AS Cambridge Physical Education: Quick Reference Handbook of Formulas and Principles | AS剑桥体育：公式定理速查手册

📚 AS Cambridge Physical Education: Quick Reference Handbook of Formulas and Principles | AS剑桥体育：公式定理速查手册

This handbook compiles the key formulas, principles and quick-reference constants that every AS Physical Education student needs for Anatomy & Physiology, Biomechanics and Training. Memorise these equations and use them to analyse sporting performance, answer calculation questions and support your written arguments.

本手册汇集了AS体育课程中解剖学与生理学、生物力学及运动训练部分核心公式、定理和常用数值。熟记这些方程，用于分析运动表现、解答计算题、支撑论述。

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1. Cardiovascular Formulas | 心血管公式

= Heart Rate (HR) × Stroke Volume (SV). Q = HR × SV. A typical resting Q is about 5 L/min.

心输出量 = 心率 × 每搏输出量。Q = HR × SV。安静时典型值约5升/分。

MHR) ≈ 220 − age (years). This estimate provides an upper limit for exercise prescription.

最大心率 ≈ 220 − 年龄（岁）。该估算值为制定运动处方提供上限。

Karvonen Formula: Target HR = (MHR − Resting HR) × Desired Intensity (%) + Resting HR. It accounts for fitness level.

卡氏公式：目标心率 = (最大心率 − 安静心率) × 目标强度(%) + 安静心率。该公式考虑了体能水平。

Stroke Volume increases during exercise but plateaus at about 40–60% of VO₂ max. Endurance training shifts the plateau to higher intensities.

运动时每搏输出量上升，但在约40–60%最大摄氧量强度时达到平台。耐力训练可使平台向更高强度推移。

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2. Respiratory Formulas and Values | 呼吸公式与数值

Minute Ventilation (V̇E) = Tidal Volume (TV) × Breathing Frequency (f). V̇E = TV × f. Resting V̇E ≈ 6–7.5 L/min.

每分通气量 = 潮气量 × 呼吸频率。V̇E = TV × f。安静时约6–7.5升/分。

Alveolar Ventilation = (Tidal Volume − Dead Space) × f. Dead space is typically ~150 mL in adults.

肺泡通气量 = (潮气量 − 死腔量) × 呼吸频率。成人死腔量通常约150毫升。

Respiratory Quotient (RQ) = Volume of CO₂ produced / Volume of O₂ consumed (V̇CO₂ / V̇O₂). RQ indicates fuel mix: 1.0 for carbohydrate, ~0.7 for fat.

呼吸商 = 二氧化碳生成量 / 氧耗量 (V̇CO₂ / V̇O₂)。RQ值指示燃料配比：碳水化合物1.0，脂肪约0.7。

s: mL/kg/min or L/min. It reflects aerobic capacity. Tests include multi-stage fitness test or direct gas analysis.

最大摄氧量 = 最大氧摄取量；单位 mL/kg/min 或 L/min。反映有氧能力。测试包括多阶段体能测试或直接气体分析。

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3. Energy Systems and Metabolism | 能量系统与代谢

ATP-PC system: Power = very high, Capacity = very low (≈ 8–10 s). No oxygen required. Used for explosive jumps, sprints.

ATP-PC系统：功率极高，容量极低（约8–10秒）。无需氧气。用于爆发性跳跃、短跑。

Anaerobic Glycolytic system: Power = high, Capacity = low (≈ 60–90 s). Produces lactate. Fuels 200–400 m sprint.

无氧糖酵解系统：功率高，容量低（约60–90秒）。产生乳酸。供能200–400米冲刺。

city = very high (hours). Uses oxygen. Dominant in endurance events. Gross efficiency ≈ 20–25%.

有氧系统：功率低，容量极高（数小时）。利用氧气。在耐力项目中占主导。总效率约20–25%。

METs × body mass (kg) × time (h). MET is metabolic equivalent (1 MET = 3.5 mL O₂/kg/min).

能量消耗（千卡）= MET值 × 体重(kg) × 时间(h)。MET为代谢当量（1 MET = 3.5 mL O₂/kg/min）。

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4. Linear Kinematics | 直线运动学

peed = Distance / Time. v = d / t. Measured in m/s. Average speed uses total displacement.

