📚 Year 13 Edexcel PE: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
Master the essential formulas, equations and theoretical principles for the Year 13 Edexcel Physical Education specification. This quick reference handbook covers biomechanics, exercise physiology, sport psychology and skill acquisition – everything you need for your linear exams in one place. Keep it handy for revision, coursework analysis and exam question practice.
掌握 Year 13 Edexcel 体育课程的核心公式、方程与理论知识。这本速查手册涵盖生物力学、运动生理学、运动心理学和技能习得，是你应对线性考试的一站式复习利器。随时翻阅，助你攻克考试分析与答题训练。
1. Cardiac Output & Heart Rate | 心输出量与心率公式
. At rest for an untrained individual, typical values might be HR ≈ 70 bpm and SV ≈ 70 mL, giving Q ≈ 4.9 L/min. During maximal exercise, elite endurance athletes can achieve Q values above 35 L/min.
心输出量（Q）指左心室每分钟泵出的血液量。基础公式为 Q = 心率（HR）× 每搏输出量（SV）。未经训练的个体安静时，典型数值为心率约 70 次/分，每搏输出量约 70 mL，心输出量约 4.9 L/min。在最大强度运动中，优秀耐力运动员的心输出量可达 35 L/min 以上。
Stroke volume itself is influenced by venous return, myocardial contractility and afterload. According to Starling's law of the heart, increased venous return leads to greater stretch of cardiac muscle fibres, resulting in a more forceful contraction and increased SV. This relationship underpins cardiovascular drift during prolonged exercise.
每搏输出量受静脉回心血量、心肌收缩力和后负荷影响。根据 Starling 心脏定律，静脉回心血量增加导致心肌纤维拉长，收缩更有力，从而使每搏输出量增加。这一关系是长时间运动中“心血管漂移”的基础。
2. Arteriovenous Oxygen Difference & Oxygen Consumption | 动静脉氧差与摄氧量
v)O₂ diff. Here, (a-v)O₂ diff is the difference in oxygen content between arterial and mixed venous blood, reflecting the amount of oxygen extracted by the tissues. At rest, (a-v)O₂ diff is about 5 mL O₂ per 100 mL of blood; during intense exercise, it can rise to 15–17 mL O₂/100 mL.
摄氧量（VO₂）由 Fick 原理确定：VO₂ = Q × (a-v)O₂ 差。其中 (a-v)O₂ 差为动脉血与混合静脉血氧含量差值，反映组织摄氧量。安静时，(a-v)O₂ 差约为 5 mL O₂/100 mL 血液；剧烈运动时可升至 15–17 mL O₂/100 mL。
Maximal oxygen uptake (VO₂max) is the gold standard measure of cardiorespiratory fitness. It can be expressed in absolute terms (L/min) or relative to body mass (mL/kg/min). A typical untrained young adult male may have a VO₂max around 40–45 mL/kg/min, while an elite male cross-country skier might exceed 90 mL/kg/min. VO₂max is limited by central factors (Qmax) and peripheral factors (muscle oxidative capacity).
最大摄氧量（VO₂max）是衡量心肺适能的黄金标准，既可用绝对值（L/min）表示，也可用相对值（mL/kg/min）表示。未经训练的年轻男性 VO₂max 约为 40–45 mL/kg/min，而优秀男子越野滑雪选手可超过 90 mL/kg/min。VO₂max 受中枢因素（最大心输出量）与外周因素（肌肉氧化能力）的限制。
3. Minute Ventilation & Respiratory Parameters | 每分通气量与呼吸参数
, TV ≈ 0.5 L and f ≈ 12 breaths/min, giving VE ≈ 6 L/min. During maximal exercise, TV can increase to 3–4 L and f to 40–60 breaths/min, pushing VE above 150 L/min in well-trained athletes.
