📚 AS Eduqas PE: Quick Reference Formula & Theorem Handbook | AS Eduqas 体育：公式定理速查手册
Welcome to your essential quick-reference guide for the AS Eduqas Physical Education specification. This handbook gathers every critical formula, equation, and key theoretical principle you need to master across physiology, biomechanics, skill acquisition, and sport psychology. Each entry is presented clearly, with paired English and Chinese explanations, to make revision efficient and exam-focused. Keep this resource close for rapid recall of definitions, calculations, and the fundamental laws that underpin sporting performance.
欢迎阅读这份专为 AS Eduqas 体育教育考试准备的必备速查手册。本手册汇集了生理学、生物力学、技能习得和运动心理学中所有关键的公式、方程和重要理论原理。每个知识点均采用中英文对照讲解，确保复习高效且紧扣考试重点。请随身携带这份资料，以便快速回忆定义、计算以及支撑运动表现的基本定律。
1. Cardiovascular Formulae | 心血管公式
Cardiovascular calculations are central to understanding exercise intensity and cardiac function. The most frequently used formulas estimate maximal heart rate and target training zones. Maximal Heart Rate (HRmax) is traditionally predicted using a simple age-based equation:
心血管计算是理解运动强度和心脏功能的核心。最常用的公式用来估算最大心率和目标训练区间。最大心率（HRmax）传统上通过一个简单的基于年龄的方程进行预测：
HRmax = 220 − age (years)
While this estimate has individual variability, it remains a foundation for prescribing training intensities. A more refined formula, the Karvonen method, incorporates resting heart rate (RHR) to calculate a target heart rate based on a desired percentage of heart rate reserve (HRR):
尽管这个估算值存在个体差异，但它仍是制定训练强度的基础。更精确的卡沃南公式融入了静息心率（RHR），根据心率储备（HRR）的期望百分比来计算目标心率：
Target HR = RHR + (Desired Intensity % × (HRmax − RHR))
Cardiac output (Q), the volume of blood pumped by the heart per minute, is the product of heart rate (HR) and stroke volume (SV). An average resting Q is about 5 L/min but can increase dramatically during maximal exercise.
心输出量（Q），即心脏每分钟泵出的血量，是心率（HR）与每搏输出量（SV）的乘积。安静时平均心输出量约为 5 升/分，但在最大强度运动时可急剧增加。
Q (L/min) = HR (bpm) × SV (L/beat)
Note that stroke volume is often expressed in millilitres; divide by 1000 to obtain litres. These relationships link directly to aerobic performance and are frequently examined.
注意每搏输出量常用毫升表示，需除以 1000 换算成升。这些关系与有氧表现直接相关，是考试中的常考点。
2. Respiratory Calculations | 呼吸系统计算
Pulmonary ventilation (VE) represents the volume of air moved into and out of the lungs each minute. It depends on two factors: the depth of each breath (tidal volume, TV) and the number of breaths per minute (breathing frequency, f).
肺通气量（VE）表示每分钟进出肺部的空气量。它取决于两个因素：每次呼吸的深度（潮气量，TV）和每分钟呼吸的次数（呼吸频率，f）。
VE (L/min) = TV (L) × f (breaths/min)
During exercise, both tidal volume and breathing frequency rise, driving ventilation from a resting value of about 6 L/min to over 150 L/min in trained athletes. Respiratory exchange ratio (RER) is another commonly used metric, defined as the ratio of carbon dioxide produced (VCO₂) to oxygen consumed (VO₂). It indicates fuel utilisation; an RER of 1.0 suggests predominantly carbohydrate oxidation, while 0.7 indicates fat oxidation.
运动时，潮气量和呼吸频率均上升，使通气量从安静时的约 6 升/分增加到训练有素运动员的 150 升/分以上。呼吸交换比（RER）是另一个常用指标，定义为二氧化碳产生量（VCO₂）与氧气消耗量（VO₂）之比。它能指示燃料利用情况；RER 为 1.0 提示以碳水化合物氧化为主，0.7 则提示脂肪氧化。
RER = VCO₂ / VO₂
Understanding these values helps link pulmonary function to energy metabolism and endurance performance.
理解这些数值有助于将肺功能与能量代谢和耐力表现联系起来。
3. Linear Motion & Kinematics | 线运动与运动学
The basic kinematic quantities – speed, velocity, and acceleration – describe how an athlete or object moves along a straight or curved path. Speed is a scalar quantity, while velocity is a vector that includes direction.
基本的运动学量——速率、速度和加速度——描述了运动员或物体沿直线或曲线路径的运动情况。速率是标量，而速度是包含方向的矢量。
Average Speed = total distance / time taken
Average Velocity = displacement / time
Acceleration = change in velocity / time taken
Momentum is a fundamental quantity representing the ‘quantity of motion’ and is the product of mass and velocity. In sport, a heavier, faster-moving athlete or object carries greater momentum, making it harder to stop or change direction.
