📚 Formula & Theorem Quick Reference Handbook for Year 11 Cambridge Physical Education | Year 11 剑桥体育公式定理速查手册
This quick reference handbook brings together all the essential formulas, key principles, and biomechanical theorems that Year 11 Cambridge IGCSE Physical Education students need to master. Each section pairs a clear English explanation with its Chinese counterpart, presenting the core equations and scientific laws that underpin human movement, training effects, and health indicators. Use this handbook to revise the numerical relationships, apply biomechanics to sporting examples, and build confidence for both paper 1 and paper 2 examinations.
本速查手册汇总了Year 11剑桥IGCSE体育课程中学生必须掌握的所有重要公式、关键原理与生物力学定理。每个部分都以英文解释与对应中文阐述成对出现，呈现支撑人体运动、训练效应和健康指标的核心方程与科学定律。利用本手册复习数量关系，将生物力学运用于运动实例，同时为试卷一和试卷二考试积累信心。
1. Maximum Heart Rate and Target Heart Rate Zones | 最大心率与靶心率区间
Maximum heart rate (HRmax) is estimated using the simple formula: HRmax the upper limit of cardiac beats per minute during exhaustive exercise. Target heart rate zones for aerobic training are typically 60–80% of HRmax. For interval training the upper zone may reach 85–90% of HRmax.
最大心率（HRmax）使用简单公式估算：HRmax = 220 − 年龄。该预测给出力竭运动时每分钟心跳的上限。有氧训练的靶心率区间通常为HRmax的60–80%。间歇训练的上限区间可达HRmax的85–90%。
HRmax = 220 − age
Target HR for aerobic training: 60%–80% × (220 − age)
ormula uses heart rate reserve to refine target zones: Target HR = Resting HR + (Intensity %) × (HRmax − Resting HR). This method gives more personalised training loads and is widely used in prescription of endurance programmes.
卡沃宁公式利用储备心率细化靶区：靶心率 = 静息心率 + (强度百分比) × (最大心率 − 静息心率)。该方法提供更加个性化的训练负荷，广泛用于耐力训练计划制定。
2. Body Mass Index (BMI) | 身体质量指数 (BMI)
e by relating mass to height. The formula is: BMI = mass (kg) ÷ height² (m). A BMI of 18.5– dered healthy for adults. Athletes with high muscle mass may have a misleadingly high BMI.
身体质量指数通过体重与身高的关系将个体分为偏瘦、正常、超重或肥胖。公式为：BMI = 体重 (kg) ÷ 身高² (m)。对成年人而言，18.5–24.9 kg/m² 视为健康。肌肉量高的运动员可能呈现误导性的高BMI值。
BMI = mass (kg) / height² (m)
s within the normal range. Coaches use BMI alongside waist-to-hip ratio and skinfold measurements to assess body composition more accurately.
例如，一名体重70 kg、身高1.75 m的运动员，其BMI为70 ÷ (1.75 × 1.75) = 22.9 kg/m²，处于正常范围。教练将BMI与腰臀比及皮褶厚度测量结合使用，以更准确地评价身体成分。
3. Energy Expenditure and METs | 能量消耗与代谢当量 (MET)
3.5 ml of oxygen per kg of body mass per minute. Energy expenditure (kcal) = MET × mass (kg) × time (h). This formula allows fitness professionals to estimate caloric burn from different exercises.
代谢当量（MET）表示身体活动的能量消耗。1 MET定义为静息代谢率，约每公斤体重每分钟消耗3.5毫升氧气。能量消耗（千卡）= MET × 体重 (kg) × 时间 (h)。此公式让健身专业人士能估算不同运动的卡路里消耗。
Energy expenditure (kcal) = MET × mass (kg) × time (h)
ning at 10 km/h = 10 METs) dramatically increase energy expenditure, which is crucial for weight management programmes.
以5 km/h步行时MET值约为3.5，那么一名60 kg的人步行1.5小时大约消耗3.5 × 60 × 1.5 = 315千卡。较高的MET值（如10 km/h跑步 = 10 MET）会显著提高能量消耗，这对体重管理方案至关重要。
4. Speed, Distance, and Time | 速度、距离与时间
t events, athletes and coaches use this relationship to calculate split times, predict finish times, and analyse pacing strategies. Instantaneous speed may vary during a race, but average speed provides a simple performance indicator.
基本运动学方程将平均速度、距离和时间联系起来：速度 = 距离 ÷ 时间。在短跑项目中，运动员和教练利用此关系计算分段用时、预测完赛时间并分析配速策略。比赛中瞬时速度可能变化，但平均速度是一个简单的表现指标。
Average speed (m/s) = distance (m) / time (s)
utes per kilometre, which is the inverse relationship: Pace (min/km) = Time (min) ÷ Distance (km). Understanding both forms helps in interpreting sports performance data.