每分通气量（VE）是每分钟进出肺的空气总量：VE = 潮气量（TV）× 呼吸频率（f）。安静时 TV ≈ 0.5 L，f ≈ 12 次/分，VE ≈ 6 L/min。最大运动时 TV 可增至 3–4 L，f 至 40–60 次/分，训练有素的运动员 VE 可超过 150 L/min。
Alveolar ventilation (VA) accounts for dead space (VD): VA = (TV – p breathing more efficient for gas exchange. Pulmonary diffusion capacity for oxygen increases during exercise because of recruitment of pulmonary capillaries and increased blood flow.
肺泡通气量（VA）需扣除死腔（VD）：VA = (TV – VD) × f。成人生理解剖死腔约为 150 mL。浅快呼吸时，每次呼吸浪费比例更大，因此深慢呼吸对气体交换更高效。运动时肺毛细血管的募集与血流加快使肺换气弥散能力增强。
4. Metabolic Equations & Energy Expenditure | 代谢方程与能量消耗
nergy expenditure from METs: kcal/min = (MET × body mass in kg × 3.5) ÷ 200. For example, jogging at 8 km/h (~8 METs) for a 70 kg person expends about 9.8 kcal/min.
代谢当量（MET）用于量化运动强度：1 MET = 3.5 mL O₂/kg/min，代表安静代谢率。由 MET 推算能量消耗的公式为：千卡/分钟 = (MET × 体重 kg × 3.5) ÷ 200。例如，体重 70 kg 的人以 8 km/h 慢跑（约 8 MET），能量消耗约为 9.8 千卡/分钟。
Respiratory exchange ratio (RER) is the ratio of CO₂ produced to O₂ consumed (VCO₂/VO₂). An RER of 0.7 indicates pure fat oxidation, 0.85 a mixture, and 1.0 pure carbohydrate oxidation. During high-intensity exercise, RER can exceed 1.0 because of buffering of lactic acid, reflecting non-metabolic CO₂ production.
呼吸交换率（RER）是二氧化碳生成量与摄氧量的比值（VCO₂/VO₂）。RER 0.7 表示纯脂肪氧化，0.85 为混合供能，1.0 为纯碳水化合物氧化。大强度运动时由于乳酸缓冲作用，RER 可超过 1.0，反映出非代谢性 CO₂ 的产生。
5. Training Intensity & Heart Rate Zones | 训练强度与心率区间
n formula calculates target heart rate using heart rate reserve (HRR): Target HR = [(HRmax – HRrest) × %intensity] + HRrest. HRmax can be estimated as 220 – age, though the formula 208 – ividual fitness level and avoids prescribing the same absolute working heart rate for all.
Karvonen 公式利用心率储备（HRR）计算目标心率：目标心率 = [(最大心率 – 安静心率) × 强度%] + 安静心率。最大心率可用 220 – 年龄估算，但 208 – (0.7 × 年龄) 对年长者更准确。该方法考虑了个人体能水平，避免给所有人设定相同的绝对工作心率。
For aerobic training, zones are typically set at 60–80% HRR. For high-intensity interval training (HIIT), work intervals may reach 85–95% ing factor, where the weighting factor increases exponentially with intensity to reflect physiological strain.
有氧训练通常设定在 60–80% HRR 区间。高强度间歇训练的工作段可达 85–95% 最大心率。训练冲量（TRIMP）综合时长与强度，利用心率数据：TRIMP = 训练时长 × Δ心率比 × 权重因子，权重因子随强度呈指数增长，以反映生理负荷。
6. Linear Kinematics & Projectile Motion | 直线运动学与抛射运动
= ut + ½at²; v² = u² + 2as; s = ½(u + v)t. Here u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement. These assume constant acceleration and linear motion.
运动学方程（SUVAT）用于体育与生物力学中的运动分析。基本关系为：v = u + at；s = ut + ½at²；v² = u² + 2as；s = ½(u + v)t。其中 u = 初速度，v = 末速度，a = 加速度，t = 时间，s = 位移。上述方程假设匀加速直线运动。
s given by R = (v₀² sin 2θ)/g. Maximum height = (v₀² sin²θ)/(2g). The optimal angle for maximum range on a flat surface is 45° when release and landing heights are equal; however, in many sports such as shot put, release height exceeds landing height, so the optimum angle is slightly less than 45°.
