📚 Year 10 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 10 CAIE 体育：公式定理速查手册
Mastering the key formulas and principles in CAIE IGCSE Physical Education (0413) is essential for analysing human performance, understanding movement and applying training concepts. This handbook compiles the calculations, equations and biomechanical theorems you will need for Year 10, arranged by topic for quick revision and classroom use.
掌握 CAIE IGCSE 体育（0413）中的关键公式和原理，对于分析人体表现、理解运动以及应用训练概念至关重要。本手册按照专题整理了 Year 10 阶段你需要掌握的计算、方程和生物力学定理，方便你快速复习和课堂使用。
1. Heart Rate Formulas | 心率公式
Maximum Heart Rate (MHR) is the highest number of beats per minute your heart can achieve and is estimated as: MHR = 220 − age. This is a fundamental value used to set training intensities.
最大心率（MHR）是指心脏每分钟能够达到的最高搏动次数，估算公式为：MHR = 220 − 年龄。这是用于设定训练强度的基础数值。
For a 15‑year‑old student, estimated MHR = 220 − 15 = 205 bpm. This predicted maximum is used in both Karvonen and percentage‑based methods, though individual variation exists.
对一名 15 岁学生来说，估算最大心率 = 220 − 15 = 205 次/分。该预测值既用于卡沃宁公式，也用于百分比法，但个体之间存在差异。
The Karvonen formula calculates target heart rate using heart rate reserve (HRR): Target HR = (HRR × % intensity) + Resting HR, where HRR = MHR − Resting HR. This accounts for fitness level better than simple %MHR.
卡沃宁公式利用储备心率（HRR）计算目标心率：目标心率 = (HRR × 强度%) + 安静心率，其中 HRR = 最大心率 − 安静心率。它比单纯的%MHR更能体现体能水平。
Karvonen: Target HR = (MHR − RHR) × % + RHR
2. Body Composition: BMI and Waist‑to‑Hip Ratio | 身体成分：BMI 与腰臀比
Body Mass Index (BMI) estimates body fat based on weight and height: BMI = weight (kg) ÷ height² (m²). It categorises individuals as underweight, normal, overweight or obese according to standard thresholds.
身体质量指数（BMI）根据体重和身高估算体脂：BMI = 体重（kg）÷ 身高²（m²）。它根据标准阈值将个体划分为偏瘦、正常、超重或肥胖。
A student of 65 kg and 1.75 m tall has BMI = 65 ÷ (1.75 × 1.75) = 65 ÷ 3.0625 ≈ 21.2 kg/m², which falls in the healthy weight range (18.5–24.9).
一名体重 65 kg、身高 1.75 m 的学生，BMI = 65 ÷ (1.75 × 1.75) = 65 ÷ 3.0625 ≈ 21.2 kg/m²，属于健康体重范围（18.5–24.9）。
Waist‑to‑hip ratio (WHR) indicates fat distribution: WHR = waist circumference ÷ hip circumference. A higher ratio suggests greater central adiposity and increased health risk.
腰臀比（WHR）反映脂肪分布：腰臀比 = 腰围 ÷ 臀围。比值较高表示中心性肥胖程度更高，健康风险增加。
BMI = mass / height²
WHR = waist cm / hip cm
3. Speed, Velocity and Acceleration | 速率、速度与加速度
Speed is a scalar quantity measuring how fast an object moves: speed (m/s) = distance (m) ÷ time (s). It does not consider direction, so a 100 m sprinter covering the distance in 12 s has a speed of 8.33 m/s.
速率是标量，衡量物体运动的快慢：速率（m/s）= 距离（m）÷ 时间（s）。它不考虑方向，因此一名 100 m 短跑运动员用 12 s 跑完全程，其速率为 8.33 m/s。
Velocity is the vector equivalent, giving rate of change of displacement: velocity (m/s) = displacement (m) ÷ time (s). If a swimmer completes a 50 m lap returning to the start, displacement is 0, so average velocity = 0.
速度是矢量，表示位移的变化率：速度（m/s）= 位移（m）÷ 时间（s）。如果游泳者游完 50 m 回到起点，位移为 0，因此平均速度 = 0。
Acceleration measures change in velocity: acceleration (m/s²) = (final velocity − initial velocity) ÷ time taken. A positive value means speeding up; a negative value (deceleration) means slowing down.
加速度衡量速度的变化：加速度（m/s²）=（最终速度 − 初始速度）÷ 所用时间。正值表示加速，负值（减速度）表示减速。
v = d / t a = Δv / t
4. Momentum and Impulse | 动量与冲量
Momentum is the product of mass and velocity: p = m × v (kg·m/s). In tackling or collision sports, a heavier and faster‑moving player has greater momentum and is harder to stop.