若100米短跑选手以10.5秒完赛，平均速度为100 ÷ 10.5 ≈ 9.52 m/s。对于马拉松等长距离项目，配速常以每公里分钟数表示，这是倒数关系：配速 (min/km) = 时间 (min) ÷ 距离 (km)。理解这两种形式有助于解读运动表现数据。
5. Force, Mass, and Acceleration (Newton's Second Law) | 力、质量与加速度（牛顿第二定律）
Newton' × a. In sport, this principle explains why a footballer kicking a ball generates greater acceleration when a larger force is applied, and why lighter athletes accelerate more quickly when the same force is produced.
牛顿第二定律指出，物体的加速度与所受合外力成正比，与其质量成反比：F = m × a。在运动中，该原理解释了为什么足球运动员施加更大的力时能产生更大的球的加速度，以及为什么相同力量下较轻的运动员加速更快。
F = m × a
2632 m/s². Understanding this relationship also governs resistance training: to develop maximal strength, athletes must increase the force they can exert, which requires handling progressively heavier masses.
当网球运动员以150 N的力击球（质量约0.057 kg），加速度为 a = F / m = 150 / 0.057 ≈ 2632 m/s²。理解这层关系也指导着抗阻训练：为了发展最大力量，运动员必须提高能施加的力，这需要处理逐步增加的更大质量。
6. Work, Energy, and Power | 功、能量与功率
s, power (P) is the rate of doing work: P = W / t. These equations quantify the mechanical efficiency of sporting actions such as jumping, throwing, and cycling.
功（W）等于力与沿力方向移动距离的乘积：W = F × d。能量是做功的能力。力学上，功率（P）是做功的速率：P = W / t。这些方程量化了跳跃、投掷和骑行等运动动作的机械效率。
Work = Force × distance W = F × d
Power = Work / time P = W / t
In a vertical jump, the work done equals the product of body weight (force due to gravity) and the vertical displacement of the centre of mass. Power output is critical in explosive sports; measuring it helps to design plyometric training programmes aimed at improving the rate of force development.
在垂直跳跃中，所做的功等于体重（重力引起的力）与质心垂直位移的乘积。功率输出在爆发力项目运动中至关重要；测量它有助于设计旨在提高力量发展速率的增强式训练方案。
7. Torque and Levers | 力矩与杠杆
nce from axis. In the human body, joints act as fulcrums, muscles produce effort forces, and bones serve as lever arms. The three classes of levers alter the trade-off between force and speed of movement.
力矩（力之转动效应）是由力在距旋转轴一定距离处作用产生的转动效果：力矩 = 力 × 到轴的垂直距离。在人体中，关节充当支点，肌肉产生动力，骨骼作为杠杆臂。三类杠杆改变了力量与运动速度之间的取舍。
Torque (Nm) = Force (N) × Distance (m)
A first-class lever (e.g. neck extension) has the fulcrum between effort and load; a second-class lever (e.g. standing calf raise) places the load between fulcrum and effort, favouring force; a third-class lever (e.g. biceps curl) places the effort between fulcrum and load, favouring speed and range of motion. Most skeletal muscles operate as third-class levers.
第一类杠杆（如颈部伸展）支点在动力与阻力之间；第二类杠杆（如站姿提踵）阻力在支点和动力之间，有利于省力；第三类杠杆（如肱二头肌弯举）动力在支点和阻力之间，有利于速度和运动幅度。大多数骨骼肌作为第三类杠杆工作。
8. Momentum and Impulse | 动量与冲量
Momentum (p) is the product of an object' e concepts are essential for analysing collisions, tackles, and striking actions in sport.
动量（p）是物体质量与其速度的乘积：p = m × v。冲量是由一段时间内施加的力引起的动量变化：冲量 = 力 × 时间 = Δp。这些概念对分析运动中的碰撞、擒抱和击打动作至关重要。
Momentum = mass × velocity p = m × v
Impulse = F × t = m × (v − u)
In a golf drive, the club head momentum is transferred to the ball. Increasing the follow-through time increases the impulse for the same average force, resulting in greater ball velocity. Coaches use this principle to teach the importance of accelerating through impact rather than decelerating at contact.
在高尔夫开球中，杆头动量传递给球。增加随挥时间可在相同平均力下增大冲量，从而得到更大的球速。教练运用此原理讲解在击球瞬间加速通过而非减速接触的重要性。
9. VO₂ Max and Aerobic Capacity | 最大摄氧量与有氧能力
ch as the Cooper 12-minute run test: VO₂ max (ml/kg/min) ≈ (distance covered in metres − 504.9) ÷ 44.73. Laboratory measurement uses the Fick equation: VO₂ = cardiac output × arteriovenous oxygen difference.
