📚 AS CIE Physical Education: Formula & Theorem Quick Reference Handbook | AS CIE 体育：公式定理速查手册
This quick reference handbook compiles the essential formulas, theorems and key equations needed for the AS CIE Physical Education syllabus. Whether you are analysing sprint kinematics or cardiac output, these equations underpin exam success and practical application in sport science.
本速查手册汇总了AS CIE体育课程所需的核心公式、定理和关键方程。无论你是分析短跑运动学还是心输出量，这些方程都是考试成功和运动科学实际应用的基础。
1. Kinematic Equations for Linear Motion | 直线运动的运动学方程
Linear motion equations are fundamental for analysing athletic performance in events such as sprinting, jumping and throwing. They describe the relationships between displacement, velocity, acceleration and time under uniform acceleration.
直线运动方程是分析短跑、跳跃和投掷等运动表现的基础。它们描述了匀加速条件下位移、速度、加速度和时间之间的关系。
v = u + at
Final velocity (v) is calculated from initial velocity (u), acceleration (a) and time (t). Useful for estimating the speed of a sprinter at any phase of the race.
末速度(v)由初速度(u)、加速度(a)和时间(t)计算得出。可用于估算短跑运动员在比赛任何阶段的速度。
s = ut + ½at²
Displacement (s) when time and acceleration are known. Coaches use this to determine the distance covered during the acceleration drive phase out of the blocks.
当已知时间和加速度时的位移(s)。教练用它来确定起跑加速阶段所覆盖的距离。
v² = u² + 2as
This equation connects velocity and displacement without requiring time. It is applied to evaluate the deceleration needed to stop within a certain distance, e.g. in a tackle or when landing from a jump.
该方程在无需时间的情况下联系速度与位移。用于评估在特定距离内停止所需的减速度，例如擒抱或跳跃落地时。
2. Newton’s Laws and Momentum | 牛顿定律与动量
Newton’s laws of motion explain how forces change an athlete’s state of motion. Momentum and impulse are vital in contact sports where collisions occur.
牛顿运动定律解释了力如何改变运动员的运动状态。在发生碰撞的接触性运动中，动量和冲量至关重要。
F = ma
Newton’s second law states that the net force (F) equals mass (m) times acceleration (a). This governs how a footballer accelerates when kicking or how a sprinter generates force against the blocks.
牛顿第二定律指出净力(F)等于质量(m)乘以加速度(a)。这决定了足球运动员踢球时如何加速，或短跑运动员如何对起跑器施加力量。
p = mv
Linear momentum (p) is the product of mass and velocity. Athletes with greater mass and speed are harder to stop, a principle exploited in rugby and American football.
线动量(p)是质量与速度的乘积。质量和速度较大的运动员更难被拦停，这一原理在橄榄球和美式足球中被广泛利用。
Ft = Δp = m(v – u)
Impulse (Ft) equals the change in momentum. It explains why following through in a tennis stroke or golf swing increases the contact time, thereby enhancing momentum transfer.
冲量(Ft)等于动量的变化。它解释了为什么网球击球或高尔夫挥杆中的随挥动作增加了接触时间，从而增强动量传递。
3. Work, Energy and Power | 功、能和功率
Understanding mechanical work, energy transformations and power output allows sports scientists to evaluate athlete efficiency and performance during explosive and endurance activities.
理解机械功、能量转换和功率输出，使运动科学家能够评估运动员在爆发力和耐力活动中的效率和表现。
W = Fd cosθ
Work done (W) is the product of force (F), displacement (d) and the cosine of the angle (θ) between them. When force and motion are parallel, W = Fd. Applied to work done against gravity in weightlifting.
做功(W)是力(F)、位移(d)及两者夹角(θ)余弦的乘积。当力与运动方向一致时，W=Fd。适用于举重时克服重力做功。
KE = ½mv²
Kinetic energy (KE) depends on mass and velocity squared. Sprinters convert chemical energy into KE; doubling speed quadruples KE, highlighting the importance of reaction forces.
动能(KE)取决于质量和速度的平方。短跑运动员将化学能转化为动能；速度加倍则动能增至四倍，凸显了反作用力的重要性。
PE = mgh
Gravitational potential energy (PE) gained when ascending a height (h). High jumpers and pole vaulters use this relationship to calculate the energy required to clear a bar.
重力势能(PE)是上升高度(h)时获得的能量。跳高和撑竿跳运动员利用这一关系计算越过横杆所需的能量。
P = W/t = Fv
Power (P) is the rate of doing work or the product of force and velocity. Maximal power output is crucial in activities like jumping, throwing and sprint cycling.
功率(P)是做功的速率，也是力与速度的乘积。最大功率输出在跳跃、投掷和短距离自行车等活动中至关重要。
4. Torque and Angular Motion | 力矩与角运动
Angular motion principles describe how objects rotate around an axis, essential for understanding gymnastics spins, diving somersaults and joint biomechanics.
角运动原理描述了物体围绕轴旋转的方式，对理解体操旋转、跳水翻腾和关节生物力学至关重要。
τ = F × d
Torque (τ) is the rotational effect of a force applied at a distance (d) from the axis. A larger moment arm produces greater rotation, explaining why longer golf clubs or tennis racquets can generate higher torque.
