📚 IGCSE Cambridge Physical Education: Formula & Theorem Quick Reference Handbook | IGCSE 剑桥体育：公式定理速查手册
Welcome to your essential quick-reference guide for IGCSE Cambridge Physical Education. This handbook consolidates every key formula, theorem, and biomechanical principle you need to memorise for the examination. From cardiovascular calculations to Newton's Laws of Motion, from lever systems to fluid dynamics in sport, each entry is presented with clear definitions, worked applications, and sporting examples. Use this revision tool to reinforce your understanding, check your recall, and approach Paper 1 and Paper 2 with confidence. Remember: in IGCSE PE, precision matters—knowing the correct formula and its units can make the difference between grades.
欢迎使用 IGCSE 剑桥体育必备速查手册。本手册汇总了考试所需记忆的每一项关键公式、定理与生物力学原理。从心血管计算到牛顿运动定律，从杠杆系统到运动中的流体力学，每个条目均配有清晰定义、应用示例和运动实例。请利用这份复习工具巩固理解、检验记忆，自信地应对试卷一和试卷二。请牢记：在 IGCSE 体育中，精确性至关重要——掌握正确的公式及其单位可能就是提分的关键。
1. Maximum Heart Rate (MHR) | 最大心率 (MHR)
The most fundamental cardiovascular formula in sports science is the estimation of Maximum Heart Rate. The widely accepted equation subtracts your age from a fixed constant. This value underpins the calculation of all training zones and is essential for designing aerobic and anaerobic exercise programmes.
运动科学中最基础的心血管公式是最大心率的估算。广泛接受的公式是用一个固定常数减去你的年龄。该数值是所有训练区间计算的基础，对于设计有氧和无氧运动计划至关重要。
MHR = 220 − Age (in years)
- Application: A 16-year-old athlete would have an estimated MHR of 204 beats per minute (bpm).
- 应用：一名 16 岁运动员的预估最大心率约为每分钟 204 次 (bpm)。
- Exam Tip: Always state 'beats per minute' or 'bpm' precise estimates, but the Cambridge IGCSE syllabus standardises on the 220 − age formula.
- 考试提示：务必注明单位为“次/分钟”或 bpm。部分课程体系也接受 206.9 − (0.67 × 年龄) 这一更精确的估算，但剑桥 IGCSE 教学大纲统一使用 220 − 年龄公式。
2. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen Formula provides a more personalised method for calculating target heart rate zones by incorporating resting heart rate. This method accounts for individual fitness levels, as a lower resting heart rate typically indicates greater cardiovascular efficiency. It is used to establish aerobic and anaerobic training thresholds with greater accuracy.
卡沃宁公式通过引入静息心率，提供了一种更个性化的目标心率区间计算方法。该方法考虑了个体体能水平差异，因为较低的静息心率通常意味着更高的心血管效率。它用于更准确地确定有氧和无氧训练阈值。
Target HR = Resting HR + (Required % × [MHR − Resting HR])
- Worked Example: A 20-year-old athlete with a resting HR of 60 bpm wants to train at 70% intensity. MHR = 220 − 20 = 200 bpm. Heart Rate Reserve = 200 − 60 = 140 bpm. 70% of 140 = 98 bpm. Target HR = 60 + 98 = 158 bpm.
- 计算示例：一名静息心率为 60 bpm 的 20 岁运动员希望以 70% 强度训练。MHR = 220 − 20 = 200 bpm。心率储备 = 200 − 60 = 140 bpm。140 的 70% = 98 bpm。目标心率 = 60 + 98 = 158 bpm。
- Aerobic Zone: 60-80% of Heart Rate Reserve. Anaerobic Zone: 80-90% of Heart Rate Reserve.
- 有氧区间：心率储备的 60%-80%。无氧区间：心率储备的 80%-90%。
3. Body Mass Index (BMI) | 身体质量指数 (BMI)
Body Mass Index is a widely used screening tool for categorising weight status in populations. Despite its limitations—it does not distinguish between muscle mass and fat mass—BMI remains a core metric in the IGCSE PE Health, Fitness and Well-being topic. It provides a quick, non-invasive indication of whether an individual is underweight, healthy weight, overweight, or obese.
