📚 GCSE OCR PE: Formula & Theorem Quick Reference Handbook | GCSE OCR 体育：公式定理速查手册
This quick reference handbook collects all the essential formulas, calculations, and biomechanical theorems required for the OCR GCSE Physical Education syllabus. From heart rate training zones to lever mechanics, each formula is explained with a worked example to support your revision and data analysis skills.
本速查手册汇集 OCR GCSE 体育课程所需的所有关键公式、计算和生物力学定理。从心率训练区间到杠杆力学，每个公式都配有实例解释，以帮助巩固复习和提升数据分析能力。
1. Maximum Heart Rate (HRmax) | 最大心率
Maximum heart rate is the highest number of beats per minute (bpm) your heart can achieve during all‑out exercise. The simplest and most widely used estimate is HRmax = 220 − age. For a 15‑year‑old student, that gives 205 bpm.
最大心率是进行极限运动时心脏每分钟所能达到的最高次数。最简单且最常用的估算方法是 最大心率 = 220 − 年龄。一名 15 岁学生的最大心率约为 205 次/分。
HRmax = 220 − age
You must memorise this formula because it underpins all subsequent training‑zone calculations. Remember that this is an estimate – true maximum heart rate varies between individuals.
你必须牢记这个公式，因为它是所有训练区间计算的基础。请记住这只是估算值——真正的最大心率因人而异。
2. Karvonen Formula | 卡氏公式
The Karvonen formula uses resting heart rate (RHR) to calculate a more personalised target heart rate. It accounts for fitness level by working with heart rate reserve (HRR): HRR = HRmax − RHR.
卡氏公式利用安静心率（RHR）来计算更为个性化的目标心率。它通过心率储备（HRR）将体能水平纳入考量：心率储备 = 最大心率 − 安静心率。
Target HR = [(HRmax − RHR) × % intensity] + RHR
For example, a 16‑year‑old (HRmax = 204 bpm) with RHR = 60 bpm wants to train at 70% intensity: HRR = 204 − 60 = 144 bpm. Target HR = (144 × 0.70) + 60 = 160.8 ≈ 161 bpm.
例如，一名 16 岁学生（最大心率 204 次/分）安静心率为 60 次/分，想以 70% 强度训练：心率储备 = 204 − 60 = 144 次/分。目标心率 = (144 × 0.70) + 60 = 160.8 ≈ 161 次/分。
3. Target Heart Rate Zones | 目标心率区间
OCR candidates must be able to identify and calculate aerobic and anaerobic training zones using both the straight‑percentage method and the Karvonen method. The typical zones are:
OCR 考生必须能运用百分比法和卡氏公式两种方法，识别并计算有氧与无氧训练区间。常见的区间划分如下：
- Aerobic zone: 60–80% HRmax (or using Karvonen at 60–80% HRR)
- Anaerobic zone: 80–90% HRmax (or using Karvonen at 80–90% HRR)
- 有氧区间：60–80% 最大心率（或使用卡氏公式对应 60–80% 心率储备）
- 无氧区间：80–90% 最大心率（或使用卡氏公式对应 80–90% 心率储备）
Using the Karvonen method for a person with HRmax 190 and RHR 50: aerobic floor = (140 × 0.60) + 50 = 134 bpm; ceiling = (140 × 0.80) + 50 = 162 bpm. Always state the training zone in bpm or as a range.
对于最大心率为 190、安静心率为 50 的人，使用卡氏公式：有氧下限 = (140 × 0.60) + 50 = 134 次/分；上限 = (140 × 0.80) + 50 = 162 次/分。作答时务必以次/分或区间形式给出目标心率。
4. Cardiac Output (Q) | 心输出量
Cardiac output is the volume of blood pumped by the heart in one minute. It links stroke volume (SV) and heart rate (HR). The relationship is fundamental to understanding how the cardiovascular system responds to exercise.
心输出量是心脏每分钟泵出的血液量。它将每搏输出量（SV）和心率（HR）联系起来。这种关系是理解心血管系统对运动反应的基础。
Q = SV × HR
Stroke volume is the amount of blood pumped per beat (ml/beat), and HR is in beats per minute. If SV = 70 ml/beat and HR = 72 bpm, Q = 70 × 72 = 5040 ml/min (5.04 L/min). During vigorous exercise, both SV and HR rise, dramatically increasing cardiac output.