力矩(τ)是力在距转轴距离(d)处施加的转动效果。力臂越长，产生的旋转越大，这就解释了为什么更长的高尔夫球杆或网球拍能产生更大的扭矩。
ω = Δθ / Δt
Angular velocity (ω) is the rate of change of angular displacement (θ). In a pirouette, a skater increases ω by pulling the arms in, reducing the moment of inertia.
角速度(ω)是角位移(θ)的变化率。在旋转舞步中，滑冰者通过收拢手臂减小转动惯量来提高角速度。
α = Δω / Δt
Angular acceleration (α) measures how quickly angular velocity changes. Coaches monitor angular acceleration in throwing events to enhance spin rate.
角加速度(α)衡量角速度变化的快慢。教练在投掷项目中监测角加速度以提高旋转速率。
I = mr²
Moment of inertia (I) for a point mass depends on mass and the square of its distance from the axis. Tucking into a ball during a dive reduces I, thereby increasing spin speed.
点质量的转动惯量(I)取决于质量和到转轴距离的平方。跳水时抱膝可以减小转动惯量，从而增加旋转速度。
5. Bernoulli’s Principle and Fluid Forces | 伯努利原理与流体作用力
Bernoulli’s principle explains lift and drag forces on projectiles such as balls, javelins and discs. It is central to understanding the Magnus effect and aerodynamics in sport.
伯努利原理解释了球、标枪和铁饼等抛射物上的升力和阻力。它是理解运动中的马格努斯效应和空气动力学的核心。
P + ½ρv² + ρgh = constant
Along a streamline, the sum of static pressure (P), dynamic pressure (½ρv²) and hydrostatic pressure (ρgh) remains constant. Faster airflow over a spinning ball’s top surface lowers pressure, creating a lift force (Magnus effect).
沿流线，静压(P)、动压(½ρv²)和静水压(ρgh)之和恒定。旋转球顶部气流较快导致压力降低，从而产生升力（马格努斯效应）。
When the ball spins, it drags air around it; on one side air moves in the same direction as the flight path, increasing velocity and decreasing pressure. The resultant pressure difference produces a sideways force, explaining curving free kicks in football or dipping topspin in tennis.
当球旋转时，它会带动周围空气；在某一侧空气与飞行方向相同，速度加快、压力降低。由此产生的压力差产生侧向力，解释了足球中的弧线任意球或网球中的上旋下坠。
6. Muscle Force and Lever Systems | 肌肉力量与杠杆系统
The body’s musculoskeletal system functions as a series of levers. Understanding the mechanical advantage of different lever classes helps analyse movement efficiency and injury risk.
人体肌肉骨骼系统就像一系列杠杆。了解不同杠杆类别的机械优势有助于分析动作效率和损伤风险。
Fₑ × dₑ = Fₗ × dₗ
At rotational equilibrium, the effort force (Fₑ) times its moment arm (dₑ) equals the load force (Fₗ) times its moment arm (dₗ). In a bicep curl, the muscle insertion point close to the elbow means a high force is required for a small load.
在转动平衡时，动力(Fₑ)乘以其力臂(dₑ)等于阻力(Fₗ)乘以其力臂(dₗ)。在肱二头肌弯举中，肌肉附着点靠近肘部意味着需要很大的力来举起小负荷。
MA = dₑ / dₗ
Mechanical advantage (MA) is the ratio of effort arm to load arm. Most human levers are third‑class levers (MA<1), favouring speed and range of motion over force production.
机械优势(MA)是动力臂与阻力臂之比。人体大多数杠杆为第三类杠杆(MA<1)，有利于速度和运动范围，而非力量输出。
7. Cardiac Output and Stroke Volume | 心输出量与每搏输出量
Cardiovascular endurance depends on the heart’s ability to deliver oxygenated blood. These formulas are essential for interpreting data in laboratory tests and training programmes.
心血管耐力取决于心脏输送含氧血液的能力。这些公式对于解读实验室测试和训练计划中的数据至关重要。
Q = SV × HR
Cardiac output (Q) is the volume of blood pumped by the heart per minute. It equals stroke volume (SV) multiplied by heart rate (HR). During maximal exercise, Q can increase up to five times the resting value.
心输出量(Q)是心脏每分钟泵出的血量。它等于每搏输出量(SV)乘以心率(HR)。在最大强度运动中，Q可增加至静息值的五倍。
SV = EDV – ESV
Stroke volume is the difference between end‑diastolic volume (EDV) and end‑systolic volume (ESV). Endurance training increases EDV, leading to a higher SV and greater aerobic capacity.
每搏输出量是舒张末期容积(EDV)与收缩末期容积(ESV)之差。耐力训练增加EDV，从而提高SV和有氧化能力。
HRₘₐₓ = 220 – age
A simple estimate of maximum heart rate (HRₘₐₓ) helps prescribe training zones. While individual variation exists, it remains widely used in exercise physiology.
