📚 Formula and Theorem Quick Reference Guide | 公式定理速查手册
This quick reference guide compiles the essential formulae and theorems needed for Year 12 OCR Physical Education, covering biomechanics, exercise physiology and training principles. Each entry presents a concise explanation in English immediately followed by its Chinese translation, ensuring you can cross-reference concepts rapidly during revision.
本速查手册汇编了 Year 12 OCR 体育学科所需的核心公式与定理，涵盖运动生物力学、运动生理学和训练原则。每一条目先给出英文说明，紧跟着中文翻译，便于你在复习时快速对照查阅。
1. Linear Kinematics | 直线运动学
Speed is the rate of change of distance: v = d / t, where v is speed, d is distance and t is time. Scalar quantity.
速率是距离的变化率：v = d / t，其中 v 为速率，d 为距离，t 为时间。标量。
Velocity is displacement divided by time: v = s / t, where s is displacement. Vector quantity, direction matters.
速度是位移除以时间：v = s / t，其中 s 是位移。矢量，方向很重要。
Acceleration is the rate of change of velocity: a = Δv / Δt = (v − u) / t, where a is acceleration, v is final velocity, u is initial velocity and t is time.
加速度是速度的变化率：a = Δv / Δt = (v − u) / t，其中 a 是加速度，v 是末速度，u 是初速度，t 是时间。
2. SUVAT Equations | 匀加速运动方程
These five equations describe uniformly accelerated linear motion: v = u + a t, s = (u + v) t / 2, s = u t + 1/2 a t², v² = u² + 2 a s and s = v t − 1/2 a t².
这五个方程描述匀加速直线运动：v = u + a t、s = (u + v) t / 2、s = u t + 1/2 a t²、v² = u² + 2 a s 以及 s = v t − 1/2 a t²。
They are only valid when acceleration is constant. Positive and negative signs must be used consistently for direction.
它们仅在加速度恒定时有效。必须始终一致地使用正负号表示方向。
Use s = u t + 1/2 a t² to calculate the flight height of a vertical jump if a and u are known.
若已知加速度和初速度，可用 s = u t + 1/2 a t² 计算垂直起跳的高度。
3. Newton’s Laws of Motion | 牛顿运动定律
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a resultant external force.
第一定律（惯性）：除非受到合外力作用，物体将保持静止或匀速直线运动状态。
Second Law: F = m a, the resultant force F equals mass m multiplied by acceleration a. This links force to change in motion.
第二定律：F = m a，合外力 F 等于质量 m 乘以加速度 a。将力与运动变化联系起来。
Third Law: For every action there is an equal and opposite reaction. In sprinting, the foot pushes backward against the ground; the ground pushes the body forward.
第三定律：每一个作用力都有一个大小相等、方向相反的反作用力。短跑时脚向后蹬地，地面将身体向前推动。
4. Linear Kinetics & Momentum | 线性动力学与动量
Momentum: p = m v, where p is momentum, m is mass and v is velocity. It is a vector measured in kg·m·s⁻¹.
动量：p = m v，p 为动量，m 为质量，v 为速度。矢量，单位为 kg·m·s⁻¹。
Impulse: Impulse = F Δt = Δp = m v − m u. Impulse equals the change in momentum. A greater impulse applied to a shot put increases its release velocity.
冲量：冲量 = F Δt = Δp = m v − m u。冲量等于动量的变化。施加在铅球上的冲量越大，出手速度越大。
Conservation of momentum: In a closed system total momentum before collision equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Used to analyse impacts in boxing or rugby tackles.
动量守恒：在封闭系统中碰撞前的总动量等于碰撞后的总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。用于分析拳击或橄榄球擒抱中的碰撞。
5. Work, Energy & Power | 功、能与功率
Work done: W = F d cos θ, where F is force, d is displacement and θ is the angle between force and displacement. When force acts along the line of motion, W = F d.
功：W = F d cos θ，F 为力，d 为位移，θ 为力与位移的夹角。当力沿运动方向作用时，W = F d。
Kinetic energy: KE = 1/2 m v². A doubling of velocity quadruples kinetic energy, explaining why faster collisions cause far more damage.
动能：KE = 1/2 m v²。速度加倍，动能增至四倍，这解释了为什么速度更快的碰撞造成的损伤大得多。
Gravitational potential energy: GPE = m g h arth, h is height above a reference level. Climbers and high jumpers convert GPE for work.
重力势能：GPE = m g h，地球上 g = 9.81 m·s⁻²，h 为相对于参考面的高度。攀岩者和跳高运动员将重力势能用于做功。
Power: P = W / t = (F d) / t = F v. Power is the rate of doing work, measured in watts. In a Wingate test, peak power reflects anaerobic capacity.
