📚 GCSE CAIE Physical Education: Quick-Reference Formula & Theorem Handbook | GCSE CAIE 体育：公式定理速查手册
Welcome to your ultimate quick-reference guide for GCSE CAIE Physical Education. This handbook compiles all the essential formulas, theorems, and principles you need to master the theoretical aspects of the course. Use it to reinforce your understanding, solve numerical problems, and connect physiological concepts to real-world sport and exercise.
欢迎来到 GCSE CAIE 体育学科的终极速查手册。本手册汇集了课程理论部分你需要掌握的所有核心公式、定理和原理。用它来巩固理解、解决数值问题，并将生理学概念与真实的运动锻炼场景联系起来。
1. Maximum Heart Rate (MHR) | 最大心率（MHR）
Maximum heart rate is the highest number of beats per minute your heart can achieve during maximal exercise. It is commonly estimated using the simple age-based formula, which provides a baseline for setting training zones.
最大心率是指你在极限运动中心脏每分钟所能达到的最高搏动次数。通常基于年龄的简单公式进行估算，为设定训练区间提供基准。
MHR = 220 − Age (in years)
For example, a 16‑year‑old athlete would have an estimated MHR of 220 − 16 = 204 bpm. Remember this is an estimation, and individual variation exists.
例如，一名 16 岁运动员的估算最大心率为 220 − 16 = 204 次/分。请记住这只是估算值，个体之间存在差异。
2. Aerobic and Anaerobic Training Zones | 有氧与无氧训练区间
Training zones are expressed as percentages of MHR. The aerobic zone (60–80% MHR) improves cardiovascular endurance by mainly using oxygen to break down fuels. The anaerobic zone (80–90% MHR) develops high-intensity tolerance and lactate threshold.
训练区间以最大心率的百分比表示。有氧区间（MHR 的 60–80%）主要通过有氧代谢改善心血管耐力。无氧区间（MHR 的 80–90%）能提高高强度耐受力和乳酸阈。
Aerobic target HR range = MHR × 0.6 to MHR × 0.8
Anaerobic target HR range = MHR × 0.8 to MHR × 0.9
For a 220‑age MHR of 200 bpm, aerobic training would occur between 120 bpm and 160 bpm. Staying within these zones ensures specific adaptation to the training goal.
若按 220−年龄 计算 MHR 为 200 次/分，有氧训练的心率区间则在 120 至 160 次/分之间。保持在对应区间内才能保证训练目标的专项适应。
3. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen method is a more precise way to calculate target heart rate because it accounts for resting heart rate (RHR) and individual fitness level. It uses heart rate reserve (HRR) instead of a flat percentage of MHR.
卡沃宁公式是计算目标心率更精准的方法，因为它考虑了静息心率（RHR）和个人体能水平。它使用心率储备（HRR）而非简单地乘以最大心率的百分比。
HRR = MHR − RHR
Target HR = (HRR × Training %) + RHR
If a performer has MHR 200 bpm and RHR 60 bpm, HRR = 140 bpm. For 70% intensity: Target HR = (140 × 0.7) + 60 = 98 + 60 = 158 bpm. This formula better reflects the true physiological strain.
若某运动员 MHR 为 200 次/分，RHR 为 60 次/分，则 HRR = 140 次/分。以 70% 强度计算：目标心率 = (140 × 0.7) + 60 = 98 + 60 = 158 次/分。该公式能更准确地反映真实的生理负荷。
4. Body Mass Index (BMI) | 身体质量指数（BMI）
BMI is a screening tool used to classify individuals as underweight, healthy weight, overweight, or obese. It compares mass to height squared and is widely used in health and fitness assessments.
BMI 是一种用于划分体重偏轻、健康体重、超重或肥胖的筛查工具。它通过体重与身高平方的比值来评估，广泛用于健康与体适能测评中。
BMI = mass (kg) ÷ height² (m²)
Example: A person weighing 70 kg with a height of 1.75 m has a BMI = 70 ÷ (1.75²) = 70 ÷ 3.0625 ≈ 22.9 kg/m². Note that BMI does not distinguish between muscle and fat.
示例：体重 70 kg，身高 1.75 m 的人，BMI = 70 ÷ (1.75²) = 70 ÷ 3.0625 ≈ 22.9 kg/m²。注意 BMI 无法区分肌肉和脂肪。
5. Basal Metabolic Rate (BMR) | 基础代谢率（BMR）
BMR represents the minimum energy required to sustain vital body functions at rest. For GCSE PE, a simplified calculation based on body mass is often used. More precise estimates use the Schofield or Mifflin‑St Jeor equations, but the key principle is a linear relationship with lean mass.
BMR 代表静息状态下维持生命基本功能所需的最低能量。GCSE 体育中常用基于体重的简化计算。更精确的估算用 Schofield 或 Mifflin‑St Jeor 方程，但核心原则是与瘦体重呈线性关系。
Estimated BMR (kcal/day) ≈ body mass (kg) × 24 (males) or × 22 (females)
This approximation reflects that males generally have a higher proportion of muscle mass. A 65 kg female might have BMR ≈ 65 × 22 = 1430 kcal/day. Actual BMR is influenced by age, genetics and body composition.
该近似反映男性通常肌肉量比例更高。一名 65 kg 女性 BMR 约为 65 × 22 = 1430 千卡/天。实际 BMR 还受年龄、遗传和身体成分影响。
6. Energy Balance | 能量平衡
Energy balance explains changes in body mass. It is the relationship between energy intake (calories consumed) and energy expenditure (calories burned through BMR, physical activity, and the thermic effect of food).