身体质量指数是一种广泛用于群体体重状况分类的筛查工具。尽管存在局限性——它无法区分肌肉质量与脂肪质量——BMI 仍然是 IGCSE 体育“健康、体能与福祉”主题中的核心指标。它能快速、无创地判断个体是否偏瘦、体重正常、超重或肥胖。
BMI = Weight (kg) ÷ [Height (m)]²
| BMI Range | BMI 范围 | Classification | 分类 |
|---|---|
| Below 18.5 | 低于 18.5 | Underweight | 体重过轻 |
| 18.5 – 24.9 | Healthy weight | 健康体重 |
| 25.0 – 29.9 | Overweight | 超重 |
| 30.0 and above | 30.0 及以上 | Obese | 肥胖 |
- Worked Example: 70 ÷ (1.75)² = 70 ÷ 3.0625 ≈ 22.9 kg/m², placing them within the healthy weight category.
- 计算示例：一名体重 70 kg、身高 1.75 m 的人，BMI = 70 ÷ (1.75)² = 70 ÷ 3.0625 ≈ 22.9 kg/m²，属于健康体重范畴。
4. Cardiac Output, Stroke Volume, and Heart Rate | 心输出量、每搏输出量与心率
The relationship between cardiac output, stroke volume, and heart rate is a cornerstone of exercise physiology. Cardiac output represents the total volume of blood pumped by the heart per minute and increases dramatically during exercise to meet the elevated oxygen demand of working muscles. This equation demonstrates how both heart rate and stroke volume contribute to overall circulatory capacity.
心输出量、每搏输出量与心率之间的关系是运动生理学的基石。心输出量表示心脏每分钟泵出的血液总量，在运动期间会显著增加，以满足工作肌肉对氧气的更高需求。该公式展示了心率和每搏输出量如何共同影响整体循环能力。
Cardiac Output (Q) = Stroke Volume (SV) × Heart Rate (HR)
- Units: Cardiac output is measured in litres per minute (L/min), stroke volume in millilitres per beat (mL/beat), and heart rate in beats per minute (bpm).
- 单位：心输出量以升/分钟 (L/min) 为单位，每搏输出量以毫升/次 (mL/beat) 为单位，心率以次/分钟 (bpm) 为单位。
- Resting Values: A typical resting Q is approximately 5 L/min (SV ~70 mL × HR ~72 bpm). During maximal exercise, Q can rise to 20-40 L/min in trained athletes.
- 静息值：典型的静息心输出量约为 5 L/min (每搏输出量约 70 mL × 心率约 72 bpm)。在最大强度运动时，训练有素的运动员心输出量可升至 20-40 L/min。
5. Newton's Three Laws of Motion | 牛顿运动三定律
Newton's Laws form the theoretical foundation of biomechanics in sport. Every movement—from a sprinter exploding out of the blocks to a footballer striking a free kick—can be analysed through these three principles. The IGCSE PE specification expects you to state each law, explain its meaning, and apply it to a sporting context with precise examples.
牛顿定律构成了运动生物力学的理论基础。每一个动作——从短跑运动员蹬离起跑器到足球运动员主罚任意球——都可以通过这三条原理进行分析。IGCSE 体育教学大纲要求能陈述每条定律、解释其含义，并结合具体运动实例进行应用。
First Law (Law of Inertia): A body remains at rest or in uniform motion in a straight line unless acted upon by an external force.
第一定律（惯性定律）：任何物体都将保持静止或匀速直线运动状态，直到有外力迫使它改变这种状态为止。
Second Law (Law of Acceleration): The acceleration of a body is directly proportional to the force applied and inversely proportional to its mass. F = m × a
第二定律（加速度定律）：物体的加速度与所受外力成正比，与其质量成反比。F = m × a
Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.
第三定律（作用与反作用定律）：每个作用力都会产生一个大小相等、方向相反的反作用力。
- Sporting Application – First Law: A hockey ball remains stationary on the pitch until a player applies a force with the stick. A curling stone continues gliding across the ice due to low friction, demonstrating inertia in motion.
