📚 IGCSE CIE Physical Education: Formula & Theorems Quick Reference Handbook | IGCSE CIE 体育：公式定理速查手册
To excel in IGCSE CIE Physical Education, you need more than just practical skill – you must be able to apply key formulas and scientific principles to analyse performance, training and the human body. This quick reference handbook collects every essential equation, theorem and calculation you are expected to know, from heart‑rate training zones and body composition to levers and fluid dynamics. Use it alongside your practical work and revision notes to reinforce your understanding and pick up marks in the written examination.
要想在 IGCSE CIE 体育考试中取得优异成绩,你不仅需要实践技能,还必须能够运用关键公式和科学原理来分析运动表现、训练过程和人体机能。这本速查手册汇集了你需要掌握的每一个重要公式、定理和计算方法,从心率训练区间和身体成分,到杠杆原理和流体动力学。请结合你的实践活动和复习笔记使用它,以巩固理解,在笔试中斩获分数。
1. Maximum Heart Rate | 最大心率
The simplest and most widely used method to estimate a person’s maximum heart rate (MHR) is the age‑based formula: MHR = 220 − age. This value represents the theoretical upper limit of how many times the heart can beat in one minute during all‑out exercise. Coaches and athletes rely on it to set training zones, although individual variations do exist – genetics, fitness level and cardiac size can push the true maximum slightly higher or lower. Even so, the 220−age rule remains the foundation for cardiovascular prescription in IGCSE PE.
估算个人最大心率(MHR)最简单且最常用的方法是基于年龄的公式：最大心率 = 220 – 年龄。该数值代表心脏在全天最高强度运动时一分钟内理论上的搏动上限。教练和运动员常用它来设定训练区间,尽管存在个体差异——基因、体能水平和心脏大小都可能使真实最大值略高或略低。即便如此,220 – 年龄法则仍然是 IGCSE 体育中心血管处方的基础。
Max HR = 220 − age
2. Karvonen Formula (Heart‑Rate Reserve) | 卡沃宁公式(心率储备)
A more personalised way to determine training heart rates is the Karvonen formula, which takes into account resting heart rate (RHR) and heart‑rate reserve (HRR). The formula is: Target HR = (Max HR − RHR) × desired intensity + RHR. Because RHR reflects an individual’s aerobic fitness, the calculated zone is tailored rather than relying solely on a percentage of maximum heart rate. For example, an athlete with RHR = 60 bpm and Max HR = 190 bpm who wants to train at 70% intensity would compute: (130 × 0.70) + 60 = 151 bpm.
确定训练心率的一种更加个性化的方式是卡沃宁公式,它考量了静息心率(RHR)和心率储备(HRR)。公式为：目标心率 = (最大心率 − 静息心率) × 期望强度 + 静息心率。由于静息心率反映了个体的有氧健身水平,计算出的区间更具针对性,而非单纯依赖最大肝率的百分比。例如,一名静息心率60次/分、最大心率190次/分的运动员若想以70%强度训练,则计算为：(130 × 0.70) + 60 = 151次/分。
Target HR = (Max HR − RHR) × Intensity% + RHR
3. Body Mass Index (BMI) | 身体质量指数
Body Mass Index is a quick screening tool that classifies individuals as underweight, normal weight, overweight or obese based on mass relative to height. The formula is BMI = mass (kg) ÷ height² (m²). Although BMI does not distinguish between fat mass and lean (muscle) mass, it is routinely used in population studies and initial health checks. In PE lessons, you may use BMI to discuss the limitations of a one‑size‑fits‑all measure for athletes with large muscle bulk.
身体质量指数是一种快速筛查工具,根据体重与身高的比例将人划分为体重不足、正常体重、超重或肥胖。公式为：BMI = 体重(kg) ÷ 身高²(m²)。虽然 BMI 无法区分脂肪重量和瘦体重(肌肉),但它常用于群体研究和初步健康检查。在体育课上,你可以用 BMI 来讨论这一“一刀切”指标对于肌肉发达的运动员有何局限性。
BMI = Weight (kg) ÷ [Height (m)]²
4. Percentage Body Fat Estimation (Skinfold Method) | 体脂率估算(皮褶厚度法)
Rather than relying on BMI, sport scientists often estimate body fat percentage from skinfold measurements at specific sites (e.g. biceps, triceps, subscapular, suprailiac). The sum of skinfolds is fed into a prediction equation, such as the Durnin‑Womersley or Jackson‑Pollock formula, which varies by age and gender. While the exact equation is beyond IGCSE scope, you need to understand that lower skinfold sums indicate a leaner physique. A typical conversion for a young male might be: %Fat = (0.29288 × sum of four skinfolds) − (0.0005 × sum²) + (0.15845 × age) − 5.76377. This highlights that body composition assessment is a multi‑step calculation.