速度 = 距离 / 时间。v = d / t。单位m/s。平均速度使用总位移计算。

elocity / Time taken. a = (v − u) / t. Units: m/s². Deceleration is negative acceleration.

加速度 = 速度变化量 / 时间。a = (v − u) / t。单位m/s²。减速为负加速度。

SUVAT Equations (constant acceleration):

匀加速运动方程（SUVAT）：

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

re u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time.

其中 u = 初速度, v = 末速度, a = 加速度, s = 位移, t = 时间。

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5. Force, Momentum and Impulse | 力、动量与冲量

Newton’s Second Law: Force = mass × acceleration. F = m a. Units: Newtons (N).

牛顿第二定律：力 = 质量 × 加速度。F = m a。单位：牛顿 (N)。

ty. p = m v. Units: kg·m/s. Momentum is conserved in collisions if no external force acts.

动量 = 质量 × 速度。p = m v。单位：kg·m/s。无外力时碰撞前后动量守恒。

Impulse = Δp = m v − m u. Increasing contact time reduces peak force (e.g., landing mats).

冲量 = 力 × 时间。冲量 = 动量变化 = m v − m u。延长作用时间可减小峰值力（如落地垫）。

fficient of friction × normal reaction force. f = μ N. Static friction > sliding friction.

摩擦力 = 摩擦系数 × 正压力。f = μ N。静摩擦力 > 滑动摩擦力。

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6. Work, Power and Energy | 功、功率与能量

oved in direction of force. W = F d. Units: Joules (J). If force is angled, W = F d cos θ.

功 = 力 × 沿力方向移动的距离。W = F d。单位：焦耳(J)。若力有角度，W = F d cos θ。

Power = Work done / Time = (Force × distance) / time. P = W / t = F v. Units: Watts (W).

功率 = 功 / 时间 = (力 × 距离)/时间。P = W / t = F v。单位：瓦特(W)。

ential Energy = m g h. Total mechanical energy remains constant (ignoring air resistance).

动能 = ½ m v²。重力势能 = m g h。总机械能守恒（忽略空气阻力时）。

ed via Wingate test (peak power, mean power) or vertical jump test (Lewis formula: power = √4.9 × weight × √jump height).

运动员功率输出可通过温盖特测试（峰值功率、平均功率）或纵跳测试（刘易斯公式：功率 = √4.9 × 体重 × √纵跳高度）测量。

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7. Levers and Torque | 杠杆与力矩

t. τ = F d. Units: Nm. Equilibrium requires Σ anticlockwise moments = Σ clockwise moments.

力矩 = 力 × 力臂（垂直距离）。τ = F d。单位：牛顿·米。平衡要求逆时针力矩之和 = 顺时针力矩之和。

between effort and load (e.g., neck extension). Mechanical advantage may be >1, =1 or <1.

第一类杠杆：支点在力与负荷之间（如颈后伸）。机械效益可大于、等于或小于1。

Second-class lever: load between pivot and effort (e.g., ankle plantar-flexion). Always mechanical advantage >1 (force amplifier).

第二类杠杆：负荷在支点和力之间（如踝关节跖屈）。机械效益始终>1，属于省力杠杆。

Third-class lever: effort between pivot and load (e.g., biceps curl). Mechanical advantage <1, favours speed and range of motion.

第三类杠杆：力在支点和负荷之间（如肱二头肌弯举）。机械效益<1，但有利于速度和运动幅度。

cal Advantage = Effort arm / Load arm. In the body, most levers are third-class for speed.

机械效益 = 力臂 / 阻力臂。人体多数杠杆为第三类，以速度为主。

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8. Projectile Motion | 抛体运动

Horizontal component: constant velocity, vₓ = u cos θ. Range = vₓ × total flight time.

水平分量：匀速，vₓ = u cos θ。射程 = vₓ × 总飞行时间。

Vertical component: affected by gravity. v_y = u sin θ − g t. Height reached: h = (u sin θ)² / (2 g).

垂直分量：受重力影响。v_y = u sin θ − g t。最大高度：h = (u sin θ)²/(2g)。

Optimum release angle for maximum range is 45° in a vacuum. In sport, actual optimum often 30–36° for shot put due to release height and air resistance.

