📚 Pre-U Cambridge Physical Education: Formula & Theorem Quick Reference Handbook | Pre-U Cambridge 体育：公式定理速查手册
In Pre-U Cambridge Physical Education, mastery of key formulas and theoretical principles is essential for analysing human movement, physiological responses, and skill acquisition. This quick reference handbook consolidates the core equations, laws, and conceptual frameworks that underpin the syllabus, spanning biomechanics, exercise physiology, and sport psychology. Use this guide as a revision tool to reinforce your understanding and ensure precision in exam responses.
在 Pre-U Cambridge 体育课程中，掌握关键公式和理论原理对于分析人体运动、生理反应和技能习得至关重要。本速查手册整合了支撑教学大纲的核心方程、定律和概念框架，涵盖运动生物力学、运动生理学和运动心理学。将本指南用作复习工具，以加深理解并确保考试答题的准确性。
1. Biomechanical Principles: Newton's Laws of Motion | 运动生物力学原理：牛顿运动定律
Newton's First Law (Law of Inertia) states that a body remains at rest or in uniform motion in a straight line unless acted upon by an external force. In sport, a football remains stationary until kicked, and a sprinter continues moving forward unless friction or air resistance slows them down. The inertia of an object is directly proportional to its mass.
牛顿第一定律（惯性定律）指出，除非受到外力作用，否则物体将保持静止或沿直线匀速运动。在运动中，足球在被踢之前保持静止，短跑运动员除非受到摩擦力或空气阻力减速，否则将继续向前移动。物体的惯性与其质量成正比。
Newton' , m is mass (kg), and a is acceleration (m/s²). This explains why greater force is required to accelerate a heavier shot put compared to a lighter tennis ball. The law also underpins impulse-momentum relationships in striking and collision sports.
牛顿第二定律表达了力、质量和加速度之间的关系：F = m × a，其中 F 为净力（牛顿），m 为质量（千克），a 为加速度（米/秒²）。这解释了为什么加速更重的铅球比加速更轻的网球需要更大的力。该定律还为击打和碰撞运动中的冲量-动量关系提供了基础。
Newton's Third Law (Action-Reaction) states that for every action, there is an equal and opposite reaction. When a swimmer pushes water backward, the water exerts an equal forward force on the swimmer. This principle is fundamental to propulsion in running, jumping, and aquatic sports.
牛顿第三定律（作用力与反作用力）指出，每一个作用力都有一个大小相等、方向相反的反作用力。当游泳运动员向后推水时，水对运动员施加相等的向前力。这一原理是跑步、跳跃和水上运动中推进力的基础。
2. Linear Kinematics: Displacement, Velocity, and Acceleration | 线性运动学：位移、速度和加速度
ance is a scalar quantity measuring total path length in metres, while speed is measured in m/s. In a 400-metre lap, distance is 400 m regardless of direction changes.
平均速率用行进距离除以所用时间计算：速率 = 距离 / 时间（v = s / t）。距离是标量，以米为单位测量总路径长度，速率以米/秒为单位。在 400 米一圈中，无论方向如何变化，距离都是 400 米。
lacement (metres) is the straight-line distance from start to finish in a specified direction. If a 400 m runner finishes at the start line, displacement is 0 m, so average velocity is 0 m/s despite high average speed.
速度是矢量，描述位移的变化率：速度 = 位移 / 时间。位移（米）是从起点到终点的直线距离，并带有指定方向。如果 400 米跑运动员在起跑线完成比赛，位移为 0 米，因此尽管平均速率很高，平均速度也为 0 米/秒。
l velocity, u is initial velocity, and t is time. Units are m/s². Negative acceleration (deceleration) occurs when velocity decreases, such as during braking in cycling or landing from a jump.
加速度是速度的变化率：a = (v - u) / t，其中 v 为末速度，u 为初速度，t 为时间。单位为米/秒²。当速度减小时发生负加速度（减速度），例如骑自行车刹车或跳跃落地时。
SUVAT Equation: v² = u² + 2as
SUVAT 方程：v² = u² + 2as
Where s is displacement, this equation links velocity, acceleration, and displacement without requiring time. It is useful for analysing vertical jump height where take-off velocity and gravitational acceleration (-9.81 m/s²) are known.