与依赖 BMI 不同,运动科学家常通过特定部位(如肱二头肌、肱三头肌、肩胛下、髂棘上)的皮褶厚度来估算体脂率。皮褶厚度之和被代入预测方程,如 Durnin‑Womersley 或 Jackson‑Pollock 公式,方程因年龄与性别而异。尽管精确的方程超出了 IGCSE 的范畴,但你需要理解：皮褶总和越低,意味着体型越精瘦。对于年轻男性,一种常见的换算为：体脂% = (0.29288 × 四处皮褶之和) − (0.0005 × 总和²) + (0.15845 × 年龄) − 5.76377。这凸显出身体成分测评是一个多步计算过程。
5. Energy Expenditure and METs | 能量消耗与代谢当量
Metabolic equivalent (MET) is a unit that represents the energy cost of physical activity. One MET is the rate of energy expenditure at rest, equivalent to an oxygen consumption of 3.5 ml·kg⁻¹·min⁻¹. To find the energy burned during an activity: kcal = MET × body mass (kg) × duration (hours). For instance, jogging at 7 METs for 30 minutes by a 70 kg person burns approximately 245 kcal (7 × 70 × 0.5). This formula helps design exercise programmes and balance dietary intake with output.
代谢当量(MET)是表示体力活动能耗的单位。1 MET 为静息时的能量消耗率,相当于每公斤体重每分钟摄氧 3.5 毫升。计算活动中消耗的热量：千卡 = MET × 体重(kg) × 时长(小时)。例如,一个体重 70 公斤的人以 7 MET 的强度慢跑 30 分钟,大约消耗 245 千卡(7 × 70 × 0.5)。该公式有助于制定运动计划,并使膳食摄入与消耗达到平衡。
Energy (kcal) = MET × Mass (kg) × Time (h)
6. Levers and Moment (Torque) | 杠杆与力矩(扭矩)
The human musculoskeletal system operates as a series of levers. A lever consists of a fulcrum, an effort force and a resistance load. The moment of a force about a joint equals the magnitude of the force multiplied by its perpendicular distance from the fulcrum: Moment = Force × Distance. Class 1, 2 and 3 levers distribute the fulcrum, load and effort differently, affecting mechanical advantage (MA = effort arm ÷ resistance arm). In sports, a third‑class lever (e.g. biceps curl) favours speed and range of movement over strength, while a second‑class lever (e.g. calf raise) favours force production.
人体肌肉骨骼系统相当于一系列的杠杆。杠杆由支点、动力和阻力构成。力对关节产生的力矩等于力的大小乘以该力到支点的垂直距离：力矩 = 力 × 力臂。第 1、2、3 类杠杆对支点、负荷与动力的安排各不相同,从而影响机械利益(机械利益 = 动力臂 ÷ 阻力臂)。在运动中,第三类杠杆(如肱二头肌弯举)以牺牲力量为代价换取速度和活动范围,而第二类杠杆(如提踵)则有利于力量输出。
Moment (Nm) = Force (N) × Perpendicular Distance (m)
7. Linear and Angular Motion | 线运动和角运动
Basic kinematic equations underpin every movement analysis. For linear motion: speed = distance ÷ time, velocity = displacement ÷ time, and acceleration = (final velocity − initial velocity) ÷ time. For angular motion, analogous quantities are angular velocity (ω = angular displacement ÷ time) and angular acceleration. Momentum (mass × velocity) is conserved in collisions, which explains principles of tackling and ball impact. Understanding these allows you to calculate and compare performance variables during sprinting, throwing or cycling.
基本运动学方程是每一项动作分析的基石。在线运动中：速率 = 路程 ÷ 时间,速度 = 位移 ÷ 时间,加速度 = (末速度 − 初速度) ÷ 时间。在角运动中,类似的物理量有角速度(ω = 角位移 ÷ 时间)和角加速度。动量(质量 × 速度)在碰撞中守恒,这解释了擒抱和球体冲击的原理。理解了这些,你就能在短跑、投掷或骑行中计算并比较各项表现变量。
Speed = d ÷ t | Velocity = Δs ÷ Δt | Acceleration = (v − u) ÷ t
8. Force, Work, Power and Efficiency | 力、功、功率与效率
Force = mass × acceleration represents Newton’s second law and is the driving principle behind muscle contraction moving limbs. Work done = Force × displacement (in the direction of the force). Power, the rate of doing work, equals work ÷ time or force × velocity. Mechanical efficiency in sports is the ratio of useful work output to total energy expended, expressed as a percentage: Efficiency = (Mechanical work ÷ Energy expended) × 100. These formulas connect biomechanics with physiology when assessing cycling economy or running economy.