能量平衡解释了体重的变化。它是能量摄入（摄入的热量）与能量消耗（通过基础代谢、体力活动和食物热效应消耗的热量）之间的关系。
Energy Balance = Energy Intake − Energy Expenditure
A positive balance leads to weight gain; a negative balance leads to weight loss. Athletes manipulate this equation to achieve desired body composition for performance, often coupling it with macronutrient timing.
正值导致体重增加，负值导致体重减轻。运动员常利用此等式调整身体成分以提升表现，并结合宏量营养素摄入时机。
7. Speed, Velocity, and Acceleration | 速率、速度与加速度
These are fundamental biomechanical quantities. Speed is scalar (distance/time), velocity is vector (displacement/time), and acceleration describes the rate of change of velocity.
这些是基本的生物力学量。速率是标量（距离/时间），速度是矢量（位移/时间），而加速度描述速度变化的快慢。
Average Speed = Distance ÷ Time
Velocity = Displacement ÷ Time
Acceleration = (Final Velocity − Initial Velocity) ÷ Time
Units: speed and velocity in m/s; acceleration in m/s². Negative acceleration (deceleration) occurs when a sprinter slows after crossing the finish line.
单位：速率和速度用 m/s；加速度用 m/s²。冲刺过线后减速时便出现负加速度（减速度）。
8. Force, Mass, and Acceleration (Newton’s Second Law) | 力、质量与加速度（牛顿第二定律）
This law underpins all movement analysis in sport. The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
该定律是体育运动中所有动作分析的基础。物体的加速度与作用在其上的净合外力成正比，与其质量成反比。
F = m × a (Force = mass × acceleration)
Applying greater force to a shot put (constant mass) increases its acceleration and thus release velocity. For a given force, a lighter tennis racket accelerates faster, aiding swing speed.
对铅球（质量不变）施加更大的力能增加其加速度从而提高出手速度。在力一定时，更轻的网球拍加速更快，有利于挥拍速度。
9. Levers and Moment of Force | 杠杆与力矩
Levers magnify either speed or force. A moment (torque) is the turning effect produced by a force acting at a distance from an axis. The principle of moments applies when a lever is in equilibrium.
杠杆能放大速度或力。力矩是力在距轴心一定距离处产生的转动效应。杠杆平衡时适用力矩原理。
Moment (Nm) = Force (N) × Perpendicular distance from fulcrum (m)
For equilibrium: Total clockwise moments = Total anticlockwise moments
In the body, a third‑class lever (e.g. biceps curl) sacrifices force for speed and range of motion. The muscle force must exceed the resistance moment to produce movement.
在人体中，第三类杠杆（如肱二头肌弯举）牺牲力来换取速度和活动范围。肌肉力量必须大于阻力矩才能产生动作。
10. Projectile Motion Factors | 抛体运动影响因素
For a projectile released above the ground, the range depends on the combination of speed of release, angle of release, and height of release. The optimal angle is below 45° when release height exceeds landing height.
对于从高于地面处出手的抛体，飞行距离取决于出手速度、出手角度和出手高度的组合。当出手高度高于落地高度时，最优角度低于 45°。
Range ∝ v² × sin(2θ) / g (simplified for level release/landing)
In a javelin throw, higher release velocity has the greatest impact on distance, while steeper angles are used when air resistance and lift become significant. Practical coaching adjusts these three factors interactively.
在标枪投掷中，更高出手速度对远度影响最大，而当空气阻力和升力作用显著时会采用更陡的角度。实际训练中需综合调整这三个因素。
11. FITT Principle | FITT 原则
FITT is the foundational framework for designing an exercise programme to achieve progressive overload. It stands for Frequency, Intensity, Time, and Type. No single formula, but intensity is often measured via %MHR or %1RM.
FITT 是设计渐进超负荷训练计划的基础框架，分别代表频率、强度、时间和类型。虽无单一公式，但强度常通过 %MHR 或 %1RM 衡量。
- Frequency: How often (e.g. 3 times/week)
- Intensity: How hard (e.g. 70% MHR, 80% 1RM)
- Time: How long (e.g. 30 minutes)
- Type: Mode of exercise (e.g. cycling, weights)
- 频率：多常训练（如每周 3 次）
- 强度：多努力（如 70% MHR、80% 1RM）
- 时间：多久（如 30 分钟）
- 类型：运动方式（如骑自行车、力量训练）
Progressive manipulation of FITT variables ensures continued adaptation and avoids plateau.
逐步调整 FITT 变量可确保持续适应，避免平台期。
12. Work, Power, and Energy Expenditure | 功、功率与能量消耗
Mechanical work and power provide a link between biomechanics and physiology. Work done is the product of force and displacement in the direction of the force. Power is the rate of doing work.
机械功和功率将生物力学与生理学联系起来。功是力与沿力方向的位移的乘积。功率是做功的速率。
Work (J) = Force (N) × Distance (m)
Power (W) = Work (J) ÷ Time (s)
Climbing stairs: a 600 N person ascending 3 m vertically does 1800 J of work. Completing this in 2 seconds yields 900 W of power. Metabolic energy cost is higher due to efficiency (~25%), so total energy expended ≈ mechanical work × 4.
爬楼梯：一个 600 N 的人垂直上升 3 m 做功 1800 J。以 2 秒完成则功率为 900 W。因人体效率约 25%，代谢能量消耗更高，总能耗 ≈ 机械功 × 4。
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