每搏输出量是心脏每次收缩射出的血量（毫升/次），心率单位为次/分。若 SV = 70 毫升/次，HR = 72 次/分，则 Q = 70 × 72 = 5040 毫升/分（5.04 升/分）。剧烈运动时，SV 和 HR 同时升高，心输出量大幅增加。
5. Minute Ventilation (VE) | 每分钟通气量
Minute ventilation is the volume of air breathed in or out of the lungs in one minute. It is the product of tidal volume (TV) and breathing rate (BR), and it rises steeply during physical activity.
每分钟通气量是每分钟吸入或呼出肺部的气体量。它是潮气量（TV）与呼吸频率（BR）的乘积，在身体活动时急剧上升。
VE = TV × BR
Tidal volume is the amount of air per breath (usually in litres), and breathing rate is the number of breaths per minute. For a resting person with TV = 0.5 L and BR = 12 breaths/min, VE = 6.0 L/min. During maximal exercise, TV can reach 3 L and BR over 45, resulting in VE > 135 L/min.
潮气量是每次呼吸的气量（通常以升为单位），呼吸频率为每分钟呼吸次数。静息时若 TV = 0.5 升，BR = 12 次/分，则 VE = 6.0 升/分。最大强度运动时，TV 可达 3 升，BR 超过 45 次/分，VE 可超过 135 升/分。
6. Body Mass Index (BMI) | 身体质量指数
BMI is a simple index of weight‑for‑height used to classify individuals as underweight, normal, overweight or obese. It is a key measure in health‑related fitness assessments.
BMI 是一种简单的体重与身高之比指数，用于将个体划分为偏瘦、正常、超重或肥胖。它是健康相关体适能评估中的一项关键指标。
BMI = weight (kg) ÷ [height (m)]²
Example: a person weighing 65 kg with a height of 1.70 m. Height squared = 1.70 × 1.70 = 2.89 m². BMI = 65 ÷ 2.89 ≈ 22.5 kg/m², which falls in the healthy weight range (18.5–24.9). Remember to convert height to metres before squaring.
示例：体重 65 公斤，身高 1.70 米。身高平方 = 1.70 × 1.70 = 2.89 平方米。BMI = 65 ÷ 2.89 ≈ 22.5 千克/平方米，属于健康体重范围（18.5–24.9）。计算前请务必将身高转换为米。
7. Speed and Acceleration | 速度和加速度
Speed and acceleration describe motion. Speed is a scalar quantity (magnitude only), while acceleration is the rate at which velocity changes. These formulas support biomechanical analysis of sporting actions.
速度和加速度描述运动状态。速率是标量（仅具有大小），而加速度是速度变化的快慢。这些公式支撑体育动作的生物力学分析。
Speed = distance ÷ time
Acceleration = (final velocity − initial velocity) ÷ time taken
If a 100‑m sprinter covers the distance in 12.0 s, her average speed = 100 ÷ 12.0 = 8.33 m/s. If her foot velocity increases from 2 m/s to 10 m/s in 0.2 s during a kick, acceleration = (10 − 2) ÷ 0.2 = 40 m/s². Always include units.
若一名百米跑运动员用 12.0 秒跑完全程，其平均速率为 100 ÷ 12.0 = 8.33 米/秒。若踢球时脚的速度在 0.2 秒内从 2 米/秒增至 10 米/秒，加速度 = (10 − 2) ÷ 0.2 = 40 米/秒²。务必写出单位。
8. Newton’s Second Law and Momentum | 牛顿第二定律与动量
Newton’s second law relates force, mass and acceleration. It explains why a heavier shot‑put requires greater force to accelerate. Momentum helps analyse collisions and changes in velocity.
牛顿第二定律将力、质量和加速度联系起来。它解释了为什么更重的铅球需要更大的力才能加速。动量则有助于分析碰撞和速度变化。
F = m × a
Momentum (p) = m × v
Force is measured in newtons (N). A 70 kg rugby player accelerating at 3 m/s² experiences a force F = 70 × 3 = 210 N. Momentum is mass × velocity (kg·m/s). The same player sprinting at 8 m/s has momentum p = 70 × 8 = 560 kg·m/s. In a tackle, momentum is transferred between players.
