📚 GCSE Cambridge Physical Education: Formula & Theorem Quick Reference Handbook | GCSE剑桥体育：公式定理速查手册
This quick reference handbook compiles essential formulas, equations and biomechanical theorems that underpin the Cambridge GCSE Physical Education syllabus. From calculating target heart rates for aerobic training to analysing lever systems in the human body, mastering these quantitative concepts will strengthen your exam answers and practical understanding of sport science.
本速查手册汇编了剑桥GCSE体育课程核心的公式、方程及生物力学定理。从计算有氧训练的目标心率到分析人体的杠杆系统，掌握这些量化概念将助你强化考试答题和体育科学的实践理解。
1. Heart Rate & Training Zones | 心率与训练区间
The most widely used estimation of maximum heart rate (HRmax) for a healthy individual is the simple age‑based formula:
最常用的健康个体最大心率 (HRmax) 估算公式是基于年龄的简单算法：
HRmax = 220 – age
For a 16‑year‑old student this gives an approximate HRmax of 204 beats per minute (bpm).
对于16岁的学生，该公式得出最大心率约为204次/分 (bpm)。
To prescribe precise training intensities, the Karvonen formula uses heart rate reserve (HRR), which accounts for resting heart rate (HRrest):
为精确设定训练强度，卡沃内公式使用了心率储备 (HRR)，该指标考虑了静息心率 (HRrest)：
Target HR = HRrest + (HRmax – HRrest) × % intensity
Typical aerobic training zones are set at 60–80% of HRR, while anaerobic threshold work often targets 80–90% HRR. Understanding these zones allows athletes to tailor training to specific energy systems.
典型有氧训练区间设定为心率储备的60–80%，而无氧阈训练常瞄准80–90% HRR。理解这些区间能帮助运动员针对特定能量系统定制训练方案。
2. Body Mass Index (BMI) & Body Composition | 身体质量指数与体成分
Body Mass Index provides a simple numeric index of body mass relative to height. The universally accepted formula is:
身体质量指数给出了体重与身高关系的简易数值指标。公认的公式如下：
BMI = weight (kg) ÷ height² (m²)
A BMI between 18.5 and 24.9 is generally classified as healthy weight for adults, while the thresholds for children vary with age and gender.
成年人的健康体重范围通常为18.5–24.9，而儿童青少年的阈值随年龄和性别变化。
For assessing fat distribution linked to health risks, waist‑to‑hip ratio (WHR) is calculated:
为评估与健康风险相关的脂肪分布，可计算腰臀比 (WHR)：
WHR = waist circumference ÷ hip circumference
Both BMI and WHR are screened in fitness testing batteries to evaluate an individual’s health status and potential risk for chronic diseases.
体能测试中常筛查BMI和腰臀比，以评估个体健康状况及慢性病风险。
3. Speed, Acceleration & Newton’s Laws | 速度、加速度与牛顿定律
Speed, the rate of change of distance, is fundamental in analysing sporting movements:
速度是距离随时间的变化率，是分析运动动作的基础：
v = Δs ÷ Δt
Acceleration describes how quickly velocity changes, and it is given by:
加速度描述速度变化的快慢，公式为：
a = Δv ÷ Δt
These kinematic quantities are governed by Newton’s three laws of motion. The First Law (inertia) states that an object remains at rest or in uniform motion unless acted upon by a net external force.
这些运动学量受牛顿运动三定律支配。第一定律（惯性）指出，物体在无净外力作用时保持静止或匀速直线运动。
The Second Law quantifies the relationship between force, mass and acceleration:
第二定律量化了力、质量与加速度的关系：
F = m × a
The Third Law asserts that for every action there is an equal and opposite reaction, which explains the ground reaction force propelling a sprinter forward.
第三定律断言任何作用力都有一个等大反向的反作用力，这解释了推动短跑运动员前进的地面反作用力。
4. Momentum, Impulse & Force‑Time Graphs | 动量、冲量与力‑时间图
Linear momentum is the product of mass and velocity, a vector quantity conserved in collisions when no external force acts:
线动量是质量与速度的乘积，为矢量，在无外力作用时碰撞中动量守恒：
p = m × v
Impulse is the product of force and the time for which it acts, and it equals the change in momentum:
冲量是力与其作用时间的乘积，等于动量的变化量：
Impulse = F × Δt = Δp
A force–time graph illustrates this concept: the area under the curve represents the total impulse. Athletes can increase momentum change by applying a larger force over a longer contact time, such as during a tennis serve or a rugby tackle.
力‑时间图直观展示了这一概念：曲线下的面积等于总冲量。运动员通过延长力的作用时间或施加更大的力来增大动量变化，例如网球发球或橄榄球擒抱。
5. Levers & Mechanical Advantage | 杠杆与机械增益
The human skeleton behaves as a system of levers, with joints acting as pivots. For any lever, mechanical advantage (MA) compares the effort arm to the resistance arm:
人体骨骼相当于杠杆系统，关节充当支点。对任何杠杆，机械增益 (MA) 定义为力臂与阻力臂的比值：
MA = effort arm ÷ resistance arm
A first‑class lever (e.g. the neck joint performing extension) can have MA < 1, = 1 or > 1 depending on the relative arm lengths. A second‑class lever (e.g. the ankle during plantar flexion) always provides MA > 1, allowing a large load to be moved with a smaller effort.