动量是代表“运动量”的基本物理量，为质量与速度的乘积。在体育运动中，更重、更快的运动员或物体具有更大的动量，因此更难停下来或改变方向。
Momentum (p) = mass (m) × velocity (v)
These linear motion principles underpin analyses of sprinting, throwing, and tackling mechanics.
这些直线运动原理是分析短跑、投掷和擒抱动作力学的基础。
4. Force, Momentum & Impulse | 力、动量与冲量
Newton’s three laws of motion govern the force–motion relationship. The Second Law directly quantifies the acceleration produced by a net force:
牛顿三定律支配着力与运动的关系。第二定律直接量化了合力产生的加速度：
Force (F) = mass (m) × acceleration (a)
Impulse, the product of force and the time over which it acts, equals the change in momentum. This explains why coaches emphasise ‘follow-through’ – extending the time of force application increases impulse and therefore the change in velocity of a ball or implement.
冲量是力与其作用时间的乘积，等于动量的变化量。这解释了教练为何强调“随挥”动作——延长施力时间可增加冲量，从而增大球或器械的速度变化。
Impulse = F × Δt = Δp = m × Δv
Ground reaction forces, friction, and air resistance also influence performance; they are analysed using free-body diagrams and vector resolution.
地面反作用力、摩擦力和空气阻力也会影响表现，通常通过受力图和矢量分解进行分析。
5. Torque & Levers | 力矩与杠杆
Torque (or moment) is the turning effect produced by a force acting around an axis of rotation. It is central to understanding how muscles create joint movement.
力矩（扭矩）是力绕转动轴产生的转动效应。它是理解肌肉如何产生关节运动的核心。
Torque (τ) = Force (F) × perpendicular distance from axis (d)
The human body operates through three classes of levers, where bones act as levers, joints as fulcrums, and muscles provide the effort. Mechanical advantage (MA) compares the effort arm to the resistance arm. When MA > 1, a small effort can move a larger resistance, typical of second-class levers (e.g., standing calf raise). First-class levers can favour either speed or force depending on fulcrum placement, while third-class levers (most common in the body, such as the biceps curl) favour speed and range of motion over force.
人体通过三级杠杆运作，其中骨骼为杠杆，关节为支点，肌肉提供动力。机械利益（MA）对比动力臂与阻力臂。当 MA 大于 1 时，较小的力就能移动较大的阻力，这是第二类杠杆的典型特征（如站立提踵）。第一类杠杆根据支点位置既可偏向力量也可偏向速度，而第三类杠杆（人体中最常见，如肱二头肌弯举）力量较小但有利于速度和运动范围。
Mechanical Advantage = Effort Arm Length / Resistance Arm Length
Understanding lever systems helps explain movement efficiency and risk of injury.
理解杠杆系统有助于解释动作效率与受伤风险。
6. Energy, Work & Power | 能量、功与功率
Energy exists in mechanical forms that relate directly to sporting action. Kinetic energy (KE) is the energy due to motion, while gravitational potential energy (GPE) is the energy due to an object’s height above the ground.
能量以与运动动作直接相关的机械形式存在。动能（KE）是物体因运动而具有的能量，重力势能（GPE）是物体因高于地面而具有的能量。
KE = ½ × m × v²
GPE = m × g × h (g = 9.81 m/s²)
Work is done when a force moves its point of application; it equals force multiplied by the distance moved in the direction of the force. Power is the rate of doing work, a key determinant of explosive performance.
力使其作用点发生位移时就做了功；功等于力乘以沿力方向移动的距离。功率是做功的速率，是决定爆发力表现的关键因素。
Work = F × d × cosθ (simplified: Work = F × d for parallel force)
Power = Work / time = F × v
Mechanical efficiency assesses how much of the total energy expended is converted into useful mechanical work. It rarely exceeds 25% in human movement.
机械效率评估的是总能量消耗中有多大比例转化为有用的机械功。人体运动的机械效率很少超过 25%。
Mechanical Efficiency (%) = (Useful Work Output / Total Energy Expended) × 100
7. Angular Motion | 角运动
Rotational movements are prevalent in diving, gymnastics, and throwing actions. Angular equivalents of linear quantities describe these motions. Angular velocity (ω) measures the rate of spin.
旋转运动在跳水、体操和投掷动作中非常普遍。角运动用线性量的对应角量来描述。角速度（ω）衡量旋转的快慢。
ω = Δθ / Δt (radians per second)
Moment of inertia (I) is the resistance to angular acceleration, depending on mass and how it is distributed relative to the axis of rotation. Tucking in a somersault reduces moment of inertia, increasing angular velocity due to conservation of angular momentum (L).
转动惯量（I）是对角加速度的阻力，取决于质量及其相对于转动轴的分布。空翻中团身可减小转动惯量，根据角动量守恒（L），角速度随之增加。
Angular Momentum (L) = I × ω
In the absence of external torques, angular momentum remains constant. This principle is exploited by athletes to control rotation speed.