最大心率(HRₘₐₓ)的简易估算有助于制定训练区间。虽然存在个体差异，但在运动生理学中仍被广泛使用。
8. Pulmonary Ventilation | 肺通气量
Measuring lung ventilation helps assess an athlete’s respiratory efficiency during submaximal and maximal effort. The formulas distinguish between total ventilation and effective alveolar ventilation.
测量肺通气量有助于评估运动员在次最大和最大强度下的呼吸效率。这些公式区分了总通气量和有效肺泡通气量。
Vₑ = TV × f
Minute ventilation (Vₑ) is the total air moved in and out of the lungs per minute. It equals tidal volume (TV) multiplied by breathing frequency (f). At rest Vₑ ≈ 6 L/min, while in elite endurance athletes it may exceed 200 L/min during maximal exercise.
每分通气量(Vₑ)是每分钟进出肺部的空气总量。它等于潮气量(TV)乘以呼吸频率(f)。静息时Vₑ约为6升/分钟，而在精英耐力运动员最大运动中可超过200升/分钟。
Vₐ = (TV – VD) × f
Alveolar ventilation (Vₐ) accounts for dead space (VD) — the air that does not participate in gas exchange. Efficient breathing patterns minimise anatomical dead space and improve oxygenation.
肺泡通气量(Vₐ)扣除了不参与气体交换的死腔(VD)。有效的呼吸模式可最大限度地减少解剖死腔，改善氧合。
9. Energy Systems and ATP Yield | 能量系统与ATP产量
Muscle contraction relies on adenosine triphosphate (ATP). Three energy systems replenish ATP at different rates and capacities, and their quantitative understanding is key to designing sport‑specific training.
肌肉收缩依赖三磷酸腺苷(ATP)。三大能量系统以不同的速率和容量再合成ATP，定量理解这些系统是设计专项训练的关键。
ATP‑PC system: PC → Pi + Creatine + Energy (1 ATP per PC)
The phosphocreatine (PC) system provides immediate energy for up to 10 seconds of maximal effort. Each PC molecule yields one ATP, supporting explosive activities like a 100 m start or a vertical jump.
磷酸肌酸(PC)系统提供即刻能量，支持最长10秒的最大强度运动。每个PC分子产生1个ATP，支撑100米起跑或垂直跳等爆发性活动。
Anaerobic glycolysis: Glucose → 2 Lactic acid + 2 ATP
When oxygen is limited, glucose is broken down to lactic acid, yielding a net gain of 2 ATP per glucose. This system dominates intense efforts lasting 10–60 seconds, such as 400 m running or repeated rugby scrums.
在氧气有限时，葡萄糖分解为乳酸，每个葡萄糖净得2个ATP。该系统主导持续10-60秒的高强度运动，如400米跑或橄榄球连续争球。
Aerobic system: Glucose + 6O₂ → 6CO₂ + 6H₂O + ~36–38 ATP
Aerobic metabolism provides the highest ATP yield per glucose, fuelling prolonged endurance activity. It also oxidises fatty acids, producing even greater ATP quantities, critical for marathon running and cycling.
有氧代谢每个葡萄糖产生最高的ATP产量，为长时间耐力活动提供燃料。该系统还能氧化脂肪酸，产生更多的ATP，对马拉松和自行车运动至关重要。
10. Body Composition and Somatotype | 身体成分与体型
Body composition indices help classify athletes, monitor health risks and tailor nutritional strategies. Simple anthropometric equations are widely used in sport science.
身体成分指数有助于对运动员进行分类、监测健康风险和制定营养策略。简单的人体测量方程在运动科学中被广泛使用。
BMI = mass (kg) / height² (m²)
Body mass index (BMI) provides a rough measure of adiposity. Despite limitations in muscular athletes, it remains a quick screening tool. A BMI of 18.5–24.9 kg/m² is considered healthy for the general population.
体重指数(BMI)提供了肥胖程度的粗略测量。尽管对肌肉发达的运动员有局限性，它仍是一种快速筛查工具。一般人群健康的BMI范围为18.5–24.9 kg/m²。
Waist‑to‑hip ratio = waist circumference / hip circumference
This ratio assesses fat distribution. Higher values indicate central obesity, which is a risk factor for cardiovascular disease. Endurance athletes typically exhibit lower ratios.
该比率评估脂肪分布。数值较高表明中心性肥胖，这是心血管疾病的危险因素。耐力运动员通常表现出较低的腰臀比。
Somatotype rating: Endomorphy, Mesomorphy, Ectomorphy
Sheldon’s somatotype chart rates physique on a 1–7 scale for three components. A sumo wrestler might have a high endomorphy rating, a sprinter high mesomorphy, and a distance runner high ectomorphy.
谢尔顿体型分类法用1-7的评分从三个成分评估体格。相扑手的内胚叶评分可能很高，短跑运动员中胚叶高，长跑运动员外胚叶高。
11. Basic Statistics in Sport Analysis | 体育分析中的基础统计
Statistical tools are used to analyse performance data, compare training methods and assess the reliability of tests. Mean and standard deviation are fundamental for interpreting results.
统计工具
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