📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This handbook compiles the essential formulas, laws and biomechanical principles required for the Year 13 CIE Physical Education syllabus. Each section is presented in both English and Chinese to support bilingual revision and quick reference. Use it to reinforce your understanding of the mechanics, physiology and physics behind human movement and sporting performance.
本手册汇编了 Year 13 CIE 体育课程所必需的基本公式、定律和生物力学原理。每个部分均以英中双语呈现，方便双语复习与快速查阅。用它来巩固你对人体运动和运动表现背后的力学、生理学和物理学的理解。
1. Linear Kinematics | 直线运动学
Linear kinematics describes motion along a straight line without considering the forces that cause it. The four SUVAT equations link initial velocity (u), final velocity (v), acceleration (a), displacement (s) and time (t).
直线运动学描述沿直线的运动，而不考虑引起运动的力。四个 SUVAT 方程将初速度 (u)、末速度 (v)、加速度 (a)、位移 (s) 和时间 (t) 联系起来。
v = u + at
v² = u² + 2as
s = ut + ½at²
s = ½(u + v)t
Always check the direction of vectors: assign positive and negative consistently to quantities like velocity and acceleration to avoid sign errors.
务必检查矢量的方向：为速度和加速度等量一致地指定正负号，以避免符号错误。
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（惯性定律）： 除非受到外合力的作用，否则物体将保持静止或匀速直线运动状态。
Second Law (Acceleration): The acceleration of a body is directly proportional to the resultant force and inversely proportional to its mass: F = ma.
第二定律（加速度定律）： 物体的加速度与合外力成正比，与其质量成反比：F = ma。
Third Law (Action–Reaction): For every action there is an equal and opposite reaction. These forces act on different bodies and never cancel out on the same body.
第三定律（作用与反作用定律）： 每一个作用力都有一个大小相等、方向相反的反作用力。这两个力作用在不同的物体上，永远不会在同一物体上抵消。
3. Forces and Free Body Diagrams | 力与受力分析
Identify all forces acting on an athlete or object: weight (W = mg), normal reaction (R), friction (Ff), air resistance/drag (Fd) and applied forces. Draw arrows proportional to magnitude.
识别作用在运动员或物体上的所有力：重力 (W = mg)、法向反作用力 (R)、摩擦力 (Ff)、空气阻力 / 阻力 (Fd) 以及施加的力。按比例绘制箭头。
On an inclined plane, the component of weight parallel to the slope is mg sinθ and the component perpendicular is mg cosθ. Friction equals μR, where μ is the coefficient of friction.
在斜面上，重力平行于斜面的分量为 mg sinθ，垂直分量为 mg cosθ。摩擦力等于 μR，其中 μ 为摩擦系数。
When an athlete is in equilibrium (static or dynamic), the vector sum of all forces is zero.
当运动员处于平衡状态（静态或动态）时，所有力的矢量和为零。
4. Momentum and Impulse | 动量与冲量
Linear momentum is the product of mass and velocity: p = mv (unit: kg·m·s⁻¹). Momentum is a vector quantity.
线动量是质量与速度的乘积：p = mv（单位：kg·m·s⁻¹）。动量是一个矢量。
Impulse equals the change in momentum: Ft = Δp = mv – mu. This impulse–momentum relationship explains why ‘giving’ with an object (increasing time t) reduces the average force experienced.
冲量等于动量的变化量：Ft = Δp = mv – mu。这种冲量-动量关系解释了为什么随物体缓冲（增加时间 t）会减小平均受力。
In a closed system, total momentum before a collision equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
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。
5. Work, Energy and Power | 功、能量与功率
Work done is force × displacement in the direction of the force: W = Fd cosθ (unit: joule, J).
功是力乘以在力方向上的位移：W = Fd cosθ（单位：焦耳，J）。
Kinetic energy: KE = ½mv². Gravitational potential energy: GPE = mgh.
动能： KE = ½mv²。重力势能： GPE = mgh。
In the absence of external resistance, mechanical energy is conserved: total initial energy = total final energy. In sport, energy is transferred and degraded as heat.
在没有外部阻力的情况下，机械能守恒：初始总能量 = 最终总能量。在运动中，能量会发生转移并退化为热量。
Power is the rate of doing work: P = W/t or P = Fv (for constant velocity in direction of force). Unit: watt (W).
功率是做功的速率：P = W/t 或 P = Fv（在力方向上的恒定速度）。单位：瓦特 (W)。
6. Levers and Torque | 杠杆与力矩
A lever consists of a rigid bar, a fulcrum (pivot), an effort force and a load. The torque (moment of force) is τ = F × d, where d is the perpendicular distance from the pivot to the line of action of the force. Unit: N·m.
