📚 Year 11 WJEC PE: Quick Reference Formulas & Theorems Handbook | WJEC 体育：公式定理速查手册
This handbook compiles every essential formula, equation and theorem you need to memorise for the Year 11 WJEC Physical Education specification. From cardiovascular calculations and respiratory volumes to Newton's laws and lever mechanics, the reference is structured to mirror the demands of the exam – equations are presented with worked context, and each concept is explained in direct, bilingual fashion. Use it alongside your revision notes to secure the marks that depend on accurate numerical application.
本手册汇总了 Year 11 WJEC 体育学科需要掌握的所有核心公式、方程与定理。内容涵盖心血管计算、呼吸气量、牛顿定律以及杠杆力学，完全贴合考试要求。每条公式都配有应用情境说明，并以中英双语对照呈现。请配合复习笔记使用，确保你在需要精确数值应用的题目中稳拿分数。
1. Cardiovascular Formulas | 心血管公式
The simplest way to estimate an individual's maximum heart rate (HRmax) is to subtract their age from 220. This empirical formula provides a baseline that coaches and athletes use to define training intensities, although individual variation means it is an approximation rather than a precise physiological limit.
估算个人最大心率（HRmax）最常用的方法是用 220 减去年龄。这一经验公式为教练和运动员划分训练强度提供了基准，但由于存在个体差异，它只是一个近似值，并非精确的生理极限。
HRmax = 220 – age
Once HRmax is known, you can calculate aerobic and anaerobic training zones as percentages of that value. For general aerobic development, a zone of 60–80% HRmax is recommended; for high-intensity anaerobic work, the range is typically 80–90% HRmax. Being able to quote and apply these percentages is a common exam requirement.
得到最大心率后，可按其百分比划定有氧与无氧训练区间。一般有氧能力发展推荐使用 60%–80% HRmax 的区间；高强度无氧训练通常采用 80%–90% HRmax。能够引用并运用这些百分比值是考试中常见的要求。
2. Karvonen Formula: Heart Rate Reserve | 卡式公式：心率储备
The Karvonen method refines target heart rate calculations by taking resting heart rate (HRrest) into account. The heart rate reserve (HRR) is the difference between maximal and resting heart rate. Adding a percentage of this reserve back to HRrest gives a more personalised training zone that reflects cardiovascular fitness.
卡氏公式通过引入安静心率（HRrest）来细化靶心率的计算。心率储备（HRR）为最大心率与安静心率之差。将一定百分比的心率储备加回到安静心率，就能得出更能反映个人心血管体能的训练区间。
HRR = HRmax – HRrest
Target HR = (HRR × %intensity) + HRrest
For example, a 16-year-old with a resting heart rate of 60 bpm training at 70% e formula is highly relevant for WJEC questions on designing personal exercise programmes.
例如，一名 16 岁、安静心率为 60 bpm 的学生以 70% 强度训练，其最大心率为 204 bpm，心率储备为 144 bpm，靶心率为 (144×0.7)+60 = 160.8 bpm。该公式在 WJEC 的个人锻炼计划设计类题目中相关性极强。
3. Respiratory Calculations | 呼吸计算
Minute ventilation (VE) is the volume of air moved into and out of the lungs each minute. It is the product of tidal volume (TV) and breathing frequency (f). Understanding this relationship helps explain how ventilation adjusts to meet the oxygen demands of exercise.
每分钟通气量（VE）是指每分钟进出肺部的空气总量。它是潮气量（TV）与呼吸频率（f）的乘积。理解这一关系有助于解释通气如何随运动时的氧需求进行调整。
VE = TV × f
At rest, tidal volume might be around 0.5 litres per breath and frequency about 12 breaths per minute, giving a minute ventilation of 6 L/min. During maximal exercise, TV can increase to 3–4 litres and f to 40–60 breaths per minute, raising VE above 150 L/min. This dramatic change illustrates the respiratory system's capacity.
安静时，潮气量大约为每次呼吸 0.5 升，频率约为 12 次/分钟，每分钟通气量为 6 L/min。在最大强度运动时，潮气量可增至 3–4 升，频率升至 40–60 次/分钟，使得 VE 超过 150 L/min。这一剧烈变化体现了呼吸系统的储备能力。
4. Body Composition and Basal Metabolic Rate | 身体成分与基础代谢率
Body Mass Index (BMI) offers a quick screening tool for weight categories. It is calculated by dividing weight in kilograms by height in metres squared. Although BMI does not distinguish fat from muscle, it remains a standard measure in health and fitness screening.