- 运动应用——第一定律：曲棍球在球场上保持静止，直到运动员用球棍施加力。冰壶在冰面上持续滑行，因摩擦力低而展现出运动惯性。
- Sporting Application – Second Law: To accelerate a shot put (mass ~7.26 kg for men), a thrower must apply a large force. The greater the force, the greater the acceleration, resulting in a longer throw.
- 运动应用——第二定律：为使铅球加速（男子约 7.26 kg），投掷者必须施加巨大力量。力量越大，加速度越大，投掷距离越远。
- Sporting Application – Third Law: When a swimmer pushes water backwards with their hands (action), the water pushes the swimmer forwards (reaction). Similarly, a sprinter drives backwards into the blocks, and the blocks propel them forwards.
- 运动应用——第三定律：当游泳者用手向后推水（作用力）时，水推动游泳者前进（反作用力）。同样，短跑运动员向后蹬起跑器，起跑器推动他们向前。
6. Speed, Velocity, and Acceleration | 速率、速度与加速度
Distinguishing between speed and velocity is a crucial skill in IGCSE PE biomechanics. Speed is a scalar quantity—it has magnitude only. Velocity is a vector quantity—it has both magnitude and direction. Acceleration describes how quickly velocity changes over time. These three concepts form the basis for analysing performance in every linear sport, from track athletics to swimming.
区分速率和速度是 IGCSE 体育生物力学中的一项关键技能。速率是标量——只有大小。速度是矢量——既有大小又有方向。加速度描述速度随时间变化的快慢。这三个概念构成了分析从田径到游泳等所有线性运动表现的基础。
Speed = Distance ÷ Time
速率 = 距离 ÷ 时间
Velocity = Displacement ÷ Time
速度 = 位移 ÷ 时间
Acceleration = (Final Velocity − Initial Velocity) ÷ Time
加速度 = (末速度 − 初速度) ÷ 时间
- Units: Speed and velocity are measured in metres per second (m/s). Acceleration is measured in metres per second squared (m/s²).
- 单位：速率和速度以米/秒 (m/s) 为单位。加速度以米/秒² (m/s²) 为单位。
- Exam Distinction: speed is positive (400 m ÷ time), but their average velocity is zero (displacement = 0 m).
- 考试区分：如果一名 400 米运动员跑完标准跑道一圈并回到起点，他的平均速率是正值（400 m ÷ 时间），但平均速度为零（位移 = 0 m）。
7. Momentum and Impulse | 动量与冲量
Momentum quantifies the amount of motion possessed by a moving body, while impulse measures the change in momentum resulting from a force applied over a period of time. These concepts are especially relevant in contact sports and collision analysis. A larger momentum means it is harder to stop or change the direction of the moving body.
动量量化了运动物体所具有的运动量，而冲量则衡量力在一段时间内作用所导致的动量变化。这些概念在接触性运动和碰撞分析中尤为重要。动量越大，意味着移动物体越难被阻止或改变方向。
Momentum (p) = Mass (m) × Velocity (v)
动量 (p) = 质量 (m) × 速度 (v)
Impulse = Force (F) × Time (t) = Change in Momentum (Δp)
冲量 = 力 (F) × 时间 (t) = 动量的变化 (Δp)
- Unit: Momentum is measured in kilogram metres per second (kg·m/s). Impulse is measured in Newton seconds (N·s).
- 单位：动量以千克·米/秒 (kg·m/s) 为单位。冲量以牛顿·秒 (N·s) 为单位。
- Sporting Example: In rugby, a player with greater mass and velocity has more momentum and is more difficult to tackle. To reduce the force of catching a fast cricket ball, a fielder extends their hands backwards, increasing the time over which the momentum change occurs, thereby reducing the force experienced.
- 运动实例：在橄榄球运动中，质量和速度更大的球员具有更大的动量，更难被擒抱。为减小接快速板球时受到的力，外野手会向后收手，延长动量变化的作用时间，从而减小所受的冲击力。
8. Force, Mass, and Weight | 力、质量与重量
Force, mass, and weight are frequently confused in GCSE-level science and PE. Mass is the amount of matter in an object and remains constant regardless of location. Weight is the gravitational force acting on that mass and varies with the strength of the gravitational field. The equation linking force, mass, and acceleration is central to biomechanics and forms the basis for understanding all human movement.
