📚 Quick Reference Handbook of Formulas and Laws for Year 12 Cambridge Physical Education | Year 12 剑桥体育公式定理速查手册
This handbook offers a streamlined collection of the most important formulas, laws, and fundamental principles required for the Year 12 Cambridge Physical Education course. It covers key topics in biomechanics and exercise physiology, serving as an efficient revision aid and a reliable reference for problem-solving.
本手册提供了 Year 12 剑桥体育课程中最关键的公式、定律和基础原理的精简汇编，涵盖生物力学和运动生理学的核心内容，可作为高效的复习辅助工具和可靠的解题参考。
1. Linear Kinematics | 线性运动学
v = Δs / Δt
Average velocity (v) is the rate of change of displacement (Δs) over a time interval (Δt); it is a vector quantity indicating both speed and direction.
平均速度（v）是位移变化量（Δs）与时间间隔（Δt）的比率；它是一个矢量，表示运动的快慢和方向。
a = Δv / Δt
Acceleration (a) is defined as the rate of change of velocity (Δv) with respect to time (Δt). It can be positive (speeding up) or negative (deceleration).
加速度（a）定义为速度变化量（Δv）与时间（Δt）的比率。它可为正值（加速）或负值（减速）。
v = u + at
One of the SUVAT equations for uniform acceleration, linking final velocity (v), initial velocity (u), acceleration (a), and time (t).
匀加速运动 SUVAT 方程之一，连接末速度（v）、初速度（u）、加速度（a）和时间（t）。
s = ut + ½at²
This equation calculates displacement (s) when initial velocity, acceleration, and time are known. The term ½at² accounts for the changing speed.
该方程用于已知初速度、加速度和时间时计算位移（s）。项 ½at² 体现速度变化带来的位移。
v² = u² + 2as
Connects final velocity squared to initial velocity squared, acceleration, and displacement, useful when time is not given.
将末速度的平方与初速度平方、加速度和位移联系起来，适用于未知时间的情况。
2. Newton's Laws of Motion | 牛顿运动定律
F = ma
Newton's Second Law: the net force (F) acting on an object equals its mass (m) multiplied by the acceleration (a) produced. This explains how forces change motion in sports.
牛顿第二定律：作用于物体的净力（F）等于质量（m）乘以产生的加速度（a）。这解释了运动中力如何改变运动状态。
Law of Inertia (First Law): A body remains at rest or in uniform motion unless acted upon by an external net force. This underlies the importance of applying force to start, stop, or change direction.
惯性定律（第一定律）： 物体保持静止或匀速直线运动，除非受到外力作用。这说明了施加力来启动、停止或改变方向的重要性。
Law of Action-Reaction (Third Law): For every action force there is an equal and opposite reaction force. In a sprint start, the athlete pushes against the blocks, and the blocks push the athlete forward.
作用与反作用定律（第三定律）： 每个作用力都有一个大小相等、方向相反的反作用力。在短跑起跑中，运动员向后蹬起跑器，起跑器则向前推动运动员。
Weight = mg
= 9.81 m/s² on Earth. It is a force vector acting downward, not to be confused with mass.
重量（W）是作用在质量（m）上的引力，地球上 g=9.81 m/s²。它是向下的力矢量，不要与质量混淆。
3. Momentum and Impulse | 动量与冲量
p = mv
Linear momentum (p) is the product of mass (m) and velocity (v). It is a vector that quantifies the amount of motion an object possesses.
线性动量（p）是质量（m）与速度（v）的乘积。它是一个矢量，用于量化物体的运动量。
Impulse = FΔt = Δp
Impulse is the change in momentum (Δp) caused by a force (F) applied over a contact time (Δt). In soccer, following through with the foot increases contact time and impulse, resulting in a faster ball.
冲量是力（F）在接触时间（Δt）内作用引起的动量变化（Δp）。在足球中，随球动作延长了接触时间，增大了冲量，从而使球速更快。
Conservation of Momentum: In a closed system, total momentum before a collision equals total momentum after, assuming no external forces. This explains rebounds and tackles where players transfer momentum.
动量守恒： 在没有外力作用的封闭系统中，碰撞前的总动量等于碰撞后的总动量。这解释了反弹和抢断中运动员之间动量的转移。
4. Work, Power and Energy | 功、功率与能量
W = Fd cosθ
Work done (W) is the product of force (F) and displacement (d) in the direction of the force. The angle (θ) accounts for force not being parallel to motion; only the parallel component does work.