真空中最大射程的最佳出手角度为45°。实际运动中，铅球出手角度因出手高度和空气阻力通常为30–36°。

Factors affecting projectile range: release velocity (most important), release height, release angle, air resistance, Magnus effect (spin).

影响抛体射程的因素：出手速度（最重要）、出手高度、出手角度、空气阻力、马格努斯效应（旋转）。

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9. Fluid Mechanics and Aerodynamics | 流体力学与空气动力学

Bernoulli’s Principle: faster-moving fluid exerts lower pressure. An airfoil shape creates lift; a spinning ball follows a curved path (Magnus effect).

伯努利原理：流体流速越快，压强越低。翼形产生升力；旋转球体沿曲线轨迹运动（马格努斯效应）。

Drag Force: F_d = ½ C_d ρ A v², where C_d is drag coefficient, ρ is fluid density, A is cross-sectional area, v is velocity. Streamlining reduces drag by lowering C_d and A.

阻力：F_d = ½ C_d ρ A v²，C_d 为阻力系数，ρ 为流体密度，A 为迎风面积，v 为速度。流线型通过降低 C_d 和 A 减少阻力。

Surface drag (friction) occurs between fluid and body surface; form drag relates to shape and pressure differential; wave drag is caused by air–water interface in swimming.

表面阻力（摩擦）产生于流体与表面之间；形状阻力取决于外形与压差；波浪阻力由游泳时气水界面引起。

Lift force perpendicular to flow direction can be generated by angle of attack or asymmetric shape (discus, javelin, sail).

垂直于来流方向的升力可由攻角或非对称形状产生（铁饼、标枪、帆）。

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10. Training Principles and Fitness Testing | 训练原则与体能测试

FITT Principle: Frequency, Intensity, Time, Type. Used to design aerobic, anaerobic, resistance and flexibility programmes.

FITT原则：频率、强度、时间、类型。用于设计有氧、无氧、抗阻和柔韧性训练计划。

Training zones based on % MHR: Aerobic zone ~60–80% MHR; Anaerobic threshold ~80–90% MHR. More precise prescription uses heart rate reserve (Karvonen).

以%最大心率划分的训练区：有氧区约60–80% MHR；无氧阈约80–90% MHR。更精准的处方使用心率储备（卡氏公式）。

Rating of Perceived Exertion (RPE): Borg scale 6–20, where 6 = rest and 20 = maximal effort. Multiply by ~10 to estimate HR.

主观疲劳感觉量表 (RPE)：博格量表6–20级，6为安静，20为最大用力。分数约乘10可估算心率。

Body Mass Index (BMI) = Weight (kg) / Height² (m). Categories: <18.5 underweight, 18.5–24.9 normal, 25–29.9 overweight, ≥30 obese. Not a direct measure of body fat.

身体质量指数 = 体重(kg) / 身高²(m)。分级：<18.5偏瘦，18.5–24.9正常，25–29.9超重，≥30肥胖。不直接反映体脂量。

Fitness tests must be valid (measures what it claims) and reliable (consistent results). Common tests include: Cooper 12-min run, handgrip dynamometer, vertical jump, 30 m sprint, Illinois agility test.

体能测试须有效（测量其所声称的内容）且信度良好（结果一致）。常用测试：库珀12分钟跑、握力计、纵跳、30米冲刺、伊利诺伊灵敏性测试。

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11. Angular Kinematics | 角运动学

displacement (θ) / time (t). ω = Δθ / Δt. Units: rad/s. Clockwise often taken as negative.

角速度 = 角位移 / 时间。ω = Δθ / Δt。单位：rad/s。顺时针通常记为负。

Angular acceleration (α) = Change in angular velocity / time. α = Δω / Δt. Units: rad/s².

角加速度 = 角速度变化量 / 时间。α = Δω / Δt。单位：rad/s²。

Linear–angular relationship: v = r ω, where v is tangential velocity, r is radius of rotation. a_t = r α. Centripetal acceleration a_c = v² / r = r ω².

线量与角量的关系：v = r ω，v 为切向速度，r 为旋转半径。切向加速度 a_t = r α。向心加速度 a_c = v²/r = r ω²。

L = I ω. In the air, a gymnast reduces I to increase ω (conservation of angular momentum).

角动量 = 转动惯量 × 角速度。L = I ω。空中时，体操运动员通过减小I来提高ω（角动量守恒）。

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