身体质量指数（BMI）是一种快速筛查体重类别的方法。其计算方法为体重（千克）除以身高（米）的平方。尽管 BMI 无法区分脂肪与肌肉，但它仍是健康与体适能筛查中的一项标准指标。
BMI = weight (kg) ÷ height² (m)
) × 24 (kcal/day), adjusted by a factor of 0.9 for females. More accurate formulas exist, but this version is sufficient to illustrate daily energy needs.
基础代谢率（BMR）是机体在完全安静状态下消耗的能量。运动科学中常使用一种简化的估算方法：BMR = 体重（公斤）× 24（千卡/天），女性乘上系数 0.9。更精确的公式虽然存在，但这个版本已足以说明每日能量需求。
5. Speed, Velocity and Acceleration | 速率、速度与加速度
Speed is a scalar quantity measuring the rate of motion, while velocity is a vector that includes direction. For most GCSE movement analysis, average speed is distance divided by time. Acceleration is defined as the rate of change of velocity.
速率是标量，衡量运动的快慢；速度是矢量，包含方向。在 GCSE 运动分析中，平均速率通常为距离除以时间。加速度则定义为速度的变化率。
average speed = total distance ÷ time
acceleration = (final velocity – initial velocity) ÷ time
33 m/s. If the sprinter accelerates from 0 to 10 m/s in 3 seconds, the acceleration is (10–)/3 ≈ 3.33 m/s². These calculations underpin biomechanical analysis of sports performance.
例如，一名百米短跑运动员用 12 秒完成比赛，平均速率为 8.33 m/s。若该运动员在 3 秒内从 0 加速到 10 m/s，其加速度为 (10–0)/3 ≈ 3.33 m/s²。这些计算是运动生物力学分析的基础。
6. Newton's Laws of Motion and Force Calculations | 牛顿运动定律与力的计算
Newton’s three laws form the theoretical backbone of all movement. The first law (inertia) states that an object will remain at rest or in uniform motion unless acted upon by a resultant force. The second law quantifies this relationship, and the third law declares that every action has an equal and opposite reaction.
牛顿三大定律是所有运动的理论支柱。第一定律（惯性定律）指出，除非受到合外力作用，否则物体将保持静止或匀速直线运动。第二定律对这一定量关系进行了量化，而第三定律阐明每个作用力都有一个大小相等、方向相反的反作用力。
F = m × a
Weight = m × g (gravitational field strength, 9.8 m/s² on Earth)
th a mass of 110 kg and accelerating at 2 m/s² generates a force of 220 N. The same player' erstanding the difference between mass and weight is essential for correct force diagrams.
在 scrum 中，一名体重 110 kg 的支柱前锋以 2 m/s² 加速推进，产生的力为 220 N。同一名运动员的体重为 110 × 9.8 = 1078 N。理解质量与重量的区别是正确绘制受力分析图的关键。
7. Momentum and Impulse | 动量与冲量
Momentum is the product of an object's mass and velocity. It is a vector quantity, meaning direction matters. The impulse experienced by a body equals the change in momentum, which also equals the average force multiplied by the time for which it acts.
动量是物体质量与速度的乘积。作为矢量，方向至关重要。物体所受的冲量等于动量的变化，也等于平均作用力乘以力的作用时间。
p = m × v
Impulse = F × t = Δp
rce experienced is 400 / 0.4 = 1000 N. Understanding impulse helps explain injury prevention through padded equipment and longer stopping times.
在橄榄球擒抱中，一名质量 80 kg、以 5 m/s 速度前进的运动员动量是 400 kg·m/s。若擒抱使其在 0.4 秒内停下，则受到的平均冲力为 400 / 0.4 = 1000 N。理解冲量有助于解释如何通过护具和延长停定时间来预防受伤。
8. Levers and Moments | 杠杆与力矩
A lever system consists of a rigid bar, a pivot (fulcrum), an effort force and a load. The turning effect produced is called the moment, calculated by multiplying the force applied by the perpendicular distance from the fulcrum. Levers are classified according to the relative positions of these components.
