📚 Year 8 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 8 CAIE 体育：公式定理速查手册
This quick reference handbook brings together the most important formulas, theorems and principles you will encounter in Year 8 CAIE Physical Education. Understanding these will help you analyse human performance, plan training sessions and answer exam questions with confidence.
本速查手册汇集了 Year 8 CAIE 体育课程中最重要的公式、定理和原理。掌握这些内容将帮助你分析人体运动表现、设计训练计划，并有信心应对考试中的各类问题。
1. Maximum Heart Rate (MHR) | 最大心率
The maximum heart rate is the highest number of beats per minute your heart can achieve during all-out exercise. It is commonly estimated with the simple formula: MHR = 220 – age. For a 13-year-old student, MHR = 220 – 13 = 207 bpm. This value is used to set training zones and monitor exercise intensity.
最大心率是指心脏在极限运动中每分钟所能达到的最高搏动次数，通常用简单公式 最大心率 = 220 – 年龄 估算。例如一位13岁的学生，最大心率 = 220 – 13 = 207 次/分。该数值用来设定训练区间和监测运动强度。
- MHR = 220 – age (years)
- Actual maximum may vary; laboratory testing is more accurate.
- 最大心率 = 220 – 年龄（岁）
- 实际最大值因人而异，实验室测试更为精确。
2. Target Heart Rate Zones | 目标心率区间
To gain specific fitness benefits, you need to work within a percentage range of your MHR. Moderate-intensity exercise uses 50–70% MHR, while vigorous activity targets 70–85% MHR. Training below 50% offers minimal cardiovascular improvement; above 90% is reserved for elite sprint intervals.
为获得特定的健身效益，需要在最大心率的一定百分比区间内锻炼。中等强度运动采用 50–70% 最大心率，高强度活动则瞄准 70–85% 最大心率。低于 50% 对心血管改善甚微，高于 90% 仅用于精英冲刺间歇训练。
| Intensity Zone | % MHR | Purpose |
|---|---|---|
| Warm-up / Recovery | 50–60% | Gradual preparation, cool-down |
| Moderate (aerobic) | 60–70% | Endurance base, fat burning |
| Vigorous (aerobic/anaerobic) | 70–85% | Cardiovascular fitness, lactate threshold |
| Maximum effort | 85–100% | Speed, anaerobic power |
3. Karvonen Formula (Heart Rate Reserve) | 卡沃南公式（心率储备法）
The Karvonen method gives a more personalised training heart rate by considering resting heart rate (RHR). Target heart rate = RHR + (Intensity % × (MHR – RHR)). For example, a student with RHR of 70 bpm, MHR of 207 bpm, aiming for 65% intensity: target HR = 70 + 0.65×(207-70) = 70 + 89 = 159 bpm.
卡沃南方法通过引入安静心率 (RHR) 提供更个性化的训练心率。目标心率 = 安静心率 + (强度百分比 × (最大心率 – 安静心率))。例如，安静心率 70 次/分，最大心率 207 次/分，设定 65% 强度：目标心率 = 70 + 0.65×(207-70) = 70 + 89 = 159 次/分。
Target HR = RHR + (I × (MHR – RHR))
where I = intensity as a decimal (e.g., 0.65 for 65%).
其中 I = 强度（小数形式，如 65% 对应 0.65）。
4. Body Mass Index (BMI) | 身体质量指数
BMI is a quick screening tool for weight categories. Formula: BMI = mass (kg) / height² (m²). A person weighing 50 kg with a height of 1.60 m has BMI = 50 / (1.60×1.60) = 19.5 kg/m². Although widely used, BMI does not distinguish between muscle and fat, so athletes with high muscle mass may be misclassified as overweight.
