📚 GCSE CIE Physical Education: Formula & Theorem Quick Reference | GCSE CIE 体育：公式定理速查手册
This quick-reference guide compiles essential formulas, equations, and principles covered in GCSE CIE Physical Education. From cardiovascular calculations to biomechanical theorems, these tools help you analyse performance, design training programmes, and understand the science behind sport. Keep this handbook handy for revision and exam practice.
本速查手册汇集了GCSE CIE体育课程中的核心公式、方程和定律。从心血管计算到生物力学定理，这些工具将帮助你分析运动表现、制定训练计划并理解运动背后的科学原理。请善用本手册进行复习和备考。
1. Maximum Heart Rate Estimation | 最大心率估算
The simplest way to estimate maximum heart rate (MHR) is the age-predicted formula. Although individual variation exists, it serves as a starting point for setting training zones.
估算最大心率（MHR）最简便的方法是通过年龄预测公式。虽然存在个体差异，但它可作为设定训练区间的起点。
Estimated MHR = 220 − age (years)
For example, a 16-year-old performer would have an estimated MHR of 220 − 16 = 204 bpm. Because the prediction can deviate from true maximum by ±10–12 bpm, practical assessments such as a graded exercise test are more reliable when available.
例如，一名16岁的运动者估算MHR为220 − 16 = 204次/分。由于该预测值与真实最大值可能相差±10–12次/分，条件允许时应优先采用分级运动测试等实测方法。
2. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen method uses resting heart rate (RHR) to tailor target zones more accurately. First calculate the heart rate reserve (HRR), then apply the desired intensity.
卡沃宁法利用安静心率（RHR）更精准地制定目标区间。首先计算心率储备（HRR），再乘以目标强度。
HRR = MHR − RHR
Target HR = RHR + (Intensity × HRR)
Intensity is expressed as a decimal (e.g., 0.60 for 60%). If a 16‑year‑old has RHR = 60 bpm and estimated MHR = 204 bpm, then HRR = 144 bpm. A target zone of 60–80% intensity gives a range of 146–175 bpm.
强度以小数表示（如60%即0.60）。若一名16岁青少年RHR = 60次/分、估算MHR = 204次/分，则HRR = 144次/分。60–80%强度对应的目标心率区间为146–175次/分。
3. Body Mass Index (BMI) | 身体质量指数
BMI is a quick screening tool for weight classification. It relates body mass to height squared and is used for population health monitoring.
BMI是快速筛查体重的工具，它将体重与身高的平方相关联，常用于人群健康监测。
BMI = weight (kg) ÷ height (m)²
The table below shows common BMI categories. Note that BMI does not differentiate between fat mass and lean muscle mass, so athletic individuals may be misclassified.
下表为常见BMI分类。注意BMI无法区分脂肪重量与瘦肌肉重量，因此运动员可能被错误归类。
| Classification | 分类 | BMI (kg/m²) |
|---|---|
| Underweight | 体重过轻 | <18.5 |
| Normal | 正常 | 18.5–24.9 |
| Overweight | 超重 | 25–29.9 |
| Obese | 肥胖 | ≥30 |
4. Waist-to-Hip Ratio (WHR) | 腰臀比
WHR assesses fat distribution by comparing the circumference of the waist to that of the hip. High WHR indicates central obesity, which is linked to greater health risks.
腰臀比通过比较腰围与臀围来评估脂肪分布。腰臀比偏高提示中心性肥胖，与更高的健康风险相关。
WHR = waist circumference ÷ hip circumference
Measure both circumferences in the same unit (cm or inches). Health risk thresholds: for males, WHR ≥ 0.90 indicates high risk; for females, WHR ≥ 0.85 indicates high risk.
两个围度使用相同单位（厘米或英寸）。健康风险阈值：男性WHR ≥ 0.90为高风险；女性WHR ≥ 0.85为高风险。
5. Speed and Acceleration | 速度与加速度
Speed describes how quickly a performer moves, while acceleration quantifies the change in velocity over time. Both are fundamental in biomechanics.
速度描述运动者移动的快慢，加速度则量化速度随时间的变化率。两者都是生物力学的基础量。
Speed = distance ÷ time
Acceleration = (final velocity − initial velocity) ÷ time
Units: speed in m/s, acceleration in m/s². An athlete who covers 100 m in 12.5 s has an average speed of 8 m/s. If they accelerate from 0 to 8 m/s in 2 s, the acceleration is 4 m/s².
单位：速度为m/s，加速度为m/s²。运动员用12.5秒跑完100米，平均速度为8 m/s。若在2秒内从0加速到8 m/s，加速度为4 m/s²。
6. Newton’s Second Law and Force | 牛顿第二定律与力
Newton’s second law explains the relationship between force, mass, and acceleration. It is the cornerstone for understanding how athletes produce and control movement.
牛顿第二定律阐明了力、质量与加速度之间的关系，是理解运动员如何产生和控制动作的基石。
Force (N) = mass (kg) × acceleration (m/s²)
Weight is a specific force caused by gravity. On Earth, gravitational field strength g ≈ 10 N/kg, so weight = mass × g.
重量是由重力产生的一种特定力。在地球上，引力场强度g ≈ 10 N/kg，因此重量 = 质量 × g。
For example, a sprinter of mass 70 kg accelerating at 6 m/s² requires a horizontal force of 420 N. The same sprinter’s weight is 700 N.