其中 s 为位移，该方程将速度、加速度和位移联系起来而不需要时间。它适用于在已知起跳速度和重力加速度（-9.81 米/秒²）时分析纵跳高度。
3. Projectile Motion and Trajectory Calculations | 抛体运动与轨迹计算
Projectile motion describes the flight path of an object released into the air, governed by horizontal and vertical components. The horizontal component of velocity remains constant (ignoring air resistance), while the vertical component is subject to gravitational acceleration of -9.81 m/s². The optimal release angle for maximum range on flat ground is 45° in a vacuum.
抛体运动描述被抛入空中的物体的飞行路径，受水平和垂直分量控制。水平速度分量保持恒定（忽略空气阻力），而垂直分量受 -9.81 米/秒² 的重力加速度影响。在真空中，平坦地面上最大射程的最佳出手角为 45°。
/ g, where u is release velocity (m/s), θ is release angle (degrees), and g is gravitational acceleration (9.81 m/s²). In shot put, the release angle is typically lower (around 34-38°) because the release point is above the landing level, altering the optimal trajectory.
抛体射程：R = (u² × sin 2θ) / g，其中 u 为出手速度（米/秒），θ 为出手角度（度），g 为重力加速度（9.81 米/秒²）。在铅球中，出手角度通常较低（约 34-38°），因为出手点高于落地点，改变了最佳轨迹。
uations are essential for analysing long jump take-off, basketball shooting, and javelin flight.
从相同高度发射和落地的抛体飞行时间：t = (2u × sin θ) / g。达到的最大高度：H = (u² × sin² θ) / (2g)。这些方程对于分析跳远起跳、篮球投篮和标枪飞行至关重要。
4. Angular Kinematics: Rotational Motion Quantities | 角运动学：旋转运动量
of angular displacement: ω = Δθ / Δt, measured in rad/s. Angular acceleration (α) is the rate of change of angular velocity: α = Δω / Δt, measured in rad/s².
角位移（θ）是物体旋转的角度，以弧度（rad）或度为单位。角速度（ω）是角位移的变化率：ω = Δθ / Δt，单位为弧度/秒。角加速度（α）是角速度的变化率：α = Δω / Δt，单位为弧度/秒²。
where r is the radius of rotation (metres). This explains why a golfer with longer arms generates greater clubhead speed for the same angular velocity, increasing ball velocity at impact.
线速度与角速度之间的关系为：v = r × ω，其中 r 为旋转半径（米）。这解释了为什么手臂较长的高尔夫球手在相同角速度下能产生更大的杆头速度，从而增加击球时的球速。
Centripetal acceleration: ac rected toward the centre of rotation and is required to maintain circular motion. In hammer throwing, the athlete must exert centripetal force to keep the hammer moving in a circular path before release.
向心加速度：ac = v² / r = ω² × r。该加速度指向旋转中心，是维持圆周运动所必需的。在链球投掷中，运动员必须施加向心力，使链球在出手前保持圆周运动。
5. Moment of Inertia and Angular Momentum | 转动惯量与角动量
m × r²). The further the mass is distributed from the axis, the greater the moment of inertia. A diver in a tuck position has a smaller moment of inertia than in a layout position.
转动惯量（I）是物体对旋转运动变化的阻力，取决于质量及其相对旋转轴的分布：I = Σ (m × r²)。质量分布离轴越远，转动惯量越大。跳水运动员在团身姿势时的转动惯量比直体姿势时小。
erved in a closed system (no external torque). When a figure skater pulls arms in during a spin, I decreases, so ω must increase to conserve L, causing faster rotation.
角动量（L）是转动惯量与角速度的乘积：L = I × ω。在封闭系统中（无外力矩），角动量守恒。当花样滑冰运动员在旋转中收紧双臂时，I 减小，因此 ω 必须增大以保持 L 不变，导致旋转更快。
This principle of conservation of angular momentum governs movements in diving, gymnastics, and trampolining. Athletes manipulate body configuration to control rotation speed during flight phases, critical for executing complex somersaults and twists.