动量是质量与速度的乘积：p = m × v（kg·m/s）。在抢断或碰撞类运动中，体重更大且速度更快的运动员动量更大，更难被阻止。
Impulse equals change in momentum and also the force applied multiplied by the time it acts: Impulse = F × Δt = Δp. Coaches use this to teach ‘follow‑through’ – extending impact time reduces peak force, improving control and reducing injury risk.
冲量等于动量的变化，也等于作用力乘以其作用时间：冲量 = F × Δt = Δp。教练利用这一原理讲解‘随挥’——延长撞击时间可减小峰值力，提高控制并降低受伤风险。
A cricket ball of 0.16 kg travelling at 30 m/s has momentum p = 0.16 × 30 = 4.8 kg·m/s. To stop it in 0.1 s, the average force required is F = Δp / Δt = 4.8 / 0.1 = 48 N.
一只 0.16 kg 的板球以 30 m/s 飞行，其动量 p = 0.16 × 30 = 4.8 kg·m/s。要在 0.1 s 内将它停下，所需平均力 F = Δp / Δt = 4.8 / 0.1 = 48 N。
5. Newton’s Three Laws of Motion | 牛顿运动三定律
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a resultant external force. A football on the pitch stays still until kicked; a moving puck would glide forever on frictionless ice.
第一定律（惯性）：除非受到合外力的作用，物体将保持静止或匀速直线运动状态。足球场上静止的球不被踢就不会动；若无摩擦，冰球将永远滑行。
Second Law (Acceleration): The acceleration of an object is directly proportional to net force and inversely proportional to mass: F = m × a. The same force applied to a lighter javelin produces greater acceleration.
第二定律（加速度）：物体的加速度与净外力成正比，与其质量成反比：F = m × a。相同的力施加于更轻的标枪会产生更大的加速度。
Third Law (Action–Reaction): For every action force there is an equal and opposite reaction force. When a swimmer pushes water backwards, the water pushes the swimmer forwards with equal magnitude.
第三定律（作用力与反作用力）：每一个作用力都有一个大小相等、方向相反的反作用力。游泳者向后推水时，水以同样大小的力向前推游泳者。
F = m × a
6. Levers and Mechanical Advantage | 杠杆与机械效益
A lever system consists of a fulcrum (pivot), effort (force applied) and load (resistance). The arrangement determines whether it is a first‑, second‑ or third‑class lever. Most human limbs act as third‑class levers with the effort between fulcrum and load, favouring speed and range over force.
杠杆系统由支点、施力点和负荷组成。三者的排列方式决定了它是一级、二级还是三级杠杆。人体大部分肢体属于三级杠杆，施力点位于支点与负荷之间，以速度和活动范围换取力量优势。
Mechanical Advantage (MA) = effort arm length ÷ resistance arm length. When MA > 1, the lever amplifies force (as in a second‑class lever where the calf raises the body at the ball of the foot). When MA < 1, it amplifies speed – typical in throwing and kicking.
机械效益（MA）= 施力臂长度 ÷ 阻力臂长度。当 MA > 1 时，杠杆放大力（如二级杠杆中提踵动作）；当 MA < 1 时，放大速度——常见于投掷和踢球动作。
MA = Effort arm / Resistance arm
7. Torque and Moments | 扭矩与力矩
Torque (or moment) is the turning effect of a force: Torque (Nm) = Force (N) × perpendicular distance from pivot (m). In gymnastics, a slight lean shifts the line of gravity away from the pivot, increasing torque and making balance harder.
扭矩（或力矩）是力的转动效应：扭矩（Nm）= 力（N）× 到支点的垂直距离（m）。在体操中，身体轻微倾斜会使重力作用线偏离支点，增大扭矩，使平衡更难维持。
For a static object to remain in equilibrium, total clockwise torque must equal total anticlockwise torque about any pivot. This principle explains body positioning in scrummaging or when holding a weight at arm’s length.
要使静止物体保持平衡，绕任意支点的顺时针总扭矩必须等于逆时针总扭矩。这一原理解释了橄榄球争球或手臂平举重物时的身体姿态。
8. Pressure and Friction | 压强与摩擦力
Pressure measures force distributed over an area: P (Pa) = F (N) ÷ A (m²). Spreading the same force over a larger area reduces pressure. Studs on football boots concentrate weight onto a small area, increasing pressure and grip on soft ground.
压强衡量分布在面积上的力：P（Pa）= F（N）÷ A（m²）。同样的力分布在更大面积上会减小压强。足球鞋的鞋钉将体重集中在很小的面积上，增大压强并提高松软地面的抓地力。
Friction acts parallel to surfaces in contact and opposes motion. The maximum static friction before sliding is Ff ≤ μ × R, where μ is the coefficient of friction and R is the normal reaction force. Choosing appropriate footwear or hand grips modifies μ to enhance performance and safety.
