📚 Year 11 CAIE Physical Education: Formulas & Theorems Quick Reference Handbook | Year 11 CAIE 体育：公式定理速查手册
Mastering the essential formulas and theorems in CAIE IGCSE Physical Education is vital for analyzing movement, physiology, and training principles. This quick reference handbook provides clear explanations and practical applications for each key equation, helping you excel in both theoretical exams and practical assessments.
掌握CAIE IGCSE体育学科的核心公式与定理，对于分析运动、生理机制以及训练原理至关重要。本速查手册逐一解析每个关键公式，并提供实际应用示例，助你在理论考试与实践评估中均取得优异成绩。
1. Speed, Distance and Time | 速度、距离与时间
Speed is the rate at which an object covers distance. The fundamental equation relates speed (v), distance (d) and time (t):
速度是物体移动距离的速率，基本公式将速度(v)、距离(d)和时间(t)相关联：
v = d / t
Where speed is measured in metres per second (m/s), distance in metres (m), and time in seconds (s). In sport, this formula helps calculate sprinting velocity, swimming pace, or the motion of a ball. Remember that average speed over a race can be found by dividing total distance by total time, even if the pace varies.
其中速度单位为米/秒(m/s)，距离单位为米(m)，时间单位为秒(s)。在体育运动中，该公式可用于计算冲刺速度、游泳配速或球的运动速度。请注意，即使速度不断变化，比赛中的平均速度仍可通过总距离除以总时间求得。
For a 100 m sprint completed in 12 seconds, average speed = 100 m / 12 s ≈ 8.33 m/s. Rearranging the formula allows you to find distance (d = v × t) or time (t = d / v). These variations are useful when planning pacing strategies in endurance events like the 1500 m run.
例如，100米短跑用时12秒，平均速度 = 100米 / 12秒 ≈ 8.33米/秒。变形公式可求距离(d = v × t)或时间(t = d / v)。在1500米跑等耐力项目中，这些变式对于规划配速策略十分实用。
2. Force, Mass and Acceleration (Newton’s Second Law) | 力、质量与加速度（牛顿第二定律）
Newton’s Second Law states that the acceleration (a) of an object is directly proportional to the net force (F) acting on it and inversely proportional to its mass (m):
牛顿第二定律指出：物体的加速度(a)与作用其上的净外力(F)成正比，与自身质量(m)成反比：
F = m × a
Force is measured in newtons (N), mass in kilograms (kg), and acceleration in metres per second squared (m/s²). In sports, a sprinter applying a larger ground reaction force generates greater acceleration. Similarly, a heavier shot put requires more force to achieve the same acceleration as a lighter one. This law also explains why reducing body mass without losing strength can improve acceleration.
力以牛顿(N)为单位，质量以千克(kg)为单位，加速度以米每二次方秒(m/s²)为单位。在运动中，短跑运动员施加更大的地面反作用力便能产生更大的加速度。同理，较重的铅球需要更大的力才能获得与较轻铅球相同的加速度。该定律也解释了为何在不减少力量的前提下降低体重能提升加速度。
The impulse–momentum relationship is derived from Newton’s second law: impulse (F × t) equals change in momentum (m × Δv). Athletes apply this principle by increasing contact time during landing to reduce peak force, thereby lowering injury risk.
冲量–动量关系由牛顿第二定律推导而来：冲量(F × t)等于动量的变化(m × Δv)。运动员通过延长落地时的接触时间来降低峰值压力，从而减少受伤风险，正是运用了这一原理。
3. Torque and Lever Systems | 力矩与杠杆系统
Torque (τ), also called moment of force, is the rotational effect of a force about a pivot. It is the product of the force applied and the perpendicular distance from the pivot:
力矩(τ)也称为力对某点的转动力矩，是力与转动轴之间的垂直距离的乘积：
τ = F × d
Where F is force in newtons (N) and d is the perpendicular distance from the pivot point in metres (m). Human movement involves three classes of levers. A quick classification:
其中F为力(牛顿，N)，d为力的作用线到转动轴的垂直距离(米，m)。人体运动涉及三类杠杆，简要分类如下：
| Lever Class | Arrangement | Mechanical Advantage | Example |
|---|---|---|---|
| First-class | Fulcrum between Effort and Load | Depends on lengths | Neck extension |
| Second-class | Load between Fulcrum and Effort | Favours force | Calf raises |
| Third-class | Effort between Fulcrum and Load | Favours speed & ROM | Bicep curl |
Understanding torque helps in analyzing muscle efficiency and injury risk. For instance, lifting a weight with a bent elbow increases torque about the elbow joint, demanding greater muscular effort from the biceps.
