度。在人体中，骨相当于杠杆，关节是支点，肌肉提供动力。第二类杠杆，如踝关节跖屈（踮脚尖时），动力臂（跟腱）长于阻力臂，MA > 1，因此力量放大。理解机械效益能解释为何某些关节运动力量强但速度慢，另一些则偏重速度。
7. Lever Systems and Torque (Moment) | 杠杆系统与力矩
There are three classes of levers found in the body, classified by the relative positions of effort, load, and fulcrum. A first-class lever has the fulcrum in the middle (e.g., neck extending, with muscles at the back, load at the front, pivot at vertebrae). A second-class lever has the load in the middle (e.g., ankle plantar flexion). A third-class lever has the effort in the middle, which is most common in the body (e.g., elbow flexion when the biceps pull on the forearm). Third-class levers favour range and speed of movement at the expense of force.
人体中存在三类杠杆，根据动力、阻力与支点的相对位置划分。第一类杠杆支点在中间（如颈后伸动作，颈后肌群为动力，头前部为阻力，脊椎为支点）。第二类杠杆阻力在中间（如踝关节跖屈）。第三类杠杆动力在中间，在人体中最常见（如肱二头肌屈肘时，肌肉在近端收缩拉动前臂）。第三类杠杆牺牲力量换取运动幅度和速度。
The turning effect of a force is called torque or moment: Moment = Force × perpendicular distance from pivot. When a gymnast holds an iron cross position, the shoulder muscles must generate enormous torque to keep the moment arm (arm length) working against gravity. Lever principles are essential for understanding technique efficiency and injury prevention.
力产生的转动效应称为力矩：力矩 = 力 × 支点的垂直距离。当体操运动员呈”十字支撑”时，肩部肌群须产生巨大力矩以对抗手臂形成的力臂。杠杆原理对于理解技术效率及预防损伤至关重要。
8. Borg Rating of Perceived Exertion (RPE) Scale | 博格自觉用力程度量表
The Borg RPE scale (6–20) quantifies subjective exercise intensity. Rate your overall feeling of effort, including breathlessness and muscle fatigue. The scale is designed so that a rating of 6 corresponds to “no exertion at all”, while 20 represents “maximal exertion”. There is a simple link between RPE and heart rate: estimated heart rate ≈ RPE × 10 bpm. So RPE 13 suggests a heart rate around 130 bpm.
博格 RPE 量表（6–20 级）用于量化主观运动强度。综合评估整体用力感，包括气喘程度和肌肉疲劳度。量表设计使 6 分对应”毫不费力”，20 分代表”最大用力”。RPE 与心率之间存在简单关联：估算心率 ≈ RPE × 10 bpm。因此 RPE 13 提示心率约 130 bpm。
Using RPE is valuable when heart rate monitors are unavailable or when taking medication that affects heart rate. Record your RPE during different stages of a training session to monitor internal load and adjust future workouts.
当没有心率监测设备或因药物影响心率时，使用 RPE 极具价值。在训练不同阶段记录 RPE，可监控内部负荷并调整后续训练计划。
9. Training Zones and the Principle of Overload | 训练区间与超负荷原则
To improve fitness, you must apply the principle of overload: exercise at a greater intensity, duration or frequency than the body is used to. The FITT principle (Frequency, Intensity, Time, Type) guides programme design. Aerobic training is generally performed at 60–80% of MHR, known as the moderate intensity zone. Exercise at 80–90% MHR shifts towards anaerobic training, improving speed and tolerance of lactic acid.
要提升体适能，必须运用超负荷原则：以超出身体习惯的强度、时间或频率进行锻炼。FITT 原则（频度、强度、时间、类型）是训练计划设计的指南。有氧训练通常在最大心率的 60–80% 区间进行，即中等强度区。以 80–90% MHR 运动则偏重无氧训练，可改善速度及乳酸耐受能力。
Specific formulas for energy systems: max oxygen uptake (VO₂ max) is a key measure of aerobic fitness. A simple field estimate can use the Multi-Stage Fitness Test (bleep test) results: predicted VO₂ max ≈ 3.46 × (level + shuttle number) + 12.2 ml/kg/min (rough estimation). The overload principle requires gradual progression, typically increasing training load by no more than 10% per week to reduce injury risk.
能量系统相关公式：最大摄氧量（VO₂ max）是衡量有氧能力的关键指标。可用多阶体能测试（蜂鸣测试）结果进行简单估算：预测 VO₂ max ≈ 3.46 × (级别 + 折返次数) + 12.2 ml/kg/min（粗略估算）。超负荷原则要求逐步进阶，通常每周增加训练负荷不超过 10%，以降低受伤风险。
Hydration also follows a measurable principle: fluid loss of 2% body weight can impair performance. Pre-exercise hydration can be guided by drinking 5–7 ml per kg of body weight at least 4 hours before activity.
水合作用同样可量化：体重 2% 的体液流失就会影响运动表现。训练前补水可遵循每千克体重饮用 5–7 毫升液体的原则，在运动前至少 4 小时完成。
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