📚 Year 13 AQA PE Formula & Theorem Quick Reference | AQA 体育 Year 13 公式定理速查手册
This handbook compiles the essential formulas, equations and key theorems required for the Year 13 AQA A-level Physical Education specification. It covers biomechanics, exercise physiology, sport psychology statistics and health measurements to support revision and exam preparation.
本手册汇集了 Year 13 AQA A-level 体育课程所需的全部核心公式、方程和关键定理，涵盖运动生物力学、运动生理学、运动心理学统计以及健康测量，旨在帮助复习与备考。
1. Linear Motion | 直线运动
Speed (v) = distance (d) / time (t). Units: m/s. Speed is a scalar quantity.
速度 (v) = 距离 (d) / 时间 (t)，单位 米/秒。速度为标量。
Velocity (v) = displacement (s) / time (t). Units: m/s in a stated direction. Velocity is a vector.
速率 (v) = 位移 (s) / 时间 (t)，单位 米/秒，带有方向。速率为矢量。
Acceleration (a) = change in velocity (Δv) / time taken (t). Units: m/s².
加速度 (a) = 速度变化量 (Δv) / 时间 (t)，单位 米/秒²。
Equations of uniformly accelerated motion: v = u + a t; s = u t + ½ a t²; v² = u² + 2 a s.
匀加速直线运动方程：v = u + a t；s = u t + ½ a t²；v² = u² + 2 a s。
2. Newton’s Laws of Motion | 牛顿运动定律
First Law (Inertia): A body remains at rest or in uniform motion in a straight line unless acted upon by an external resultant force.
第一定律（惯性）：除非受到外力作用，物体将保持静止或匀速直线运动状态。
Second Law: Force (F) = mass (m) × acceleration (a). Units: Newtons (N). F = m a.
第二定律：力 (F) = 质量 (m) × 加速度 (a)，单位 牛顿 (N)。F = m a。
Third Law: For every action force there is an equal and opposite reaction force.
第三定律：作用力与反作用力大小相等、方向相反。
Weight (W) = mass (m) × gravitational field strength (g). On Earth g = 9.81 m/s².
重量 (W) = 质量 (m) × 重力场强度 (g)，地球表面 g = 9.81 米/秒²。
3. Momentum & Impulse | 动量与冲量
Momentum (p) = mass (m) × velocity (v). p = m v. Units: kg m/s. Momentum is a vector.
动量 (p) = 质量 (m) × 速度 (v)。p = m v，单位 千克·米/秒。动量是矢量。
Impulse = average force (F) × time (t) = change in momentum (Δp). F t = m v – m u.
冲量 = 平均力 (F) × 时间 (t) = 动量变化量 (Δp)。F t = m v – m u。
Conservation of linear momentum: In a closed system, total momentum before collision = total momentum after collision.
动量守恒定律：在封闭系统中，碰撞前总动量 = 碰撞后总动量。
4. Projectile Motion | 抛体运动
Horizontal component of velocity: vₓ = u cosθ. Horizontal displacement: sₓ = u cosθ × t. Air resistance ignored.
水平速度分量：vₓ = u cosθ。水平位移：sₓ = u cosθ × t。忽略空气阻力。
Vertical component of velocity: v_y = u sinθ – g t. Vertical displacement: s_y = u sinθ t – ½ g t².
竖直速度分量：v_y = u sinθ – g t。竖直位移：s_y = u sinθ t – ½ g t²。
Time of flight (for symmetrical parabola): t = 2 u sinθ / g.
飞行时间（对称抛物轨迹）：t = 2 u sinθ / g。
Maximum height: H = (u² sin²θ) / (2 g). Range: R = (u² sin 2θ) / g.
最大高度：H = (u² sin²θ) / (2 g)。射程：R = (u² sin 2θ) / g。
5. Angular Motion | 角运动
Angular displacement (θ) measured in radians (rad). 1 revolution = 2π rad = 360°.