第一类杠杆（如颈部后伸）根据力臂长度可获得MA < 1、=1或>1。第二类杠杆（如踝关节跖屈）总能提供MA > 1，能以较小的力移动较大负荷。
A third‑class lever (e.g. the biceps curl at the elbow) has MA < 1, favouring range and speed of movement over force, which is a trade‑off seen in many sporting actions.
第三类杠杆（如肘关节的肱二头肌弯举）MA < 1，牺牲力量以换取活动幅度和速度，这正是许多运动动作中的一种权衡。
6. Torque & Moments of Force | 力矩与转动效应
Torque, also called moment of force, measures the turning effect produced by a force applied at a distance from a pivot:
力矩，亦称力之矩，衡量力作用在距支点一定距离处所产生的转动效应：
Moment = force × perpendicular distance from pivot
For a body to remain in rotational equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any pivot.
物体要保持转动平衡，绕任意支点的顺时针力矩之和必须等于逆时针力矩之和。
This principle is applied in gymnastics when a performer holds a still position: the torque generated by body weight about a handhold must be balanced by muscular torque. Understanding moments also helps in analysing joint stability and injury prevention.
体操运动员保持静止姿势时便应用了这一原理：体重关于抓握点的力矩必须由肌肉力矩平衡。理解力矩还有助于分析关节稳定性与损伤预防。
7. Work, Power & Energy in Sport | 运动中的功、功率与能量
In sports biomechanics, mechanical work is done when a force moves its point of application in the direction of the force:
在运动生物力学中，当力沿其方向移动作用点时便做了机械功：
Work = F × d (parallel)
Power is the rate of doing work, critical for explosive events:
功率是做功的快慢，对爆发性项目至关重要：
P = Work ÷ time = F × v
Kinetic energy and gravitational potential energy describe the energy stored in moving and elevated bodies respectively:
动能和重力势能分别描述运动物体和高位物体储存的能量：
KE = ½ m v²
GPE = m g h
Efficiency of movement can be expressed as the percentage of useful energy output relative to total energy input:
动作效率可表示为有用能量输出与总能量输入的百分比：
Efficiency = (useful energy output ÷ total energy input) × 100%
8. Fitness Test Prediction Equations | 体能测试预测公式
The Harvard Step Test provides an index of cardiovascular fitness based on recovery heart rate:
哈佛台阶测试根据恢复心率提供心血管适能指数：
Fitness Index = (duration in seconds × 100) ÷ (2 × sum of three pulse counts)
The Cooper 12‑minute run test estimates maximal oxygen uptake (VO₂ max) from the distance covered:
库珀12分钟跑测试通过跑动距离估算最大摄氧量 (VO₂ max)：
VO₂ max (ml/kg/min) = (distance in metres – 504.9) ÷ 44.73
For the multi‑stage fitness test (bleep test), a widely used simplified regression is:
对于多阶段体能测试（哔哔跑），一个常用的简化回归方程为：
VO₂ max ≈ 3.46 × (level + shuttle fraction) + 12.2
These equations allow coaches to convert field test performances into physiological metrics and monitor training adaptations over time.
这些公式让教练能将实地测试表现转化为生理指标，并持续监测训练适应。
9. Principles of Stability | 稳定性原理
Stability is determined by the position of the centre of gravity relative to the base of support. An object or athlete is stable when the vertical line through the centre of gravity falls inside the base of support.
稳定性取决于重心相对于支撑基底的位置。当通过重心的垂直线落在支撑基底之内时，物体或运动员即处于稳定状态。
Increasing the base of support, lowering the centre of gravity, and increasing body mass all enhance stability. In contact sports, a wider stance and a lower body position are adopted to resist being pushed over.
增大支撑基底、降低重心、增加体重都能提高稳定性。在对抗性运动中，运动员采用宽站姿和低重心来抵抗被推倒。
The turning effect of a force attempting to topple an object also obeys the moment principle: the sum of destabilising torques must be smaller than the restoring torques for the position to be maintained.
试图推翻一个人的力的转动效应同样遵循力矩原理：保持姿势要求倾覆力矩的总和小于恢复力矩。
10. Energy Expenditure & Metabolic Equivalents (METs) | 能量消耗与代谢当量
One metabolic equivalent (MET) is defined as the resting oxygen consumption, approximately 3.5 ml O₂ per kg body mass per minute. Physical activities are rated as multiples of METs.
一个代谢当量 (MET) 定义为静息耗氧量，约合每公斤体重每分钟3.5毫升氧气。身体活动按MET倍数分级。
The energy cost of an activity can be estimated using:
活动能量消耗可用下式估算：
Energy (kcal) = MET × body mass (kg) × time (hours)
A rough guide: an adult weighing 70 kg cycling at 8 METs for 0.5 hours expends about 280 kcal. This helps in designing weight‑management programmes alongside sport training.
粗略换算：一名70公斤的成年人以8 MET骑行0.5小时约消耗280千卡。这有助于在体育训练同时设计体重管理计划。
The respiratory quotient (RQ) reveals which energy substrate is being metabolised:
呼吸商 (RQ) 揭示正在代谢的能量底物：
RQ = VCO₂ (CO₂ produced) ÷ VO₂ (O₂ consumed)
RQ values of 1.0, 0.7 and around 0.8 indicate predominant carbohydrate, fat and protein metabolism respectively. Knowledge of RQ helps sports nutritionists fine‑tune dietary strategies during different phases of training.
RQ值为1.0、0.7和0.8左右分别指示碳水化合物、脂肪和蛋白质为主的代谢。了解RQ有助于运动营养师在不同训练阶段微调膳食策略。
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