角动量守恒原理支配着跳水、体操和蹦床运动中的动作。运动员通过操纵身体形态来控制飞行阶段的旋转速度，这对于执行复杂的空翻和转体动作至关重要。
6. Cardiovascular Dynamics: Cardiac Output and Stroke Volume | 心血管动力学：心输出量与每搏输出量
rate. Stroke volume (SV) is the volume of blood ejected per beat (ml/beat), and heart rate (HR) is beats per minute (bpm). Typical resting Q is approximately 5 L/min for an untrained individual.
心输出量（Q）是心脏每分钟泵出的血液量，计算公式为：Q = 每搏输出量 × 心率。每搏输出量（SV）是每次心跳射出的血液量（毫升/次），心率（HR）为每分钟心跳次数（次/分）。未经训练的个体典型静息心输出量约为 5 升/分。
Q (L/min) = SV (ml/beat) × HR (bpm) / 1000
Q（升/分）= SV（毫升/次）× HR（次/分）/ 1000
During maximal exercise, Q can increase to 20-40 L/min in elite endurance athletes. This increase is achieved through elevated HR (up to ~200 bpm) and increased SV due to enhanced venous return and myocardial contractility. Training adaptations include increased left ventricular volume and stroke volume at rest and during submaximal exercise.
在最大运动强度下，精英耐力运动员的心输出量可增至 20-40 升/分。这一增加通过心率升高（最高约 200 次/分）以及静脉回流和心肌收缩力增强导致的每搏输出量增加来实现。训练适应包括静息和亚极量运动时左心室容积和每搏输出量增加。
7. The Fick Principle and Oxygen Uptake | 菲克原理与摄氧量
The Fick principle states that oxygen consumption (V̇O₂) equals cardiac output multiplied by the arteriovenous oxygen difference: V̇ ence represents the amount of oxygen extracted by tissues from each litre of blood, normally ~4-5 ml O₂ per 100 ml blood at rest, increasing to ~15-18 ml O₂ per 100 ml during intense exercise.
菲克原理指出，摄氧量（V̇O₂）等于心输出量乘以动静脉氧差：V̇O₂ = Q × (a-v O₂ diff)。动静脉氧差代表组织从每升血液中提取的氧气量，静息时通常为每 100 毫升血液约 4-5 毫升 O₂，剧烈运动时增至每 100 毫升约 15-18 毫升 O₂。
Maximal oxygen uptake (V̇O₂ max) is the gold standard measure of cardiorespiratory fitness. It represents the maximum rate at which the body can consume oxygen during exhaustive exercise. Typical V̇O₂ max values range from 35-45 ml/kg/min in untrained individuals to 70-85 ml/kg/min in elite endurance athletes.
最大摄氧量（V̇O₂ max）是心肺适能的黄金标准测量指标。它代表身体在力竭运动中消耗氧气的最大速率。典型 V̇O₂ max 值范围从未经训练个体的 35-45 毫升/千克/分到精英耐力运动员的 70-85 毫升/千克/分。
V̇O₂ max = Q max × (a-v O₂ diff) max
V̇O₂ max = Q 最大值 × (a-v O₂ diff) 最大值
8. Respiratory Mechanics: Minute Ventilation and Pulmonary Volumes | 呼吸力学：每分通气量与肺容量
Minute ventilation (V̇E) is the volume of air moved into and out of the lungs per minute, calculated as: V̇ y. Tidal volume (TV) is the volume of air per breath (litres), and breathing frequency (f) is breaths per minute. Resting V̇E is typically 6-8 L/min.
每分通气量（V̇E）是每分钟进出肺部的空气量，计算公式为：V̇E = 潮气量 × 呼吸频率。潮气量（TV）是每次呼吸的空气量（升），呼吸频率（f）为每分钟呼吸次数。静息每分通气量通常为 6-8 升/分。
V̇E (L/min) = TV (L) × f (breaths/min)
V̇E（升/分）= TV（升）× f（次/分）
y 3-5 L. Residual volume (RV) is the air remaining in the lungs after maximal exhalation (~ = VC + RV. These volumes influence respiratory efficiency during high-intensity exercise.