角位移 (θ) 以弧度 (rad) 计量。1 圈 = 2π 弧度 = 360°。
Angular velocity (ω) = change in angular displacement / time. ω = Δθ / Δt. Units: rad/s.
角速度 (ω) = 角位移变化量 / 时间。ω = Δθ / Δt，单位 弧度/秒。
Angular acceleration (α) = change in angular velocity / time. α = (ω₂ – ω₁) / t. Units: rad/s².
角加速度 (α) = 角速度变化量 / 时间。α = (ω₂ – ω₁) / t，单位 弧度/秒²。
Torque (T) = moment of inertia (I) × angular acceleration (α). T = I α.
力矩 (T) = 转动惯量 (I) × 角加速度 (α)。T = I α。
Angular momentum = moment of inertia (I) × angular velocity (ω). Angular momentum is conserved when no external torque acts.
角动量 = 转动惯量 (I) × 角速度 (ω)。无外力矩时角动量守恒。
6. Fluid Mechanics & Bernoulli’s Principle | 流体力学与伯努利原理
Drag force (air resistance / fluid resistance) opposes motion. Factors: velocity, cross‑sectional area, fluid density, shape (coefficient of drag).
阻力（空气阻力/流体阻力）与运动方向相反。影响因素：速度、横截面积、流体密度、形状（阻力系数）。
Bernoulli’s Principle: For a steady flow of an incompressible fluid, an increase in velocity occurs simultaneously with a decrease in pressure. p + ½ ρ v² + ρ g h = constant.
伯努利原理：稳定流动的不可压缩流体中，流速增加处压强减小。p + ½ ρ v² + ρ g h = 常数。
Magnus effect: A spinning object in a fluid creates a pressure difference, producing a lift force perpendicular to the flow direction.
马格努斯效应：旋转物体在流体中产生压强差，形成与来流方向垂直的升力。
Lift force (e.g. in discus, football) depends on spin rate, relative velocity and object surface characteristics.
升力（如铁饼、足球）取决于自旋速率、相对速度以及物体表面特征。
7. Centre of Mass & Stability | 重心与稳定性
Centre of mass (CoM) is the point where the mass of a body is concentrated. It may lie outside the body.
重心（质心）是物体质量集中的点，可位于身体外部。
Stability increases when: the CoM is lower, the base of support is wider, and the line of gravity falls within the base of support.
稳定性增大条件：重心更低、支撑面更宽、重力作用线落在支撑面内。
Toppling occurs when the line of gravity moves outside the base of support, producing a torque that rotates the body.
当重力作用线移出支撑面时产生转动力矩，导致倾倒。
8. Exercise Physiology: Cardiac & Respiratory | 运动生理学：心血管与呼吸
Cardiac output (Q) = heart rate (HR) × stroke volume (SV). Q = HR × SV. Units: L/min.
心输出量 (Q) = 心率 (HR) × 每搏输出量 (SV)。Q = HR × SV，单位 升/分。
Maximal heart rate (HRmax) can be estimated by: HRmax = 220 – age (years). More accurate formula: HRmax = 208 – (0.7 × age).
最大心率估算：HRmax = 220 – 年龄（岁）。更精确公式：HRmax = 208 – (0.7 × 年龄)。
Minute ventilation (VE) = tidal volume (TV) × breathing frequency (f). VE = TV × f. Units: L/min.
每分通气量 (VE) = 潮气量 (TV) × 呼吸频率 (f)。VE = TV × f，单位 升/分。
Oxygen uptake (VO₂) = cardiac output × arteriovenous oxygen difference (a-vO₂ diff). Units: mL/kg/min.
摄氧量 (VO₂) = 心输出量 × 动静脉氧差 (a-vO₂ diff)，单位 毫升/公斤/分。
VO₂max is the maximum rate of oxygen uptake during exhaustive exercise, a key indicator of aerobic endurance.
VO₂max 是力竭运动中的最大摄氧速率，是有氧耐力的核心指标。
9. Energy Systems & Efficiency | 能量系统与效率
Mechanical efficiency (%) = (useful mechanical work output / total energy expended) × 100.