例如，质量70 kg的短跑运动员以6 m/s²加速，需要420 N的水平力；其重量为700 N。
7. Momentum and Impulse | 动量与冲量
Momentum measures the ‘quantity of motion’ and is conserved in collisions. Impulse describes the effect of a force applied over time, equalling the change in momentum.
动量衡量“运动的量”，在碰撞中守恒。冲量则描述力在一段时间内作用的效果，等于动量的变化量。
Momentum (p) = mass (kg) × velocity (m/s)
Impulse = force (N) × time (s) = Δp
In contact sports, increasing the time over which a force is applied (e.g., through tackling technique or padding) reduces the peak force, lowering injury risk.
在接触性运动中，延长力作用的时间（如通过合理的擒抱技术或护具）可降低峰值力，从而减少受伤风险。
8. Work, Power, and Torque | 功、功率与力矩
Work is done when a force moves an object. Power indicates how quickly work is performed, and torque measures the turning effect of a force.
力使物体移动时便做功。功率表示做功的快慢，力矩则衡量力产生的转动效应。
Work (J) = force (N) × distance moved in direction of force (m)
Power (W) = work (J) ÷ time (s) OR power = force × velocity
Torque (Nm) = force (N) × perpendicular distance from pivot (m)
A weightlifter lifting 1000 N through 2 m performs 2000 J of work. If the lift takes 2 s, the average power output is 1000 W. Torque is critical in analysing levers and rotational movements such as a gymnast’s giant swing.
举重运动员将1000 N的重物上举2 m，做功2000 J。若推举用时2 s，平均输出功率为1000 W。力矩在分析杠杆和旋转动作（如体操大回环）时至关重要。
9. Levers and Mechanical Advantage | 杠杆与机械优势
Lever systems magnify the effect of an effort force to overcome a load. Mechanical advantage describes the efficiency of the lever.
杠杆系统可放大动力效应以克服阻力。机械优势用于描述杠杆的效率。
Mechanical Advantage (MA) = load ÷ effort
In an ideal lever (ignoring friction), MA also equals the length of the effort arm divided by the length of the load arm. The human body uses all three classes of lever: first class (e.g., neck extension), second class (e.g., ankle plantarflexion during toe-off), and third class (e.g., biceps curl). Third-class levers dominate, favouring speed and range of motion over force.
在理想杠杆（忽略摩擦）中，机械优势也等于动力臂长除以阻力臂长。人体使用全部三类杠杆：第一类（如颈部后伸）、第二类（如蹬离地面时足踝跖屈）和第三类（如肱二头肌弯举）。第三类杠杆占主导，以牺牲力的大小换取速度和活动范围。
10. Energy Expenditure and MET | 能量消耗与梅脱
MET (Metabolic Equivalent of Task) expresses the energy cost of physical activities. One MET is the resting metabolic rate (~3.5 ml O₂/kg/min). Total energy expenditure can be estimated using MET values.
梅脱（MET，代谢当量）表示体力活动的能量消耗。1 MET相当于安静代谢率（约3.5毫升氧气/千克/分钟）。可利用MET值估算总能量消耗。
Energy (kcal) = MET × body mass (kg) × time (hours)
Common MET examples: walking at 5 km/h ≈ 3 METs, jogging ≈ 7 METs, running at 10 km/h ≈ 10 METs. A 60 kg person running for 0.5 h at 10 METs expends about 300 kcal.
常见MET示例：步行5 km/h ≈ 3 MET；慢跑 ≈ 7 MET；跑步10 km/h ≈ 10 MET。一名60 kg的人以10 METs跑步0.5小时，约消耗300 kcal。
11. Training Load (Session RPE) | 训练负荷（课次主观疲劳评分）
Session RPE provides a simple way to quantify internal training load. It multiplies the perceived intensity of a session by its duration.
课次主观疲劳评分（Session RPE）是量化内部训练负荷的简便方法，用课次的感知强度乘以持续时间。
Training Load (AU) = Session RPE × duration (minutes)
RPE is typically measured on Borg’s 6–20 scale or the CR-10 scale. For example, a football training session rated 5 on CR-10 and lasting 90 minutes yields a training load of 450 arbitrary units (AU). Monitoring load helps manage injury risk and plan recovery.
RPE通常采用博格6–20级量表或CR-10量表。例如，一次足球训练课CR-10评分为5，持续90分钟，则训练负荷为450 AU。监控负荷有助于控制受伤风险并规划恢复。
12. VO₂max Estimation (Cooper Test) | 最大摄氧量估算（库珀测试）
VO₂max indicates the maximum volume of oxygen the body can utilise per minute per kilogram of body weight. It is a key indicator of aerobic fitness. Field tests such as the Cooper 12‑minute run provide an estimate without laboratory equipment.
最大摄氧量（VO₂max）代表身体每分钟每千克体重所能利用的最大氧气体积，是评价有氧能力的关键指标。库珀12分钟跑等现场测试可在无实验室设备的情况下提供估算值。
VO₂max (ml/kg/min) = (distance covered in metres − 504.9) ÷ 44.73
For example, covering 2800 m in 12 minutes gives an estimated VO₂max of (2800 − 504.9) ÷ 44.73 ≈ 51.3 ml/kg/min. Other prediction methods include the multi-stage fitness test (bleep test), but the Cooper formula is one of the most widely used in GCSE specifications.
例如，12分钟跑完2800米，估算VO₂max = (2800 − 504.9) ÷ 44.73 ≈ 51.3 ml/kg/min。其他预测方法还包括多阶段体能测试（蜂鸣测试），但库珀公式是GCSE考纲中最常用的之一。
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