功率：P = W / t = (F d) / t = F v。功率是做功的速率，单位为瓦特。在 Wingate 测试中，峰值功率反映无氧能力。
6. Levers & Torque | 杠杆与力矩
A lever consists of a fulcrum (pivot), an effort force and a resistance force. The moment (torque) is M = F × d, where F is the force applied and d is the perpendicular distance from the pivot. Measured in N·m.
杠杆由支点、动力和阻力组成。力矩为 M = F × d，F 为作用力，d 为从支点出发的垂直距离。单位为 N·m。
In the body, joints act as fulcrums, muscles provide effort and limb weight or external loads provide resistance. Most sporting levers are third‑class levers (effort between fulcrum and resistance) to favour speed and range of motion over force.
人体中，关节作为支点，肌肉提供动力，肢体重量或外部载荷提供阻力。运动中的杠杆大多为第三类杠杆（动力在支点和阻力之间），以牺牲力来换取速度和活动范围。
Principle of moments: For a lever to be balanced, the sum of clockwise moments equals the sum of anticlockwise moments: Σ M_clockwise = Σ M_anticlockwise.
力矩平衡原理：杠杆平衡时，顺时针力矩之和等于逆时针力矩之和：Σ M_顺时针 = Σ M_逆时针。
7. Angular Motion | 角运动
Angular velocity: ω = Δθ / Δt velocity in rad·s⁻¹ and θ is angular displacement in radians. One revolution = 2π radians.
角速度：ω = Δθ / Δt，ω 为角速度 (rad·s⁻¹)，θ 为角位移 (弧度)。一整圈 = 2π 弧度。
Angular acceleration: α = Δω / Δt. Describes how quickly an object spins up or slows down, critical for diving or gymnastics rotations.
角加速度：α = Δω / Δt。描述物体旋转加速或减速的快慢，对跳水或体操转体十分关键。
Moment of inertia: I = Σ m r². Resistance to angular acceleration depends on mass and its distribution from the axis. A tucked somersault reduces I, increasing ω due to conservation of angular momentum.
转动惯量：I = Σ m r²。对角加速度的阻抗取决于质量及其离轴分布。团身空翻可减小转动惯量，因角动量守恒而增大角速度。
Angular momentum: L = I ω. If no external torque acts, L remains constant. This principle explains how a figure skater spins faster when pulling arms in.
角动量：L = I ω。若无外力矩作用，L 保持不变。该原理解释了花样滑冰运动员收臂时为何旋转更快。
8. Fluid Mechanics & Drag | 流体力学与阻力
Drag force: Fd = 1/2 Cd ρ A v², where Cd is the drag coefficient, ρ is fluid density, A is cross‑sectional area and v is velocity. Reducing area or streamlining Cd lowers drag.
阻力：Fd = 1/2 Cd ρ A v²，Cd 为阻力系数，ρ 为流体密度，A 为横截面积，v 为速度。减小面积或优化流线型 Cd 可降低阻力。
Bernoulli’s principle: In a horizontal fluid flow, an increase in velocity occurs together with a decrease in pressure. This generates lift on an aerofoil or a discus, and explains the Magnus effect on spinning balls.
伯努利原理：在水平流体中，流速增加处压强降低。这在翼型或铁饼上产生升力，并解释了旋转球体的马格努斯效应。
Magnus effect: A spinning ball creates a pressure difference, causing it to curve in flight. A top‑spin volleyball dips sharply; back‑spin produces a flatter trajectory.
马格努斯效应：旋转球体产生压强差，导致飞行中弧线运动。上旋排球急剧下坠；下旋则使轨迹更平。
9. Cardiovascular & Respiratory Formulae | 心血管与呼吸公式
Maximum heart rate: HR_max ≈ 220 − age. This estimate allows training zones to be set. Individual variation is significant; laboratory testing is more accurate.
最大心率：HR_max ≈ 220 − 年龄。该估算可设定训练区间。个体差异显著，实验室测试更准确。
Karvonen formula for target heart rate: Target HR = (HR_max − HR_rest) × % intensity + HR_rest. This heart rate reserve method personalises aerobic training zones.
Karvonen 目标心率公式：靶心率 = (HR_max − HR_rest) × 强度百分比 + HR_rest。这种心率储备法能个性化设定有氧训练区间。
Cardiac output: Q = HR × SV, where Q is cardiac output (L·min⁻¹), HR is heart rate and SV is stroke volume. Endurance training increases stroke volume, raising Q at rest and exercise.
心输出量：Q = HR × SV，Q 为心输出量 (L·min⁻¹)，HR 为心率，SV 为每搏输出量。耐力训练可增加每搏输出量，提升安静和运动时的心输出量。
Minute ventilation: VE = f × TV, where VE is minute ventilation (L·min⁻¹), f is breathing frequency and TV is tidal volume. During exercise both f and TV rise dramatically.
每分通气量：VE = f × TV，VE 为每分通气量 (L·min⁻¹)，f 为呼吸频率，TV 为潮气量。运动中 f 和 TV 均急剧上升。
VO₂ max equation: VO₂ = Q × (a−v)O₂ difference. Maximal oxygen uptake depends on the product of cardiac output and arteriovenous oxygen extraction.