杠杆由刚性杆、支点（枢轴）、用力点和负载组成。力矩（力的力矩）为 τ = F × d，其中 d 是支点到力作用线的垂直距离。单位：N·m。
Three classes of levers exist in the body:
人体内存在三类杠杆：
- First class (effort–fulcrum–load): e.g. neck extension (atlanto-occipital joint).
- 第一类（用力点–支点–负载）：例如颈部伸展（寰枕关节）。
- Second class (fulcrum–load–effort): e.g. standing on tiptoes (ball of foot as fulcrum).
- 第二类（支点–负载–用力点）：例如踮脚尖站立（脚掌球部为支点）。
- Third class (fulcrum–effort–load): majority of body levers, e.g. biceps curl at elbow. This favours speed and range of motion over strength.
- 第三类（支点–用力点–负载）：大多数人体杠杆，例如肘关节肱二头肌弯举。这类杠杆利于速度和运动幅度，而非力量。
Mechanical advantage = effort arm ÷ load arm. When MA > 1, the lever amplifies force; when MA < 1, it amplifies speed.
机械效益 = 力臂 ÷ 负载臂。当 MA > 1 时，杠杆增力；当 MA < 1 时，杠杆增速。
7. Angular Motion | 角运动
Angular displacement (θ) is measured in radians. Angular velocity: ω = Δθ / Δt (rad·s⁻¹). Angular acceleration: α = Δω / Δt (rad·s⁻²).
角位移 (θ) 以弧度计量。角速度： ω = Δθ / Δt (rad·s⁻¹)。角加速度： α = Δω / Δt (rad·s⁻²)。
Conversion between linear and angular quantities for a point on a rotating body at distance r from the axis:
旋转体上距离转轴 r 的点的线量与角量之间的换算：
v = ωr
a = αr (tangential acceleration)
Centripetal acceleration: ac = v² / r = ω² r, directed towards the centre. Centripetal force: Fc = mv² / r = mω² r.
向心加速度： ac = v² / r = ω² r，方向指向中心。向心力： Fc = mv² / r = mω² r。
Angular momentum is conserved when no external torque acts: L = Iω = constant, where I is moment of inertia (= Σ mr²). An athlete tucking in during a dive reduces I and increases ω, speeding up rotation.
角动量在没有外力矩作用时守恒：L = Iω = 常量，其中 I 为转动惯量（= Σ mr²）。运动员跳水时抱紧身体会减小 I 并增大 ω，加速旋转。
8. Fluid Mechanics (Bernoulli & Drag) | 流体力学（伯努利与阻力）
Bernoulli’s principle: Where the velocity of a fluid (or air) is high, the pressure is low, and vice versa. This explains lift on an aerofoil or a spinning ball (Magnus effect).
伯努利原理： 流体（或空气）流速高的地方压强低，反之亦然。这解释了翼型或旋转球体上的升力（马格努斯效应）。
The Magnus effect arises when a spinning ball drags air faster on one side, creating a pressure differential that curves the flight path.
马格努斯效应产生于旋转的球体使一侧空气流动更快，从而产生压强差，使飞行路径弯曲。
Drag force opposing motion through a fluid: Fd = ½ Cd ρ A v², where Cd is drag coefficient, ρ fluid density, A cross-sectional area, v velocity. Reducing A (tucking in cycling) or streamlining shape lowers drag.
阻力是流体中阻碍运动的力：Fd = ½ Cd ρ A v²，其中 Cd 为阻力系数，ρ 为流体密度，A 为截面积，v 为速度。减小 A（自行车赛中团身）或流线型造型可降低阻力。
9. Centre of Mass and Stability | 重心与稳定性
The centre of mass (CoM) is the point where the entire mass of a body can be considered concentrated for translational motion. In a uniform gravitational field, it coincides with the centre of gravity.
质心 (CoM) 是物体全部质量在平移运动中可以视为集中的一点。在均匀重力场中，它与重心重合。
Stability depends on: height of CoM (lower increases stability), size of base of support (larger base increases stability), and the line of gravity’s position relative to the base. An object topples when the line of gravity falls outside the base.
稳定性取决于：质心高度（越低越稳定）、支撑面大小（越大越稳定）以及重力线相对于支撑面的位置。当重力线落在支撑面之外时，物体会倾倒。
In sport, lowering CoM (e.g. defensive stance) and widening the base (e.g. sumo stance) improve stability, while raising CoM aids quick direction changes.