BMI 是快速筛查体重类别的工具。公式：BMI = 体重 (kg) / 身高² (m²)。一名体重 50 kg、身高 1.60 m 的人，BMI = 50 / (1.60×1.60) = 19.5 kg/m²。尽管被广泛使用，BMI 并不区分肌肉和脂肪，因此肌肉量高的运动员可能被误判为超重。
| BMI (kg/m²) | Weight Status |
|---|---|
| < 18.5 | Underweight |
| 18.5 – 24.9 | Healthy weight |
| 25.0 – 29.9 | Overweight |
| ≥ 30.0 | Obese |
5. Speed and Velocity | 速率与速度
Speed is a scalar quantity measuring how fast an object moves: Speed = distance / time. Velocity is a vector; it includes direction. In sports, average speed is often used: v = d/t. A sprinter completing 100 m in 12.5 s has an average speed of 8 m/s. Instantaneous speed may be higher during the middle of the race.
速率是标量，衡量物体移动的快慢：速率 = 距离 / 时间。速度是矢量，包含方向。在体育中常使用平均速率：v = d/t。一名短跑运动员用 12.5 秒跑完 100 米，平均速率为 8 m/s。比赛中段的瞬时速率可能更高。
Speed = distance / time
Units: metres per second (m/s). For acceleration, a = Δv / Δt, where Δv is change in velocity and Δt is time taken.
单位：米每秒 (m/s)。加速度 a = Δv / Δt，其中 Δv 为速度变化量，Δt 为所用时间。
6. Force, Work and Power | 力、功和功率
Force (F) causes an object to accelerate. Newton’s second law states F = m × a, where m is mass (kg) and a is acceleration (m/s²). Work (W) is done when a force moves an object: W = F × d (joules). Power (P) is the rate of doing work: P = W / t or P = F × v (watts). A weightlifter lifting 1000 N through 2 m in 1 s produces 2000 W of power.
力 (F) 使物体加速。牛顿第二定律：F = m × a，其中 m 为质量 (kg)，a 为加速度 (m/s²)。力使物体移动时即做功 (W)：W = F × d（焦耳）。功率 (P) 是做功的速率：P = W / t 或 P = F × v（瓦特）。一位举重运动员用 1 秒将 1000 牛的杠铃上举 2 米，产生的功率为 2000 瓦。
F = m × a
W = F × d
P = W / t = (F × d) / t
7. Levers and Moments | 杠杆与力矩
A lever consists of a rigid bar rotating around a fulcrum. The turning effect is called a moment: Moment = Force × perpendicular distance from the fulcrum. In the body, muscles, bones and joints form lever systems. There are three classes: first class (e.g., neck nodding), second class (standing on tiptoes) and third class (biceps curl). Third-class levers are most common in the body, favouring speed and range of motion over force.
杠杆由刚性杆绕支点旋转构成。转动效应称为力矩：力矩 = 力 × 力臂（力到支点的垂直距离）。人体中，肌肉、骨骼和关节组成杠杆系统。杠杆分为三类：第一类（如点头动作）、第二类（如踮脚尖）和第三类（如哑铃弯举）。第三类杠杆在人体中最常见，以速度和活动范围换取更小的力。
Moment (Nm) = Force (N) × Distance (m)
For equilibrium, total clockwise moments = total anticlockwise moments. This principle explains why a longer lever arm allows a smaller effort to move a larger load.
平衡时，顺时针合力矩 = 逆时针合力矩。这一原理解释了为何更长的力臂可使较小的力移动较大负荷。
8. Balance and Centre of Mass | 平衡与重心
The centre of mass is the point where all body mass seems to be concentrated. An object is stable if a vertical line through its centre of mass falls inside its base of support. To improve stability, lower the centre of mass, widen the base, or shift the line of gravity towards the oncoming force. Sprinters crouch low at the start; rugby players adopt a wide stance before contact.
重心是身体所有质量看似集中的点。如果穿过重心的垂直线落在支撑面内，物体就稳定。为提高稳定性，可降低重心、增大支撑面、或使重力线靠近即将承受的力。短跑选手起跑时身体下蹲，橄榄球运动员在接触前则采用宽站位。
There is no single formula, but the principle can be quantified: Stability ∝ (base width × mass) / height of centre of mass. This relationship helps explain why taller athletes often need a wider stance to achieve similar balance.