力的单位是牛顿（N）。一名 70 公斤的橄榄球运动员以 3 米/秒² 加速时，所受的力 F = 70 × 3 = 210 牛。动量 = 质量 × 速度（千克·米/秒）。同一运动员以 8 米/秒冲刺时，动量 p = 70 × 8 = 560 千克·米/秒。擒抱时，动量在球员间传递。
9. Work Done and Power | 做功与功率
Work done measures energy transferred when a force moves an object. Power is the rate at which work is performed. Both concepts are essential for evaluating athletic performance, especially in weightlifting and cycling.
做功衡量力使物体移动时所传递的能量。功率是做功的速率。这两个概念对评估运动表现至关重要，尤其是在举重和自行车运动中。
Work = force × distance moved in the direction of the force
Power = work done ÷ time taken
Work is measured in joules (J): 1 J = 1 N·m. If a weightlifter applies an upward force of 800 N over 0.6 m, work = 800 × 0.6 = 480 J. If the lift takes 0.5 s, power = 480 ÷ 0.5 = 960 W. Power can also be expressed as Power = force × velocity (for steady motion).
功的单位为焦耳（J）：1 焦 = 1 牛·米。若举重运动员向上施加 800 牛的力，移动 0.6 米，则做功 = 800 × 0.6 = 480 焦。若该次上举耗时 0.5 秒，功率 = 480 ÷ 0.5 = 960 瓦。功率也可表示为 功率 = 力 × 速度（匀速运动时）。
10. Moments and Lever Systems | 力矩与杠杆系统
A moment (torque) is the turning effect of a force around a pivot. The musculoskeletal system acts through three classes of levers, and the principle of moments explains how force and effort arms influence movement efficiency.
力矩是力绕支点产生的转动效应。肌肉骨骼系统通过三类杠杆发挥作用，力矩原理解释了力臂和阻力臂如何影响动作效率。
Moment = force × perpendicular distance from pivot
For a first‑class lever (e.g., neck extension), the effort and load are on opposite sides of the pivot. For a second‑class lever (calf raise), the load lies between pivot and effort. For a third‑class lever (bicep curl), the effort is between pivot and load – requiring more force but allowing greater speed and range.
对于第一类杠杆（如颈部后伸），动力和阻力位于支点两侧。第二类杠杆（如提踵），阻力位于支点和动力之间。第三类杠杆（如肱二头肌弯举），动力位于支点和阻力之间——需要更大的力，但可获得更快的速度和更大的活动范围。
Mechanical advantage = effort arm length ÷ resistance arm length
A mechanical advantage > 1 means less effort is needed to move a load (second‑class levers). A value < 1 means more effort is required but speed is gained (third‑class). In a third‑class bicep curl, if effort arm = 4 cm and resistance arm = 30 cm, mechanical advantage = 4/30 = 0.13.
机械利益 = 动力臂长 ÷ 阻力臂长。机械利益 > 1 表示用较小的力即可移动负荷（第二类杠杆）；< 1 表示需较大的力但能获得速度优势（第三类杠杆）。在第三类肱二头肌弯举中，若动力臂 4 厘米，阻力臂 30 厘米，机械利益 = 4/30 = 0.13。
11. Percentage Calculations for Training Loads | 训练负荷的百分比计算
Strength and power programmes often rely on a percentage of one‑repetition maximum (1RM). Calculating these percentages allows coaches to prescribe safe and effective training loads for different fitness goals.
力量和爆发力训练常依据单次最大重量（1RM）的百分比进行。计算这些百分比有助于教练为不同健身目标设定安全有效的训练负荷。
Training load (kg) = 1RM × (desired percentage ÷ 100)
For muscular endurance, loads of 50–65% 1RM are typical; for hypertrophy, 70–80%; for maximum strength, 85–95%. Example: an athlete’s bench press 1RM is 80 kg. To train at 75% for hypertrophy: load = 80 × 0.75 = 60 kg. Sets and reps are then designed around this load.
发展肌耐力通常使用 50–65% 1RM；肌肥大 70–80%；最大力量 85–95%。示例：运动员的卧推 1RM 为 80 公斤，若以 75% 强度进行肌肥大训练，负荷 = 80 × 0.75 = 60 公斤。随后围绕该负荷设计组数和次数。
Always apply these percentages after an appropriate warm‑up and under supervision. Percentage‑based programming is a core component of OCR’s principles of training and periodisation.
务必在充分热身后并在监督下应用这些百分比。基于百分比的训练设计是 OCR 训练原则和周期化理论的核心内容之一。
Published by TutorHao | Physical Education Revision Series | aleveler.com
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