理解力矩有助于分析肌肉工作效率与受伤风险。例如，屈肘抬举重物会增大肘关节的力矩，从而要求肱二头肌付出更大的力量。
4. Calculating Moments in Static Equilibrium | 静力平衡的力矩计算
For a body to be in rotational equilibrium, the sum of clockwise moments about any point must equal the sum of anticlockwise moments:
物体若要保持转动平衡，则绕任意点的顺时针力矩之和必须等于逆时针力矩之和：
∑ clockwise moments = ∑ anticlockwise moments
This principle is crucial when analyzing balance and stability in sports like gymnastics, dance, or starting positions in athletics. For example, a gymnast on a beam adjusts limb position to keep the centre of mass over the base of support.
这一原理对于分析体操、舞蹈或田径起跑时的平衡与稳定性至关重要。例如，平衡木上的体操运动员通过调整四肢位置使重心保持在支撑面内。
Consider a diver: if the anticlockwise moment created by extended arms equals the clockwise moment of the trunk, the diver maintains a static pose before take-off. Coaches use this theorem to optimize body alignment and prevent falls.
以跳水运动员为例：若伸展手臂产生的逆时针力矩与躯干的顺时针力矩相等，起跳前即可保持静态姿势。教练员运用该定理来优化身体姿态，防止摔倒。
5. Centre of Mass and Stability | 重心与稳定性
The position of the centre of mass (CoM) affects stability. A body is more stable when the CoM is low, the base of support is wide, and the line of gravity falls within the base. While there is no single formula, the relationship can be conceptualized by:
重心的位置影响稳定性。重心低、支撑面宽且重力线落在支撑面内时，身体更稳定。虽然并无单一公式，但可通过以下关系加以理解：
Stability ∝ (Base width × CoM height⁻¹)
In practice, athletes lower their CoM by bending knees in defensive stances (e.g., basketball defence, wrestling) to resist being moved. The line of gravity must stay within the base to avoid toppling; this is why a fencer leans forward only as far as the rear leg can counterbalance.
实践中，运动员通过屈膝降低重心（如篮球防守、摔跤）以增强抵抗位移的能力。重力线必须保持在支撑面内才能避免倾倒，这解释了为何击剑运动员仅能前倾至后腿可以平衡的极限。
6. Maximum Heart Rate (HRmax) | 最大心率
The simplest estimation of maximum heart rate is achieved through the age-predicted formula:
最大心率最简便的估算可通过年龄预测公式得出：
HRmax = 220 – age
For a 16-year-old, predicted HRmax is 204 beats per minute (bpm). However, this formula has a standard error of about ±10-12 bpm, so more accurate equations like HRmax = 208 – (0.7 × age) are often recommended. HRmax is fundamental for prescribing aerobic training intensities and should be periodically reassessed with field tests.
16岁青少年的预测最大心率为204次/分(bpm)。但该公式的标准误差约为±10–12 bpm，因此通常推荐使用更精确的公式，如HRmax = 208 – (0.7 × 年龄)。最大心率是制订有氧训练强度的基础，并应定期通过场地测试重新评估。
7. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen method calculates target heart rate based on heart rate reserve (HRR), which accounts for resting heart rate (RHR):
卡沃宁法以心率储备(HRR)为基础计算靶心率，心率储备考虑了静息心率(RHR)：
Target HR = (HRmax – RHR) × % intensity + RHR
For a 15-year-old with RHR of 60 bpm aiming at 70% intensity: HRmax = 205 bpm, HRR = 145 bpm, target = 145 × 0.7 + 60 = 161.5 bpm. If the same athlete wanted to train at 85% intensity, target = 145 × 0.85 + 60 = 183.25 bpm. This yields a more personalised training zone than using flat percentages of HRmax alone, because it respects individual fitness levels.
以一位15岁、静息心率60 bpm的青少年为例，设定70%强度：HRmax = 205 bpm，HRR = 145 bpm，靶心率 = 145 × 0.7 + 60 = 161.5 bpm。若同一运动员以
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