功（W）是力（F）与沿力方向的位移（d）的乘积。角度（θ）用于力不平行于运动的情况；只有平行分量做功。
P = W / t
Power (P) is the rate of doing work or transferring energy over time (t). In sport, athletes with greater power outputs can generate more explosive movements.
功率（P）是做功或转移能量的速率（时间 t）。在体育运动中，输出功率更大的运动员能够产生更具爆发力的动作。
KE = ½mv²
Kinetic energy (KE) is the energy of motion, dependent on mass and the square of velocity. A small increase in speed leads to a large gain in KE, highlighting the importance of velocity in striking and sprinting.
动能（KE）是运动能量，取决于质量和速度的平方。速度小幅增加会导致动能大幅提高，这凸显了速度在击打和短跑中的重要性。
GPE = mgh
Gravitational potential energy (GPE) is energy stored due to an object's height (h) above a reference point. In pole vaulting, a fast run-up is converted into GPE to clear the bar.
重力势能（GPE）是物体因高度（h）高于参考点而储存的能量。在撑竿跳高中，快速的助跑转化为重力势能以越过横杆。
5. Levers and Moments | 杠杆与力矩
Moment = F × d
A moment (torque) is the turning effect of a force (F) acting at a perpendicular distance (d) from the pivot. It is measured in newton-metres (Nm).
力矩（扭转效应）是力（F）在相对于支点的垂直距离（d）上产生的旋转效果，单位为牛顿·米（Nm）。
Principle of Moments: For a lever system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. This principle is used to analyse force production around a joint.
力矩原理： 在转动平衡的杠杆系统中，顺时针力矩之和等于逆时针力矩之和。该原理用于分析关节周围的力生成。
Lever Classes: First class (effort–fulcrum–load, e.g., neck extension), second class (effort–load–fulcrum, e.g., ankle plantar flexion during jumping), third class (fulcrum–effort–load, e.g., bicep curl). Most human movements utilise third-class levers, which favour speed and range of movement over force.
杠杆类别： 第一类（力–支点–阻力，如颈部后伸），第二类（力–阻力–支点，如跳跃时的踝关节跖屈），第三类（支点–力–阻力，如肱二头肌弯举）。多数人体动作使用第三类杠杆，牺牲力量以换取速度和运动幅度。
6. Angular Motion | 角运动
ω = Δθ / Δt
Angular velocity (ω) is the rate of change of angular displacement (Δθ) over time (Δt). It is measured in radians per second (rad/s) and describes how fast an object rotates.
角速度（ω）是角位移（Δθ）随时间（Δt）的变化率，单位为弧度/秒（rad/s），描述物体旋转的快慢。
v = rω
The linear velocity (v) of a point on a rotating body equals the radius (r) multiplied by angular velocity. This relationship explains why a longer tennis racket head travels faster and generates more ball speed.
旋转体上一点的线速度（v）等于半径（r）乘以角速度。该关系解释了为什么较长的网球拍头移动更快并产生更大的球速。
L = Iω
Angular momentum (L) is the product of moment of inertia (I) and angular velocity. In diving, a tuck position reduces I, so ω increases to conserve angular momentum, allowing faster somersaults.
角动量（L）是转动惯量（I）与角速度的乘积。在跳水中，团身姿势减小 I，因此 ω 增大以保持角动量守恒，从而实现更快的空翻。
τ = Iα
Torque (τ) equals moment of inertia (I) times angular acceleration (α). This is the rotational analogue of Newton's Second Law and is essential for analysing movements like a discus throw.
力矩（τ）等于转动惯量（I）乘以角加速度（α）。这是牛顿第二定律的旋转版本，对于分析铁饼投掷等动作至关重要。
7. Projectile Motion | 抛体运动
Horizontal component: uₓ = u cosθ
The horizontal velocity (uₓ) of a projectile remains constant (ignoring air resistance). It is calculated from the initial velocity (u) and the angle of release (θ).
抛体的水平速度（uₓ）保持不变（忽略空气阻力），由初速度（u）和出手角（θ）计算得出。
Vertical component: uᵧ = u sinθ
The initial vertical velocity (uᵧ) determines how high and how long the projectile stays in the air. It is affected by gravity, causing symmetric parabolic flight without air resistance.