肺容量包括肺活量（VC），即最大吸气后能呼出的最大空气量，通常为 3-5 升。残气量（RV）是最大呼气后留在肺内的空气（约 1.2 升）。肺总量 = VC + RV。这些容量影响高强度运动时的呼吸效率。
9. Energy Systems: ATP Yield and Metabolic Equations | 能量系统：ATP 产量与代谢方程
TP + Cr, catalysed by creatine kinase. This system yields approximately 1 ATP per PCr molecule and is the primary energy source for explosive movements such as sprint starts and powerlifting.
ATP-PC（磷酸肌酸）系统为持续长达 10 秒的高强度活动提供即时能量。反应式：PCr + ADP → ATP + Cr，由肌酸激酶催化。该系统每分子 PCr 约产生 1 个 ATP，是冲刺起跑和力量举等爆发性运动的主要能量来源。
Anaerobic glycolysis yields a net 2 ATP per glucose molecule through the breakdown of glucose to pyruvate, which is converted to lactate when oxygen is insufficient. This system predominates in efforts lasting 30-90 seconds, such as 400 m sprinting. The accumulation of hydrogen ions from lactic acid dissociation contributes to metabolic acidosis and fatigue.
无氧糖酵解通过将葡萄糖分解为丙酮酸，每分子葡萄糖净产 2 个 ATP；当氧气不足时，丙酮酸转化为乳酸。该系统在持续 30-90 秒的运动中占主导地位，如 400 米冲刺。乳酸解离产生的氢离子积累导致代谢性酸中毒和疲劳。
The aerobic system yields approximately 36-38 ATP per glucose molecule through glycolysis, the Krebs cycle, and the electron transport chain. This system dominates in endurance activities lasting longer than 2-3 minutes. The complete oxidation of a fatty acid molecule can yield over 100 ATP, making fat a high-energy-density fuel for prolonged exercise.
有氧系统通过糖酵解、克雷布斯循环和电子传递链，每分子葡萄糖约产生 36-38 个 ATP。该系统在持续超过 2-3 分钟的耐力活动中占主导地位。一个脂肪酸分子的完全氧化可产生超过 100 个 ATP，使脂肪成为长时间运动的高能量密度燃料。
10. Anthropometric Measurements: Body Composition and Somatotype | 人体测量：身体成分与体型分类
ight² (m²). While widely used, BMI does not distinguish between muscle and fat mass, limiting its accuracy for athletic populations. A rugby forward with high muscle mass may be classified as 'overweight' or 'obese' despite low body fat percentage.
身体质量指数（BMI）是一种筛查工具，计算公式为：BMI = 体重（千克）/ 身高²（米²）。虽然广泛使用，但 BMI 不能区分肌肉和脂肪质量，限制了其在运动员人群中的准确性。肌肉质量高的橄榄球前锋可能被归类为"超重"或"肥胖"，尽管体脂率较低。
Body fat percentage can be estimated using skinfold callipers and regression equations such as the Durnin and Womersley formula or Jackson and Pollock equations. Measurements are taken at specific anatomical sites (e.g., triceps, subscapular, suprailiac) and converted to body density, then to body fat percentage using the Siri or Brozek equations.
体脂率可用皮褶卡尺和回归方程（如 Durnin 和 Womersley 公式或 Jackson 和 Pollock 方程）进行估算。在特定解剖部位（如肱三头肌、肩胛下、髂上）进行测量，并转换为身体密度，然后使用 Siri 或 Brozek 方程转换为体脂率。
Siri Equation: Body Fat % = (495 / Body Density) - 450
Siri 方程：体脂率 % = (495 / 身体密度) - 450
Somatotype ratings use a three-number scale (endomorphy, mesomorphy, ectomorphy) rated 1-7. Endomorphy reflects relative fatness, mesomorphy reflects musculoskeletal robustness, and ectomorphy reflects linearity. Gymnasts typically exhibit high mesomorphy (5-6) with moderate ectomorphy, while marathon runners show higher ectomorphy (4-6).