最大摄氧量（VO₂ max）是身体每分钟每公斤体重所能利用的最大氧气量。通常借助预测公式估算，例如库珀12分钟跑测试：VO₂ max (ml/kg/min) ≈ (跑步距离米数 − 504.9) ÷ 44.73。实验室测量使用菲克方程：VO₂ = 心输出量 × 动静脉氧差。
VO₂ max ≈ (Distance (m) − 504.9) / 44.73 (Cooper test)
VO₂ = Q × (a-v O₂ diff)
Improving VO₂ max is a primary goal in endurance training. Higher values indicate superior oxygen delivery and utilisation, enabling athletes to sustain higher intensities. Altitude training, interval training, and continuous running all target this physiological variable.
提高最大摄氧量是耐力训练的首要目标。数值越高表明氧输送和利用能力越强，使运动员能维持更高强度。高原训练、间歇训练和持续跑都以此生理变量为目标。
10. Fluid Mechanics: Drag and Buoyancy | 流体力学：阻力与浮力
Drag force opposing motion through a fluid (air or water) is given by: Fdrag = ½ × Cd × ρ × A × v², where Cd is the drag coefficient, ρ is fluid density, A is frontal area, and v is velocity. Reducing frontal area or streamlining body positions decreases drag, improving speed in cycling, swimming, and sprinting.
在流体（空气或水）中阻碍运动的阻力表示为：Fdrag = ½ × Cd × ρ × A × v²，其中Cd为阻力系数，ρ为流体密度，A为迎风面积，v为速度。减小迎风面积或流线型身体姿态可降低阻力，从而提高自行车、游泳和短跑的速度。
Fdrag = ½ × Cd × ρ × A × v²
Buoyancy is determined by Archimedes' principle: an object immersed in a fluid experiences an upward force equal to the weight of fluid displaced. Swimmers with higher body fat percentage experience greater buoyancy, reducing energy cost to maintain horizontal position. Wetsuits increase buoyancy and lower drag, enhancing performance.
浮力由阿基米德原理决定：浸入流体中的物体受到一个向上的力，等于被排开流体的重量。体脂百分比较高的游泳者会感受到更大的浮力，降低了保持水平体位的能量消耗。防寒泳衣增加浮力并降低阻力，提升表现。
11. Centre of Mass and Stability | 质心与稳定性
The centre of mass (COM) is the point where the body's mass is concentrated. Stability increases when the line of gravity falls within the base of support, when the base of support is widened, when the COM is lowered, and when body mass is increased. The principle of moments applies: for stability, the clockwise torque must equal anticlockwise torque about any pivot.
质心（COM）是身体质量集中点。当重力线落在支撑面之内、支撑面加宽、重心降低、身体质量增加时，稳定性增大。力矩原理在此适用：为保持稳定，绕任何支点的顺时针力矩必须等于逆时针力矩。
Stability: line of gravity within base of support; low COM; wide base; greater mass
In contact sports, players adopt a crouched stance to lower COM and widen base, increasing resistance to tackles. In gymnastics, a handstand requires the athlete to control the COM directly above a narrow base, demanding fine motor skill. Understanding these principles helps coaches teach safe and effective techniques.
在身体对抗体育项目中，运动员采用蹲姿以降低质心、加宽支撑面，增强对抗铲球的稳定性。在体操中，倒立要求运动员在极窄的支撑面上方直接控制质心，需要精细运动技巧。理解这些原理有助于教练教授安全有效的技术。
12. Training Load and TRIMP | 训练负荷与训练冲量 (TRIMP)
t rate reserve: TRIMP = duration (min) × ΔHR ratio × a weighting factor (y) where ΔHR ratio = (HRavg − HRrest) / (HRmax − HRrest) and y typically equals 0.64e1.92 × ΔHR ratio for men and 0.86e1.67 × ΔHR ratio for women. This nonlinear formula reflects the exponentially higher physiological stress at higher intensities.
训练冲量（TRIMP）通过将训练课持续时间与平均心率响应相乘来量化训练负荷。常用版本使用储备心率：TRIMP = 持续时间 (min) × ΔHR比率 × 权重因子 (y)，其中ΔHR比率 = (HRavg − HRrest) / (HRmax − HRrest)，对于男性y通常为0.64e1.92 × ΔHR比率，女性为0.86e1.67 × ΔHR比率。此非线性公式反映出高强度下呈指数增加的生理应激。
TRIMP = t × ΔHR ratio × y
Coaches and sports scientists use TRIMP to periodise training, avoid overtraining, and ensure progressive overload. A similar concept, session RPE load, uses the rating of perceived exertion multiplied by duration to give a simple field-based alternative when heart rate monitors are unavailable.
教练和运动科学家使用TRIMP进行训练周期规划，避免过度训练并确保渐进超负荷。一个类似概念——课次RPE负荷——使用主观疲劳感觉评分乘以持续时间，在没有心率监测器时提供了一个简便的现场替代方案。
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