抛射运动中，水平和垂直分量相互独立。若出手速度为 v₀、角度为 θ，水平射程 R = (v₀² sin 2θ)/g，最大高度 = (v₀² sin²θ)/(2g)。当出手与落地高度相同时，平地最大射程的最佳角度为 45°；但在铅球等项目中出手高度高于落地高度，最佳角度略小于 45°。
7. Angular Kinematics & Torque | 角运动学与力矩
er is v = ωr, where r is the radius of rotation. In a golf swing or tennis serve, a longer lever (extended arm or racket) generates greater tangential velocity at impact for the same angular velocity.
角速度（ω）为角位移的变化率：ω = Δθ/Δt，单位为 rad/s。转动杠杆上某点的线速度（切向速度）为 v = ωr，其中 r 为转动半径。在高尔夫挥杆或网球发球中，相同角速度下，更长的杠杆（伸展手臂或球拍）可在击球点产生更大的切向速度。
ce (moment arm). In the human body, muscles create torque about joints. For example, the quadriceps generate an extensor torque at the knee, and the moment arm of the patella increases the effective torque. Angular impulse is τ × t and equals the change in angular momentum (ΔL).
力矩（τ）是力的旋转对应量：τ = F × d，其中 d 为从转轴到力作用线的垂直距离（力臂）。人体中，肌肉围绕关节产生力矩。例如股四头肌在膝关节产生伸膝力矩，而髌骨的力臂可增大有效力矩。角冲量为 τ × t，等于角动量的变化量（ΔL）。
8. Momentum, Impulse & Impact Mechanics | 动量、冲量与碰撞力学
ss and velocity: p = mv. Impulse (J) is the product of force and time: J = Ft. The impulse– impulse equals change in momentum: Ft = Δ(mv). Coaches apply this by teaching athletes to “follow through” – time of force application increases impulse and thus change in velocity of the projectile.
线性动量（p）是质量与速度的乘积：p = mv。冲量（J）是力与时间的乘积：J = Ft。冲量–动量关系指出冲量等于动量的变化：Ft = Δ(mv)。教练员运用此原理指导运动员“随挥”——延长力的作用时间增加冲量，从而增大物体的速度变化。
Coefficient of restitution (e) measures the elasticity of a collision: e = √rebound height/drop height nelastic collision has e = 0 (objects stick together). In football, FIFA regulations require a ball to have e between 0.82 and 0.88 when dropped from 2 m, influencing rebound behaviour.
恢复系数（e）衡量碰撞弹性：e = √回弹高度 / 下落高度 或 e = (碰撞后相对速度) / (碰撞前相对速度)。完全弹性碰撞 e = 1（如理想超级球）；完全非弹性碰撞 e = 0（物体粘合）。足球竞赛规则要求从 2 m 落下时，球体的恢复系数在 0.82–0.88 之间，从而影响反弹表现。
9. Fluid Mechanics & Drag Forces | 流体力学与阻力
Drag force (FD) opposes motion through a fluid and is calculated as: FD = ½ CD ρ A v². Here CD is the drag coefficient (shape dependent), ρ is fluid density, A is frontal cross-sectional area, and v is velocity relative to the fluid. Because drag is proportional to v², even a small increase in speed results in a large increase in drag – crucial in cycling, swimming and sprinting.
阻力（FD）阻碍物体在流体中的运动，计算公式为 FD = ½ CD ρ A v²。其中 CD 为阻力系数（与形状相关），ρ 为流体密度，A 为正面横截面积，v 为相对于流体的速度。由于阻力与 v² 成正比，速度略微增加便会导致阻力显著上升——这对自行车、游泳和短跑极为关键。
Bernoulli's principle explains the lift force on an aerofoil: faster fluid flow over a curved surface creates lower pressure, generating a net lift force perpendicular to flow. In discus and javelin, athletes exploit aerodynamic lift to extend flight time. The Magnus effect occurs when a spinning ball deflects due to pressure differences, explaining the curved trajectory of a bending free kick in football.