体型评分使用三数字量表（内胚型、中胚型、外胚型），评分为 1-7。内胚型反映相对肥胖度，中胚型反映肌肉骨骼强健度，外胚型反映线性度。体操运动员通常表现出高中胚型（5-6）和中度外胚型，而马拉松运动员表现出更高的外胚型（4-6）。
11. Motor Unit Recruitment and Force Production | 运动单位募集与力量产生
The Henneman size principle states that motor units are recruited in order of increasing size and excitability threshold. Smaller, slow-twitch (Type I) motor units are recruited first for low-force, endurance-based activities. As force demand increases, larger fast-twitch (Type IIa, then Type IIx) motor units are progressively recruited.
亨尼曼大小原则指出，运动单位按大小和兴奋阈值递增的顺序募集。较小的慢肌（I 型）运动单位首先被募集，用于低力量、耐力型活动。随着力量需求增加，较大的快肌（IIa 型，然后是 IIx 型）运动单位逐步被募集。
The force-velocity relationship describes that the force a muscle can produce decreases as contraction velocity increases. Maximum isometric force (zero velocity) exceeds concentric force at any speed. Peak power output occurs at approximately 30-40% of maximum shortening velocity and ~30% of maximum isometric force.
力量-速度关系描述肌肉产生的力量随收缩速度增加而减小。最大等长力量（零速度）超过任何速度下的向心力量。峰值功率输出发生在最大缩短速度约 30-40% 和最大等长力量约 30% 处。
The length-tension relationship explains that a muscle produces maximum force at its resting (optimal) length, where actin and myosin filament overlap is ideal. Force decreases when the muscle is excessively shortened or lengthened beyond this optimum, reflecting reduced cross-bridge formation potential.
长度-张力关系解释肌肉在其静息（最佳）长度时产生最大力量，此时肌动蛋白和肌球蛋白丝重叠最为理想。当肌肉过度缩短或拉长超过此最佳长度时力减小，这反映了横桥形成潜力的降低。
12. Levers and Mechanical Advantage in the Human Body | 人体中的杠杆与机械优势
t arm length / resistance arm length. When MA > 1, the system favours force production; when MA < 1, it favours range and speed of movement. Most human levers are third-class, sacrificing force for speed and range.
杠杆系统由支点（关节）、动力（肌力）和阻力（负荷）组成。机械优势（MA）计算公式为：MA = 动力臂长度 / 阻力臂长度。当 MA > 1 时，系统有利于力量产生；当 MA < 1 时，有利于运动范围和速度。人体大多数杠杆是第三类杠杆，牺牲力量以换取速度和范围。
| Lever Class | Arrangement | Example |
|---|---|---|
| First Class | Fulcrum between effort and resistance | Neck extension (atlanto-occipital joint) |
| Second Class | Resistance between fulcrum and effort | Calf raise (ball of foot) |
| Third Class | Effort between fulcrum and resistance | Biceps curl (elbow joint) |
| 杠杆类型 | 排列方式 | 示例 |
|---|---|---|
| 第一类 | 支点在动力和阻力之间 | 颈部伸展（寰枕关节） |
| 第二类 | 阻力在支点和动力之间 | 小腿提踵（脚掌） |
| 第三类 | 动力在支点和阻力之间 | 肱二头肌弯举（肘关节） |
In a third-class lever, the effort arm is shorter than the resistance arm, so MA is always less than 1. This arrangement allows the distal end of the limb to move faster and further than the muscle contraction distance, advantageous for throwing and kicking actions where limb speed is prioritised over raw force.
在第三类杠杆中，动力臂比阻力臂短，因此 MA 始终小于 1。这种排列使肢体远端比肌肉收缩距离移动得更快、更远，有利于投掷和踢腿动作，在这些动作中，肢体速度优先于单纯力量。
13. Agility, Balance, and Centre of Mass | 灵敏性、平衡与质心
The centre of mass (COM) is the point at which the body's mass is equally distributed in all directions. In anatomical position, the COM lies approximately at the level of the second sacral vertebra, but it shifts with body position changes. Stability is maximised when the line of gravity falls within the base of support.