虽然没有单一的公式，但这一原理可量化：稳定性 ∝ (支撑面宽度 × 质量) / 重心高度。该关系式可解释为何身材较高的运动员常需要更宽的站立姿势以保持同等平衡。
9. The FITT Principle | FITT 原则
The FITT principle provides a framework for designing a safe and effective training programme. It stands for Frequency, Intensity, Time and Type. To improve aerobic fitness, a student might exercise 3‑5 times per week (Frequency), at 65–75% MHR (Intensity), for 20–30 minutes (Time), using continuous running or cycling (Type). Progressive overload is applied by adjusting one or more FITT variables.
FITT 原则为设计安全有效的训练计划提供了一个框架。它代表频率（Frequency）、强度（Intensity）、时间（Time）和类型（Type）。为提高有氧耐力，一名学生可每周锻炼 3‑5 次（频率），强度为 65–75% 最大心率，每次 20‑30 分钟（时间），采用持续跑或骑自行车（类型）。通过调整一个或多个 FITT 变量实现渐进超负荷。
- Frequency – how often (days per week)
- Intensity – how hard (heart rate, RPE)
- Time – duration (minutes per session)
- Type – mode of exercise (aerobic, anaerobic, flexibility)
- 频率 – 多久一次（每周天数）
- 强度 – 难度多大（心率、主观疲劳感觉）
- 时间 – 时长（每次训练分钟数）
- 类型 – 运动方式（有氧、无氧、柔韧性等）
10. Energy Systems and Recovery | 能量系统与恢复
Physical activity relies on three energy systems: the ATP‑PC system (0–10 s, explosive power), the lactic acid system (10–90 s, high intensity) and the aerobic system (90+ s, endurance). While there are no simple formulas for energy production, recovery heart rate is a useful indicator: Recovery HR (%) = [(Peak HR – HR after 1 min) / (Peak HR – RHR)] × 100. A fit performer will see a rapid drop towards resting values.
身体活动依赖三种能量系统：ATP‑CP 系统（0‑10 秒，爆发力）、乳酸系统（10‑90 秒，高强度）和有氧系统（90 秒以上，耐力）。尽管能量生成没有简单的公式，恢复心率是一个有用的指标：恢复心率(%) = [(峰值心率 – 1 分钟后心率) / (峰值心率 – 安静心率)] × 100。体能良好的运动者心率会快速回落至安静水平。
Active recovery promotes blood lactate clearance. A simple rule of thumb: post-exercise heart rate should drop by at least 12 bpm in the first minute for a healthy cardiovascular response.
积极恢复可促进血乳酸清除。一条简单的经验法则：运动后第一分钟心率应至少下降 12 次/分，表明心血管反应健康。
11. Speed and Velocity in Team Sports | 团队运动中的速率与速度
In games like football or hockey, players often change direction, so calculation of velocity as a vector becomes important. Velocity = displacement / time. Displacement is the shortest straight-line distance from start to finish in a specified direction. A winger sprinting 40 m upfield in 5 s has a velocity of 8 m/s in that direction, even if the actual running distance was greater due to zig-zag movements.
在足球或曲棍球等项目中，运动员频繁改变方向，因此作为矢量的速度计算很重要。速度 = 位移 / 时间。位移是从起点到终点的最短直线距离，并带有方向。一名边锋向前场冲刺 40 米，用时 5 秒，其速度为 8 m/s（方向向前），即便实际跑动距离因之字形移动而更长。
Velocity = displacement / time
Comparing speed and velocity helps coaches evaluate game movement efficiency.
比较速率和速度有助于教练评估比赛中的移动效率。
12. Training Intensity and Talk Test | 训练强度与谈话测试
Besides heart rate, a simple subjective method is the talk test. When exercising at moderate intensity, you should be able to talk but not sing; at vigorous intensity, you can only say a few words before needing to breathe. This aligns roughly with the ventilatory threshold, offering an easy way to monitor intensity without technology. The relationship between heart rate and perceived exertion can be expressed as RPE = Heart Rate / 10 (Borg scale approximate), giving a quick intensity check.
除心率外，一个简单的主观方法是谈话测试。中等强度运动时应能交谈但不能唱歌；高强度运动时只能说几个词就需喘气。这与通气阈大致吻合，提供了一种无需设备监测强度的便捷方式。心率与主观疲劳感的关系可近似为 RPE = 心率 / 10（根据博格量表），可快速检查强度。
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