伯努利原理解释了翼型升力：曲面流速较快则压力较低，产生垂直于流动方向的净升力。在铁饼与标枪中，运动员利用空气动力升力延长飞行时间。马格努斯效应则说明旋转球体因压力差而偏转，解释了足球中香蕉球的弧线轨迹。
10. Levers & Mechanical Advantage | 杠杆与机械效益
.g. neck extension), the fulcrum lies between effort and resistance, and MA can be >1, =1 or <1. Second-class levers (e.g. calf raise) always have MA >1, magnifying force. Third-class levers (e.g. biceps curl) have MA <1, sacrificing force for speed and range of motion – the most common lever class in the body.
人体肌肉骨骼系统以杠杆系统运作。机械效益 (MA) = 力臂 / 阻力臂。第一类杠杆（如颈后伸肌）支点在力与阻力之间，MA 可大于、等于或小于 1。第二类杠杆（如提踵）MA 始终大于 1，放大力量。第三类杠杆（如肱二头肌弯举）MA 小于 1，牺牲力量换取速度与活动范围，是人体中最常见的杠杆类型。
own moment arm (dr) for motion to occur: Fm × dm > Fr × dr. Thus, insertion points far from the joint centre improve force production but reduce angular velocity, demonstrating the force–velocity trade-off.
生物力学中，肌肉产生的力（Fm）乘以其力臂（dm）必须大于外部阻力（Fr）乘以其力臂（dr）才能产生运动：Fm × dm > Fr × dr。因此，远离关节中心的附着点可改善力输出，但会降低角速度，体现力量–速度权衡。
11. Key Theoretical Models in Sport Psychology | 运动心理学核心理论模型
Weiner's attribution model classifies perceived causes of success and failure along three dimensions: locus of causality (internal vs external), stability (stable vs unstable), and controllability (controllable vs uncontrollable). An athlete attributing failure to a lack of effort (internal, unstable, controllable) is more likely to persist than one attributing it to lack of ability (internal, stable, uncontrollable).
Weiner 的归因模型将成功与失败的感知原因分为三个维度：因果源（内部/外部）、稳定性（稳定/不稳定）、可控性（可控/不可控）。将失败归因于努力不足（内部、不稳定、可控）的运动员比归因于能力不足（内部、稳定、不可控）的运动员更可能坚持。
Bandura' bal persuasion, and emotional arousal. The formula for performance is not a mathematical equation but a theoretical pathway: self-efficacy → choice of activity, effort, persistence → performance. Vealey's sport-specific model adds trait sport confidence and competitive orientation.
Bandura 的自我效能理论指出，效能期望来自四个来源：操作成就、替代经验、言语说服和情绪唤起。运动表现的通路并非数学等式而是理论路径：自我效能 → 活动选择、努力、坚持 → 表现。Vealey 的运动专项模型则加入了特质运动自信与竞赛取向。
12. Information Processing & Reaction | 信息加工与反应时
Hick' me, N is the number of alternatives, and a and b are constants. Increasing the number of choices from 2 to 4 approximately doubles the decision component of reaction time. This underlies the advantage of offside trap tactics – by reducing the attacker's N, a defender can force slower decisions.
Hick 定律描述了刺激–反应选择数量与反应时之间的关系：RT = a + b log₂(N)，其中 RT 为反应时，N 为备择数目，a 和 b 为常数。备择项由 2 个增至 4 个，反应时的决策成分大约翻倍。这正是造越位战术的优势所在——通过减少进攻方的 N，防守者可迫使对方决策变慢。
Fitts' is law has profound implications for user interface design and for sport. In a basketball free throw, the large target width (hoop) relative to distance yields a relatively low index of difficulty, allowing high success rates despite psychological pressure.
Fitts 定律将运动时（MT）与目标距离（D）和目标宽度（W）联系起来：MT = a + b log₂(2D/W)。这一定律对用户界面设计和体育运动均有深远启示。在篮球罚球中，目标宽度（篮筐）相对距离较大，导致难度指数较低，使得即使有心理压力，仍能保持较高的命中率。
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