质心（COM）是身体质量在所有方向上均匀分布的点。在解剖学姿势中，质心大约位于第二骶椎水平，但随身体姿势变化而移动。当重力线落在支撑面内时，稳定性最大。
Balance is governed by three factors: base of support area (wider base increases stability), height of COM (lower COM increases stability), and the distance of the line of gravity from the edge of the base of support. A wrestler adopting a crouched stance lowers COM and widens the base, maximising resistance to being moved.
平衡受三个因素支配：支撑面面积（更宽的基底增加稳定性）、质心高度（较低的质心增加稳定性）以及重力线距支撑面边缘的距离。摔跤运动员采取蹲姿时可降低质心、加宽支撑面，最大化抗移动能力。
Agility combines speed, balance, coordination, and the ability to change direction rapidly. The Illinois agility test and T-test are standard assessments. Agility performance depends on neuromuscular coordination, reactive strength, and the efficient transfer of momentum during directional changes.
灵敏性结合了速度、平衡、协调和快速改变方向的能力。伊利诺伊灵敏性测试和 T 测试是标准评估方法。灵敏性表现取决于神经肌肉协调、反应力量以及方向改变时动量的有效转移。
14. Training Principles and Overtraining Theory | 训练原则与过度训练理论
The principle of progressive overload states that training adaptations occur when the body is subjected to a stimulus exceeding its current capacity. The FITT principles guide prescription: Frequency, Intensity, Time (duration), and Type of exercise. For aerobic improvement, intensity is often prescribed using heart rate reserve (HRR) via the Karvonen formula.
渐进超负荷原则指出，当身体受到超过其当前能力的刺激时会发生训练适应。FITT 原则指导运动处方：频率、强度、时间（时长）和运动类型。对于有氧能力改善，常使用卡沃宁公式通过心率储备（HRR）来制定强度。
Target HR = Resting HR + (Intensity % × [HR max - Resting HR])
目标心率 = 静息心率 + (强度 % × [最大心率 - 静息心率])
r older populations. The General Adaptation Syndrome (GAS) model describes three stages of response to training stress: alarm, resistance, and exhaustion. Overtraining syndrome represents maladaptation when recovery is insufficient relative to training load.
最大心率可使用标准公式 220 - 年龄 或适用于老年人群的 208 - (0.7 × 年龄) 进行估算。一般适应综合征（GAS）模型描述了训练应激反应的三个阶段：警觉、抵抗和衰竭。当恢复相对于训练负荷不足时，过度训练综合征代表适应不良。
15. Skill Acquisition: Feedback, Transfer, and Information Processing | 技能习得：反馈、迁移与信息处理
put. Reaction time (RT) = the time between stimulus onset and movement initiation. Movement time (MT) = time from movement start to completion. Response time = RT + MT. Hick' es that RT increases logarithmically as the number of stimulus-response choices increases.
信息处理使用 Welford 或 Whiting 模型，涉及感官受体输入、感知、决策和运动输出。反应时（RT）= 从刺激出现到动作开始的时间。动作时（MT）= 从动作开始到完成的时间。响应时 = RT + MT。希克定律指出，反应时随刺激-反应选择数以对数函数增加。
Feedback types: intrinsic (sensory information from within the body, e.g., proprioception), extrinsic/augmented (information from an external source, e.g., coach), knowledge of results (KR, outcome information), and knowledge of performance (KP, quality of movement information). Effective feedback should be specific, constructive, and appropriately timed.
反馈类型：内在反馈（来自身体内部的感觉信息，例如本体感觉）、外在/增强反馈（来自外部来源的信息，例如教练）、结果知晓（KR，结果信息）和表现知晓（KP，动作质量信息）。有效的反馈应当具体、建设性且时机恰当。
Transfer of learning can be positive (previous learning benefits new skill), negative (previous learning hinders new skill), bilateral (transfer between limbs), or proactive/retroactive. A tennis player learning badminton may experience negative transfer initially due to differing wrist actions despite similar racket sports contexts.
学习迁移可以是正迁移（先前学习有利于新技能）、负迁移（先前学习阻碍新技能）、双侧迁移（肢体间迁移）或前摄/倒摄迁移。网球运动员学习羽毛球时，由于手腕动作不同，尽管同属球拍类运动，最初可能经历负迁移。
Published by TutorHao | Physical Education Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图