在没有外力矩作用时，角动量保持恒定。运动员利用这一原理控制旋转速度。
8. Fluid Dynamics in Sport | 运动中的流体力学
When an object moves through air or water, fluid forces profoundly affect its trajectory. Bernoulli’s principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. This explains the lift force on an aerofoil and the curved flight of a spinning ball (Magnus effect).
当物体在空气或水中运动时，流体力会显著影响其轨迹。伯努利原理指出，流体速度增加时，压强同时降低。这解释了机翼的升力以及旋转球的弧线飞行（马格努斯效应）。
For a spinning ball, the surface that moves against the airflow creates high pressure, while the opposite side produces low pressure, generating a force towards the low-pressure zone. A topspin tennis ball dips faster, while a backspin ball floats.
对于旋转球，与气流相对运动的表面产生高压，另一侧则产生低压，从而形成指向低压区的力。上旋网球下坠更快，下旋球则会飘浮。
Drag force opposes motion and is influenced by fluid density, velocity, frontal cross-sectional area, and drag coefficient. Streamlining and surface design reduce drag, enhancing speed in cycling, swimming, and sprinting.
阻力与运动方向相反，受流体密度、速度、迎风截面积和阻力系数的影响。流线型姿势和表面设计可减少阻力，提高自行车、游泳和短跑的速度。
Drag Force (Fd) = ½ ρ v² Cd A
Where ρ is fluid density, v is velocity, Cd is drag coefficient, and A is cross-sectional area. Although not always tested numerically, the relationships are critical for understanding technique optimisation.
式中 ρ 为流体密度，v 为速度，Cd 为阻力系数，A 为横截面积。尽管不总进行数值计算，但这些关系对理解技术优化至关重要。
9. Skill Acquisition Laws | 技能习得定律
Two classic psychomotor laws explain the relationship between task difficulty, choice, and movement time. Hick’s law states that reaction time increases logarithmically as the number of stimulus–response alternatives increases. This informs coaching decisions to reduce options under pressure.
两条经典的心理运动定律解释了任务难度、选择数量与动作时间的关系。希克定律指出，随着刺激-反应选择数量的增加，反应时间呈对数增长。这为教练在压力下减少选项提供了依据。
Reaction Time (RT) = a + b log₂ (n)
Where a and b are empirically derived constants, and n is the number of possible choices. An increase from 1 to 2 choices produces a substantial RT rise, but further increases have diminishing effects.
其中 a 和 b 为经验常数，n 为可能的选择数量。从 1 个选择增加到 2 个选择时，反应时间大幅上升，但进一步增加的选择数量影响逐渐减小。
Fitts’ law predicts the time required to rapidly move to a target area, based on target width and distance. It is used to design sporting tasks, human-computer interfaces, and practice drills.
费茨定律根据目标宽度和距离预测快速移动至目标区域所需的时间。它被用于设计运动任务、人机界面和练习训练。
Movement Time (MT) = a + b log₂ (2D / W)
Where D is the distance from the starting point to the target centre, and W is the width of the target. A larger D or smaller W increases difficulty and movement time. This law highlights the speed–accuracy trade-off that athletes constantly manage.
其中 D 为起点到目标中心的距离，W 为目标宽度。D 越大或 W 越小，难度和动作时间均增加。该定律凸显了运动员需不断权衡的速度-准确性权衡关系。
10. Psychological Theories of Arousal | 心理唤醒理论
While not expressed as mathematical equations, several drive and arousal theories are fundamental principles in sport psychology, often stated as ‘laws’ or models. The Inverted-U hypothesis (Yerkes-Dodson Law) proposes that performance improves with arousal up to an optimal point, beyond which further arousal causes performance decline. The optimum depends on skill complexity and individual differences.
虽然不表现为数学方程，但若干驱力和唤醒理论是运动心理学的基本原则，常被称为“定律”或模型。倒 U 形假说（耶克斯-多德森定律）认为，表现随唤醒水平提高而改善直至最优值，超过该值后进一步唤醒会导致表现下降。最优值取决于技能复杂度和个体差异。
The Drive Theory suggests a linear relationship between arousal and performance for well-learned, simple skills, where the dominant response is more likely to be emitted. However, this oversimplifies complex sporting behaviours and is less supported in dynamic environments.
驱力理论认为，对于熟练的简单技能，唤醒与表现呈线性关系，此时主导反应更容易出现。但该理论过于简化复杂的运动行为，在动态环境中支持度较低。
More contemporary models, such as the Catastrophe Theory, incorporate both physiological arousal and cognitive anxiety, showing that performance can ‘drop off a cliff’ when anxiety is high and arousal passes a threshold. These frameworks help explain choking under pressure and inform anxiety management strategies.
较现代的理论，如突变理论，结合了生理唤醒与认知焦虑，表明当焦虑水平高且唤醒超过阈限时，表现会“陡然崩溃”。这些框架有助于解释压力下的发挥失常，并为焦虑管理策略提供依据。
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