力、质量和重量在 GCSE 阶段的科学与体育中常被混淆。质量是物体所含物质的量，在任何地点都保持不变。重量是作用在该质量上的重力，随引力场强度的变化而变化。连接力、质量和加速度的方程是生物力学的核心，也是理解所有人体运动的基础。
Force (F) = Mass (m) × Acceleration (a)
力 (F) = 质量 (m) × 加速度 (a)
Weight (W) = Mass (m) × Gravitational Field Strength (g)
重量 (W) = 质量 (m) × 重力场强度 (g)
- Units: eration is in m/s². On Earth, g ≈ 9.8 m/s² (rounded to 10 m/s² in many PE exam questions).
- 单位：力和重量以牛顿 (N) 为单位。质量以千克 (kg) 为单位。加速度以 m/s² 为单位。在地球上，g ≈ 9.8 m/s²（许多体育考题中四舍五入取 10 m/s²）。
- Example: fter lifting a barbell of mass 100 kg must overcome a weight of 100 kg × 10 m/s² = 1000 N.
- 示例：举重运动员举起质量为 100 kg 的杠铃，必须克服等效重量 100 kg × 10 m/s² = 1000 N 的重力。
9. Moment of Force (Torque) and Levers | 力矩（扭矩）与杠杆
The principle of moments explains how forces cause rotation around a pivot or fulcrum. This is directly applied in the analysis of lever systems in the human body, where bones act as levers, joints serve as fulcrums, and muscles supply the effort force. Understanding moment calculations is essential for evaluating mechanical advantage in sporting movements.
力矩原理解释了力如何使物体绕支点或转轴旋转。这直接应用于人体杠杆系统的分析，其中骨骼充当杠杆，关节作为支点，肌肉提供动力。理解力矩计算对于评估运动动作中的机械优势至关重要。
Moment (M) = Force (F) × Perpendicular Distance from Fulcrum (d)
力矩 (M) = 力 (F) × 力到支点的垂直距离 (d)
Mechanical Advantage (MA) = Effort Arm ÷ Resistance Arm
机械优势 (MA) = 动力臂 ÷ 阻力臂
- Unit: Moment is measured in Newton metres (N·m).
- 单位：力矩以牛顿·米 (N·m) 为单位。
- Principle of Moments: For a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
- 力矩原理：对于处于平衡状态的杠杆，顺时针力矩之和等于逆时针力矩之和。
- Lever Classes in Sport: First-class lever (e.g., neck extension in heading a football); Second-class lever (e.g., calf raise at the ankle—rare in the body, always provides MA > 1); Third-class lever (e.g., bicep curl—most common in the body, designed for range and speed of movement, MA < 1).
- 运动中的杠杆类别：第一类杠杆（如头球时的颈部伸展）；第二类杠杆（如踝关节提踵——在人体中较为罕见，MA 始终大于 1）；第三类杠杆（如肱二头肌弯举——在人体中最为常见，旨在获得更大的活动范围和运动速度，MA 小于 1）。
10. Bernoulli's Principle and the Magnus Effect | 伯努利原理与马格努斯效应
Bernoulli's Principle is fundamental to understanding how objects move through fluids (air and water). It states that faster-moving fluid exerts lower pressure. This principle explains how aerofoils generate lift, how discus and javelin achieve flight, and why spinning balls curve through the air—the Magnus Effect. These are key concepts in the IGCSE PE movement analysis topic.
伯努利原理是理解物体如何在流体（空气和水）中运动的基础。它指出，流速较快的流体会产生较低的压力。这一原理解释了翼型如何产生升力、铁饼和标枪如何实现飞行，以及为什么旋转的球会在空气中弯曲行进——即马格努斯效应。这些是 IGCSE 体育动作分析主题中的关键概念。
Bernoulli's Principle: As the velocity of a fluid (air/water) increases, the pressure exerted by that fluid decreases.
伯努利原理：当流体（空气/水）的速度增加时，该流体施加的压力降低。
Magnus Effect: A spinning object creates a pressure differential in the surrounding fluid, causing the object to deviate from its expected flight path.