摩擦力平行于接触面，并与运动方向相反。即将滑动前最大静摩擦力为 Ff ≤ μ × R，其中 μ 为摩擦系数，R 为法向反作用力。选择合适的鞋或手胶可以改变 μ，从而提升表现和安全性。
9. Centre of Mass and Stability | 质心与稳定性
The centre of mass (CoM) is the point where all the body’s mass appears to be concentrated. In uniform gravitational fields, it coincides with the centre of gravity. Lowering the CoM and widening the base of support increases stability – a key principle in tackling and defensive stances.
质心（CoM）是身体质量似乎集中的那一点。在均匀重力场中，它与重心重合。降低质心并增大支撑面可提高稳定性——这是抢断和防守姿势的关键原理。
A performer is stable as long as the vertical line through their CoM falls inside the base of support. The moment it moves outside, a torque is created and they will topple unless a step is taken.
只要通过质心的垂直线落在支撑面内，运动员就保持稳定。一旦落到支撑面之外，就会产生力矩，导致跌倒，除非迈出一步进行调整。
10. Work, Power and Energy | 功、功率与能量
Work done is the product of force and the distance moved in the direction of that force: W (J) = F (N) × d (m). When a weightlifter lifts 100 kg through 1.8 m, work against gravity is approximately 100 × 9.8 × 1.8 = 1764 J.
功等于力与沿该力方向移动距离的乘积：W（J）= F（N）× d（m）。当举重运动员将 100 kg 的杠铃举起 1.8 m 时，克服重力所做的功大约是 100 × 9.8 × 1.8 = 1764 J。
Power is the rate of doing work: P (W) = work done (J) ÷ time taken (s) or P = F × v for constant velocity. A sprinter who develops high power can accelerate rapidly and maintain a high top speed.
功率是做功的速率：P（W）= 所做的功（J）÷ 所用时间（s），或在匀速下 P = F × v。能够产生高功率的短跑运动员可以快速加速并保持较高的最高速度。
Kinetic energy (KE) = ½mv². This explains why a small increase in speed produces a large increase in kinetic energy – and why faster collisions in rugby carry far more energy.
动能（KE）= ½mv²。这解释了为什么速度的小幅增加会导致动能大幅增加——也说明了橄榄球中更快的碰撞携带的能量要多得多。
W = F × d P = W / t KE = ½mv²
11. Projectile Motion Principles | 抛体运动原理
When a projectile is released, its flight path is determined by release velocity, angle of release and height of release. The optimal release angle for maximum horizontal distance on level ground is 45° (ignoring air resistance), but in sports such as shot put, the release angle is lower (around 34–38°) because the release point is already above the landing surface.
抛体离手后，其飞行轨迹由出手速度、出手角度和出手高度决定。忽略空气阻力时，水平地面上最大远度的最佳出手角为 45°，但在铅球等项目中，出手角度较低（约 34–38°），因为出手点本就高于落地区。
Vertical and horizontal components of velocity are independent. Increasing release speed has the greatest influence on distance; the parabolic trajectory can be elongated by combining high velocity with an appropriate angle.
速度的垂直分量和水平分量相互独立。提高出手速度对远度的影响最大；将高速度与合适的角度结合，可延长抛物线轨迹。
12. Training and Fitness Testing Calculations | 训练与体测计算
Target training zones are often expressed using percentage of MHR: light intensity 50–60%, moderate 60–70%, vigorous 70–85% MHR. These percentages guide aerobic and anaerobic training sessions.
目标训练区间常用最大心率百分比表示：低强度 50–60%，中等强度 60–70%，大强度 70–85% MHR。这些百分比用于指导有氧和无氧训练课。
For circuit or interval training, work‑to‑rest ratios are critical: a ratio of 1:3 (e.g. 30 s work, 90 s rest) targets speed and power; a 1:1 ratio targets aerobic endurance. Calculations of repetition volume (sets × reps × load) help monitor progressive overload.
在循环或间歇训练中，运动与休息比至关重要：1:3（如 30 秒运动，90 秒休息）针对速度和爆发力；1:1 针对有氧耐力。重复次数总量计算（组数 × 次数 × 负荷）有助于监测渐进超负荷。
Multi‑stage fitness test (bleep test) results are used to estimate VO₂max (ml/kg/min) via standardised conversion tables. Predicted VO₂max provides a baseline for cardiovascular fitness and can track changes across a training programme.
多阶段体能测试（蜂鸣测试）的结果可通过标准化换算表估算 VO₂max（ml/kg/min）。预测 VO₂max 为心血管适能提供基准，并可追踪整个训练计划中的变化。
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