力 = 质量 × 加速度代表牛顿第二定律,是肌肉收缩带动肢体运动的核心原理。做功 = 力 × 位移(沿力方向)。功率,即做功的速率,等于 功 ÷ 时间,也可表示为 力 × 速度。运动中的机械效率是有用功输出与总能量消耗之比,以百分数表示：效率 = (机械功 ÷ 能量消耗) × 100。这些公式在评估骑行经济性或跑步经济性时,将生物力学与生理学联系起来。
F = m × a | W = F × d | P = W ÷ t = F × v | Efficiency = (W_useful ÷ E_total) × 100%
9. Aerobic and Anaerobic Thresholds (Training Zones) | 有氧与无氧阈值(训练区间)
Exercise intensity is often prescribed as a percentage of maximal heart rate or heart‑rate reserve. The aerobic threshold (AeT) is the point where blood lactate begins to rise above resting levels but can still be cleared; it occurs around 60-70% of Max HR. The anaerobic threshold (AT), or lactate turnpoint, at approximately 80-90% Max HR, marks the intensity where lactate accumulates rapidly. Training zones derived from these thresholds guide endurance work, tempo runs and high‑intensity intervals. Practically, you can estimate AT ≈ 85-90% HRmax for simple programming.
运动强度常以最大心率或心率储备的百分比来制定。有氧阈值(AeT)是血乳酸开始升高但仍能被清除的点,大约发生在最大心率的60-70%。无氧阈值(AT),或称乳酸拐点,约在最大心率的80-90%,标志着乳酸急剧积累的强度。从这些阈值衍生出的训练区间指导耐力训练、节奏跑和高强度间歇训练。在实践中,你可以简略地以 AT ≈ 85-90% 最大心率来进行训练计划。
AT ≈ 85−90% Max HR | AeT ≈ 60−70% Max HR
10. Fluid Mechanics in Sport (Bernoulli and Magnus) | 体育运动中的流体力学(伯努利与马格努斯)
Bernoulli’s principle states that an increase in the velocity of a fluid (air) occurs simultaneously with a decrease in pressure. This explains why an aerofoil shape generates lift – and why a spinning ball curves. The Magnus effect is the perpendicular force exerted on a rotating object moving through a fluid: the spin alters the pressure distribution, causing the ball to swerve. In football and tennis, top‑spin makes the ball dip, while back‑spin creates a floating trajectory. The ideal projectile formula, without air resistance, gives range = (v² × sin(2θ)) ÷ g, but spin complicates the real‑world flight.
伯努利原理指出,流体(空气)流速的增加与压力的降低同时发生。这解释了为何翼型会产生升力,以及旋转球体何以弯曲。马格努斯效应是作用于旋转物体的垂直力：旋转改变了压力分布,使球体偏离直线。在足球和网球中,上旋让球快速下坠,而下旋则产生飘浮轨迹。理想抛射体公式(忽略空气阻力)为：射程 = (v² × sin(2θ)) ÷ g,但旋转使实际飞行复杂化。
Range (no spin) = (v² sin2θ) ÷ g | Bernoulli: velocity ↑ → pressure ↓
11. Hydration and Sweat‑Rate Calculation | 水合与出汗率计算
Athletes can estimate fluid loss during exercise by measuring pre‑ and post‑exercise body mass. Sweat rate = (mass loss + fluid intake − urine output) ÷ exercise time. Rehydration should target 150% of the mass lost to restore fluid balance within 4–6 hours. For example, if an athlete loses 1.2 kg during a 2‑hour game and drinks 500 ml, sweat rate ≈ (1.2 + 0.5) kg ÷ 2 h = 0.85 L/h. This is a crucial practical skill in endurance sports to prevent heat illness and performance decline.
运动员可通过测量运动前后的体重来估算体液流失。出汗率 = (体重减少量 + 饮水量 − 排尿量) ÷ 运动时间。补水应以体重损失量的150%为目标,以便在4-6小时内恢复体液平衡。例如,一名运动员在2小时的比赛中减重1.2公斤并喝水500毫升,则出汗率 ≈ (1.2 + 0.5) 公斤 ÷ 2 小时 = 0.85 升/小时。这是耐力运动中预防热疾病和表现下降的关键实践技能。
Sweat rate (L/h) = (Pre‑exercise mass − Post‑exercise mass + Fluid intake) ÷ Time
12. Training Principles and the FITT Formula | 训练原则与 FITT 公式
While not a mathematical theorem, the FITT principle encapsulates the key variables of an exercise programme: Frequency, Intensity, Time and Type. Applying FITT alongside progressive overload ensures adaptation occurs safely. For cardiovascular fitness, a typical prescription might be 3–5 times per week (Frequency), at 60–85% HRR (Intensity), for 20–60 minutes (Time), using rhythmical, large‑muscle activities (Type). To calculate intensity using the Karvonen formula (see Section 2), you plug the chosen percentage into the heart‑rate reserve equation, demonstrating how FITT and physiology formulas are interlinked in training design.
虽然 FITT 不算数学定理,但它概括了运动计划的关键变量：频率(Frequency)、强度(Intensity)、时间(Time)和类型(Type)。将 FITT 与渐进超负荷结合使用,可确保身体安全地获得适应。对于心血管健康,典型的处方可能是：每周3-5次(频率),60-85%心率储备(强度),每次20-60分钟(时间),采用节律性的大肌肉群活动(类型)。在利用卡沃宁公式计算强度时(参见第2节),你把选定的百分比代入心率储备方程,这充分体现了 FITT 与生理学公式在训练设计中的内在联系。
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