机械效率 (%) = （有用机械功输出 / 总能量消耗）× 100。
1 MET = resting metabolic rate ≈ 3.5 mL O₂/kg/min. Exercise intensity can be expressed in METs.
1 MET = 静息代谢率 ≈ 3.5 毫升氧/公斤/分。运动强度可用 MET 值表示。
Energy expenditure (kcal/min) = METs × body mass (kg) × (3.5 / 200).
能量消耗（千卡/分）= METs × 体重（公斤）× (3.5 / 200)。
Excess post-exercise oxygen consumption (EPOC) reflects the O₂ debt repaid after exercise to restore homeostasis.
运动后过量氧耗 (EPOC) 代表运动后恢复稳态所需偿还的氧债。
10. Statistical Analysis for Sport Sciences | 运动科学统计分析
Mean (x̄) = Σx / n, where Σx is the sum of all data values and n is the number of data values.
平均值 (x̄) = Σx / n，Σx 为所有数据值之和，n 为数据个数。
Standard deviation (s) = √[ Σ(x – x̄)² / (n – 1) ]. Measures spread of data around the mean.
标准差 (s) = √[ Σ(x – x̄)² / (n – 1) ]。用于衡量数据围绕平均值的分散程度。
Chi-squared (χ²) = Σ[ (O – E)² / E ]. O = observed frequency, E = expected frequency. Tests goodness of fit or independence.
卡方 (χ²) = Σ[ (观察值 O – 期望值 E)² / E ]。用于拟合优度或独立性检验。
Pearson product–moment correlation coefficient (r) = Σ[(x – x̄)(y – ȳ)] / √[ Σ(x – x̄)² × Σ(y – ȳ)² ]. Ranges from -1 to +1.
皮尔逊积差相关系数 (r) = Σ[(x – x̄)(y – ȳ)] / √[ Σ(x – x̄)² × Σ(y – ȳ)² ]，取值范围 -1 至 +1。
Spearman’s rank correlation coefficient (rₛ) can be used for ordinal data: rₛ = 1 – [6 Σ d² / n(n² – 1)], where d = difference in ranks.
斯皮尔曼等级相关系数 (rₛ) 用于顺序数据：rₛ = 1 – [6 Σ d² / n(n² – 1)]，d 为等级差。
11. Anthropometry & Health Measures | 人体测量与健康指标
Body Mass Index (BMI) = mass (kg) / height² (m²). Units: kg/m². Classification: <18.5 underweight, 18.5–24.9 normal, 25–29.9 overweight, ≥30 obese.
身体质量指数 (BMI) = 体重 (公斤) / 身高² (米²)。分级：<18.5 偏瘦，18.5–24.9 正常，25–29.9 超重，≥30 肥胖。
Waist-to-hip ratio (WHR) = waist circumference (cm) / hip circumference (cm). High WHR indicates central obesity and increased cardiovascular risk.
腰臀比 (WHR) = 腰围 (厘米) / 臀围 (厘米)。高 WHR 提示向心性肥胖与心血管风险升高。
Sum of skinfolds (e.g. biceps, triceps, subscapular, suprailiac) estimates body fat percentage via predictive equations.
皮褶厚度总和（如二头肌、三头肌、肩胛下、髂上）通过预测方程估算体脂百分比。
Basal metabolic rate (BMR) prediction: e.g. Mifflin-St Jeor equation: for males, BMR = 10 × weight(kg) + 6.25 × height(cm) – 5 × age(y) + 5; for females, BMR = 10 × weight(kg) + 6.25 × height(cm) – 5 × age(y) – 161.
基础代谢率 (BMR) 预测（Mifflin-St Jeor 公式）：男性 BMR = 10×体重(kg) + 6.25×身高(cm) – 5×年龄(y) + 5；女性 BMR = 10×体重(kg) + 6.25×身高(cm) – 5×年龄(y) – 161。
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