马格努斯效应：旋转的物体会在其周围流体中产生压力差，导致物体偏离预期的飞行路径。
- Application – Topspin in Tennis: A ball hit with topspin has air moving faster over the top surface and slower underneath. The higher pressure below pushes the ball downwards, causing it to dip sharply onto the court.
- 应用——网球上旋球：击打上旋球时，球体上表面空气流速较快，下表面流速较慢。球下方的较高压力将球向下推，使其急速下坠落入球场。
- Application – Backspin (Slice): Backspin creates higher pressure above the ball, generating lift and keeping the ball in the air longer—useful in golf and defensive tennis shots.
- 应用——下旋球（削球）：下旋使球上方产生较高压力，产生升力并使球在空中停留更长时间——在高尔夫球和网球防守性击球中非常有用。
11. Projectile Motion Principles | 抛体运动原理
Projectile motion analysis in IGCSE PE focuses on the factors that determine the horizontal range and trajectory of a thrown, kicked, or struck object. The optimal release angle, release height, and release velocity are interdependent and vary by sport. Understanding these principles helps explain why javelin throwers release at 33-36 degrees rather than the theoretical 45 degrees.
IGCSE 体育中的抛体运动分析侧重于决定投掷、踢击或击打物体水平射程与轨迹的因素。最佳出手角度、出手高度和出手速度相互关联，并因运动项目而异。理解这些原理有助于解释为何标枪运动员在 33-36 度角出手，而非理论上的 45 度角。
Factors affecting projectile range: Release velocity (most significant), release angle, release height, and aerodynamic factors (air resistance, spin).
影响抛体射程的因素：出手速度（最为显著）、出手角度、出手高度以及空气动力学因素（空气阻力、旋转）。
- Optimal Angle (Theory): In a vacuum, with equal release and landing heights, 45° maximises horizontal range.
- 理论最优角度：在真空中，当出手点与落点高度相同时，45°可使水平射程最大化。
- Sport-Specific Angles: Shot put: ~38-42° (release point is above landing point due to height of the athlete); Javelin: ~33-36° (aerodynamic lift assists flight); Long jump: ~18-22° (the athlete must also maximise controlled forward rotation for landing).
- 运动专项角度：铅球：约 38-42°（因运动员身高，出手点高于落点）；标枪：约 33-36°（空气动力升力辅助飞行）；跳远：约 18-22°（运动员还必须为落地而最大化可控的前旋）。
12. Centre of Mass and Stability | 质量中心与稳定性
The centre of mass (also called centre of gravity) is the point where all the mass of a body is considered to be concentrated. In sports biomechanics, manipulating the position of the centre of mass relative to the base of support is crucial for enhancing stability or initiating movement. This concept governs everything from a wrestler's defensive stance to a high jumper's Fosbury Flop technique.
质量中心（也称重心）是物体全部质量被认为集中的点。在运动生物力学中，操控质量中心相对于支撑面的位置对于增强稳定性或启动运动至关重要。这一概念支配着一切，从摔跤手的防守姿势到跳高运动员的背越式跳高技术。
Stability increases when: Centre of mass is lower; centre of mass is central over the base of support; base of support is wider; line of gravity falls within the base of support.
稳定性在以下情况增强：质量中心较低；质量中心位于支撑面中心正上方；支撑面较宽；重力作用线落在支撑面内。
- Application – Defensive Stance: A basketball defender widens their feet (larger base of support), bends their knees (lowers centre of mass), and keeps their body weight evenly distributed. This maximises stability and allows rapid reactions.
- 应用——防守姿势：篮球防守者双脚分开（增大支撑面）、屈膝（降低质量中心）并均匀分配体重。这最大限度地提高了稳定性，并允许快速做出反应。
- Application – Sprint Start: In the 'set' position, a sprinter shifts their centre of mass forward, close to the edge of the base of support. This deliberately reduces stability to enable explosive forward acceleration upon the starting signal.
- 应用——短跑起跑：在“预备”姿势中，短跑运动员将质量中心前移，靠近支撑面边缘。这刻意降低了稳定性，以便在发令信号响起时实现爆发性前冲加速。
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