杠杆系统由刚性杆、支点、动力和阻力构成。所产生的转动效应称为力矩，其值为施加的力乘以力到支点的垂直距离。根据三者的相对位置，杠杆可分为不同类别。
Moment = force × perpendicular distance from fulcrum
Mechanical advantage = effort arm ÷ resistance arm
- First class: fulcrum between effort and load (e.g. neck extension in heading a football).
- Second class: load between fulcrum and effort (e.g. calf raises – ball of foot as fulcrum, body weight as load).
- Third class: effort between fulcrum and load (e.g. biceps curl – elbow as fulcrum, biceps effort between elbow and hand).
- 第一类杠杆：支点在力点与重点之间（如头球时颈部后伸）。
- 第二类杠杆：重点在支点与力点之间（如提踵——脚前掌为支点，体重为重）。
- 第三类杠杆：力点在支点与重点之间（如肱二头肌弯举——肘为支点，肱二头肌力在肘与手之间）。
Most skeletal muscles operate as third-class levers, which favour range and speed of movement over force. The moment of a biceps pulling at 300 N with a distance of 0.04 m from the elbow is 12 N m. Lever calculations regularly appear in biomechanics problems on the WJEC paper.
人体骨骼肌大多为第三类杠杆，以牺牲力为代价换取活动幅度和速度。如果肱二头肌以 300 N 的力在距离肘关节 0.04 m 处拉，产生的力矩为 12 N·m。杠杆计算频繁出现在 WJEC 试卷的生物力学问题中。
9. Training Thresholds and Energy Expenditure | 训练界限与能量消耗
Training zones are normally prescribed using percentages of HRmax. Aerobic training typically targets 60–80% HRmax, while anaerobic threshold work sits at 80–90%. Calculating these zones from a known HRmax allows coaches to individualise conditioning programmes.
训练区间通常依据 HRmax 的百分比来制定。有氧训练一般将心率控制在 60%–80% HRmax，无氧阈训练则在 80%–90%。根据已知最大心率计算出这些区间，能让教练针对个人调整训练方案。
Energy expenditure during physical activity can be estimated using the metabolic equivalent (MET). One MET equals a resting oxygen consumption of approximately 3.5 ml O₂ per kg of body mass per minute. The total kilocalories burned is a function of MET value, body mass and duration.
体育活动中的能量消耗可用代谢当量（MET）来估算。1 MET 相当于安静时的摄氧量，约 3.5 ml O₂/kg/min。消耗的总千卡数是 MET 值、体重和持续时间的函数。
kcal = METs × body mass (kg) × time (hours)
threshold calculations with MET-based energy expenditure gives a scientific foundation for designing weight-management and fitness programmes in the PE course.
例如，一名 60 kg 的学生以 7 METs 慢跑 0.5 小时，消耗 7 × 60 × 0.5 = 210 kcal。将训练界限计算与基于 METs 的能量消耗结合起来，就能为 PE 课程中的体重管理和健身计划设计提供科学依据。
10. Practical Application Summary | 综合应用要点
To succeed in WJEC Year 11 PE, you must be able to select the correct formula from a given scenario, substitute data accurately, and interpret the result in a sporting context. Common pitfalls include using speed instead of velocity for momentum questions or confusing vector and scalar quantities in free-body diagrams.
要在 WJEC 体育 Year 11 考试中取得好成绩，必须能够根据给定情境选择正确公式、准确代入数据，并能在运动背景下解读结果。常见错误包括在动量问题中使用速率替代速度，或在受力图中混淆矢量和标量。
Remember that units carry meaning: force is in newtons (N), moment in newton metres (N m), momentum in kilogram metres per second (kg m/s), and energy in kilocalories or joules. Always check that distances are in metres when calculating moments, and that heart rate values are in beats per minute before applying the Karvonen formula.
记住，单位本身就带有含义：力的单位是牛顿（N），力矩为牛顿·米（N·m），动量为千克·米/秒（kg·m/s），能量为千卡或焦耳。计算力矩时需确保距离以米为单位；在应用卡氏公式前，确认心率数值的单位是“次/分钟”。
This handbook provides a concise yet comprehensive reference. Practise these formulas repeatedly with varied data sets until they become second nature, and you will approach the biomechanics and physiology questions with far greater confidence.
本手册提供了精炼而又全面的速查资料。请反复使用不同数据练习这些公式，直到完全熟稔为止；届时你在应对生物力学和生理学问题时将更有信心。
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