最大摄氧量方程：VO₂ = Q × (a−v)O₂ 差。最大摄氧量取决于心输出量与动静脉氧差的乘积。
10. Body Composition & Fitness Testing | 身体成分与体能测试
Body Mass Index: BMI = mass (kg) / height² (m²). Used to classify underweight, normal, overweight and obese, but does not differentiate fat from lean mass.
身体质量指数：BMI = 体重 (kg) / 身高² (m²)。用于划分过轻、正常、超重和肥胖，但不能区分脂肪量与瘦体重。
Fat‑free mass can be estimated via skinfold equations. For males: Body density = 1.1714 − 0.063 × log₁₀(sum of four skinfolds); then % body fat = (495 / body density) − 450 (Siri equation). Females use different coefficients.
去脂体重可用皮褶厚度方程估算。男性：体密度 = 1.1714 − 0.063 × log₁₀(四个皮褶部位之和)；然后 体脂% = (495 / 体密度) − 450 (Siri 公式)。女性系数不同。
Wingate anaerobic test: Peak power (W) = (optimal revolution count × resistance) / 6.12 (for Monark cycle). Mean power and fatigue index are also calculated to assess anaerobic fitness.
Wingate 无氧测试：峰值功率 (W) = (最佳圈数 × 阻力) / 6.12 (Monark 自行车)。同时计算平均功率和疲劳指数，评估无氧适能。
11. Training Principles & Load Calculation | 训练原则与负荷计算
overns programme design: Frequency, Intensity, Time, Type. Intensity is often expressed as %HR_max or %1RM in resistance training.
FITT 原则指导训练计划设计：频度、强度、时间、类型。强度常用 %HR_max 或抗阻训练中的 %1RM 表示。
One‑repetition maximum approximation: 1RM ≈ weight lifted × (1 + (reps / 30)) (Epley formula). Used to prescribe loads for strength and power development safely without maximal testing.
一次最大重复次数估算：1RM ≈ 举起的重量 × (1 + (重复次数 / 30)) (Epley 公式)。用于安全地制定力量与爆发力负荷，无需最大力量测试。
Rate of perceived exertion: Session RPE = RPE × duration (min). Internal training load is monitored to avoid overtraining. The Borg 6‑20 scale corresponds to heart rate: HR ≈ RPE × 10.
主观疲劳感觉：课次 RPE = RPE × 持续时间 (分钟)。监测内部训练负荷以避免过度训练。Borg 6‑20 量表与心率相关：HR ≈ RPE × 10。
Acute:chronic workload ratio: ACWR = acute load (7‑day rolling) / chronic load (28‑day average). A ratio above 1.5 significantly increases injury risk.
急慢性负荷比值：ACWR = 急性负荷 (7 天滚动) / 慢性负荷 (28 天均值)。比值高于 1.5 会显著增加受伤风险。
12. Key Biomechanical Principles & Additional Theorems | 关键生物力学原理与补充定理
Coefficient of restitution: e = − (v₂ − v₁) / (u₂ − u₁) ns; e = 0 for perfectly inelastic. In sport, e governs ball bounce and racquet efficiency.
恢复系数：e = − (v₂ − v₁) / (u₂ − u₁)，v 和 u 分别为两物体碰撞后与碰撞前的速度。e = 1 为完全弹性碰撞；e = 0 为完全非弹性碰撞。运动中，e 决定球的反弹和球拍效率。
Static & dynamic friction: Ff ≤ μ_s Rn (static) and Ff = μ_k Rn (kinetic). The normal reaction Rn often equals mg on flat surfaces. Frictional force affects cutting, traction and shoe design.
静摩擦与动摩擦：Ff ≤ μ_s Rn（静摩擦）与 Ff = μ_k Rn（动摩擦）。在水平面上法向反力 Rn 通常等于 mg。摩擦力影响急停、抓地力和鞋类设计。
Projectile motion (no air resistance): Vertical motion is governed by gravity; horizontal motion is constant. Range: R = (u² sin 2θ) / g. Maximum height: H = (u² sin² θ) / (2 g). Optimum angle for maximum range is 45° in a vacuum; with release height difference it may be slightly lower.
抛体运动（无空气阻力）：垂直运动受重力支配；水平运动为匀速。射程：R = (u² sin 2θ) / g。最大高度：H = (u² sin² θ) / (2 g)。真空中最大射程的最优角度为 45°；若出手高度不同，角度可能略低。
Stability and centre of mass: The wider the base of support and the lower the centre of mass, the greater the stability. Moment of force about the edge of the base determines whether a body topples: stability moment > toppling moment for balance.
稳定性与质心：支撑面越宽、质心越低，稳定性越大。关于支撑面边缘的力矩决定物体是否倾倒：稳定力矩 > 倾倒力矩 才能保持平衡。
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