在运动中，降低质心（例如防守姿势）和加宽支撑面（例如相扑站姿）可提高稳定性，而升高质心则有利于快速变向。
10. Projectile Motion | 抛体运动
Projectile motion can be analysed by separating horizontal and vertical components of velocity. Horizontal velocity remains constant (ignoring air resistance); vertical motion is governed by gravity (g = 9.81 m·s⁻²).
抛体运动可通过分解速度的水平分量和竖直分量来分析。水平速度保持不变（忽略空气阻力）；竖直运动受重力控制 (g = 9.81 m·s⁻²)。
For a projectile launched at initial velocity u at an angle θ to the horizontal:
对于以初速度 u 、与水平方向成 θ 角发射的抛体：
ux = u cosθ, uy = u sinθ
Time of flight for symmetric launch–landing: t = 2u sinθ / g. Range: R = (u² sin 2θ) / g. Maximum height: H = (u² sin²θ) / 2g.
对称发射-着陆的飞行时间：t = 2u sinθ / g。射程：R = (u² sin 2θ) / g。最大高度：H = (u² sin²θ) / 2g。
Optimal release angle for maximum range on a flat surface is 45° (in vacuum); with air resistance, it is slightly lower. Release height above landing height changes the optimal angle.
在平坦表面上达到最大射程的最佳出手角为 45°（真空中）；有空气阻力时，最佳角度略低。出手高度高于着陆高度会改变最佳角度。
11. Biomechanical Principles in Sport | 运动生物力学原理
Summation of forces: To generate maximum force, an athlete should recruit body segments sequentially from large to small, timing each action so that the final velocity of the distal segment is maximised (e.g. javelin throw).
力的叠加： 要产生最大力量，运动员应从大到小依次动员身体环节，并安排好每个动作的时机，使末端环节的最终速度最大化（例如掷标枪）。
Impulse–momentum relationship: Applying force over a longer time (e.g. following through) increases the change in momentum, improving accuracy and velocity.
冲量-动量关系： 在较长时间内施加力（例如顺势动作）会增加动量的变化量，提高准确性和速度。
Newton’s third law in sport: A sprinter drives backwards against the blocks; the blocks push forward, providing the reaction force for acceleration.
运动中的牛顿第三定律： 短跑运动员向后蹬起跑器；起跑器向前推动，提供加速的反作用力。
Conservation of angular momentum is used in diving, gymnastics and figure skating to control rotation speed. Tightening the body decreases I and increases ω.
角动量守恒应用于跳水、体操和花样滑冰中以控制旋转速度。收紧身体会减小 I 并增加 ω。
12. Physiological Formulas (VO₂ max, Cardiac Output) | 生理学公式（最大摄氧量，心输出量）
Cardiac output (Q) is the volume of blood pumped by the heart per minute: Q = SV × HR, where SV is stroke volume (ml/beat) and HR is heart rate (bpm). Typical resting Q ≈ 5 L/min; can exceed 35 L/min in elite endurance athletes.
心输出量 (Q) 是心脏每分钟泵出的血量：Q = SV × HR，其中 SV 是每搏输出量 (ml/次)，HR 是心率 (bpm)。静息心输出量通常约为 5 L/min；优秀耐力运动员可超过 35 L/min。
VO₂ max is the maximum volume of oxygen the body can utilise per minute per kilogram of body mass. It is expressed in ml·kg⁻¹·min⁻¹ and calculated as: VO₂ max = Q × (a−v)O₂ difference, where (a−v)O₂ difference is the arteriovenous oxygen difference.
最大摄氧量 (VO₂ max) 是身体每分钟每千克体重能够利用的最大氧气量。单位是 ml·kg⁻¹·min⁻¹，计算公式为：VO₂ max = Q × (a−v)O₂ 差，其中 (a−v)O₂ 差为动静脉氧差。
The Fick equation can also be written as: VO₂ = Q × (CaO₂ – CvO₂), where CaO₂ and CvO₂ are arterial and venous oxygen contents.
菲克方程也可写作：VO₂ = Q × (CaO₂ – CvO₂)，其中 CaO₂ 和 CvO₂ 分别是动脉和静脉血氧含量。
| Variable 变量 | Formula/Relationship 公式/关系 |
|---|---|
| Cardiac Output (Q) | Q = SV × HR |
| VO₂ max | VO₂ max = Q × (a−v)O₂ diff |
| Oxygen delivery | DO₂ = Q × CaO₂ |
| Mean arterial pressure | MAP ≈ diastolic + ⅓(systolic‑diastolic) |
Understanding these physiological formulas helps explain adaptations to training, such as increased stroke volume and improved oxygen extraction.
理解这些生理学公式有助于解释训练适应性，例如每搏输出量增加和摄氧能力提高。
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