初始垂直速度（uᵧ）决定了抛体的飞行高度和滞空时间。它受重力影响，在无空气阻力时形成对称的抛物线轨迹。
Time of flight: t = 2u sinθ / g
Total time a projectile is airborne (assuming equal release and landing heights). Doubling the vertical component of velocity doubles the flight time.
抛体在空中飞行的总时间（假设出手与落地高度相同）。垂直速度分量加倍，则飞行时间加倍。
Maximum height: h = (u sinθ)² / 2g
The peak vertical displacement achieved by the projectile. For a given speed, a release angle of 90° gives maximum height, but 45° is optimal for range on flat ground without air resistance.
抛体达到的最高垂直位移。在给定速度下，90° 出手角获得最大高度，但在无空气阻力的平地上，45° 是射程最优角。
8. Fluid Mechanics and Aerodynamics | 流体力学与空气动力学
FD = ½ CD ρ A v²
The drag force (FD) opposing an object moving through a fluid depends on the drag coefficient (CD), fluid density (ρ), cross-sectional area (A), and the square of velocity (v²). Cyclists adopt a streamlined position to minimise A and CD, reducing drag.
流体中运动的物体受到的阻力（FD）取决于阻力系数（CD）、流体密度（ρ）、横截面积（A）和速度的平方（v²）。自行车手采用流线型姿势以最小化 A 和 CD，从而减少阻力。
Bernoulli Principle: In a fluid flow, an increase in velocity corresponds to a decrease in pressure. This principle helps explain the lift force on a spinning ball or an aerofoil. In sport, a well-struck football with spin experiences a pressure differential that causes it to swerve.
伯努利原理： 在流体流动中，流速增加导致压力降低。该原理有助于解释旋转球或翼型上的升力。在体育运动中，带有旋转的足球会受到压力差影响从而产生弧线。
Magnus Effect: A spinning object moving through a fluid creates a pressure difference, generating a force perpendicular to the relative flow. Topspin in tennis makes the ball dip faster, while backspin lengthens the flight.
马格努斯效应： 旋转物体在流体中运动时产生压力差，形成垂直于流动方向的力。网球中的上旋使球更快下坠，而下旋则延长球的飞行。
9. Physiological Parameters and Formulae | 生理参数与公式
Cardiac Output (Q) = Heart Rate (HR) × Stroke Volume (SV)
Cardiac output (Q), typically measured in L/min, is the volume of blood pumped by the heart per minute. Endurance training increases stroke volume, allowing a higher Q at a given heart rate.
心输出量（Q），通常以 L/min 为单位，是心脏每分钟泵出的血量。耐力训练增加每搏输出量，从而在某一给定心率下能实现更高的 Q。
HRmax = 220 - age
An estimate of maximum heart rate (HRmax) in beats per minute, often used to set training zones. Individual variation exists, so it is a general guide rather than an exact value.
最大心率（HRmax，次/分）的估算值，常用于设定训练区间。存在个体差异，因此仅为一般性指导，并非精确数值。
Minute Ventilation (VE) = Tidal Volume (TV) × Breathing Frequency (f)
Minute ventilation (VE) is the total volume of air inhaled or exhaled per minute. During intense exercise, both tidal volume and breathing frequency increase to meet oxygen demands.
每分通气量（VE）是每分钟吸入或呼出的空气总量。在剧烈运动中，潮气量和呼吸频率均增加以满足氧气需求。
BMI = mass (kg) / height² (m²)
Body Mass Index provides a rough classification of body composition. It does not distinguish between muscle and fat mass, so athletic individuals may be misclassified as overweight.
身体质量指数（BMI）提供身体成分的粗略分类。它不区分肌肉和脂肪质量，因此运动员可能被误判为超重。
10. Force Resolution and Free Body Diagrams | 力的分解与受力分析
Fx = F cosθ, Fy = F sinθ
Any force vector (F) can be resolved into perpendicular components (Fx horizontal, Fy vertical) using trigonometry. This is essential when analysing forces on a slope or during a turn.
任何力矢量（F）都可以用三角函数分解为垂直分量（Fx 水平，Fy 垂直）。这在分析斜坡或转弯时的力时至关重要。
Equilibrium: ual zero (ΣF = 0). This concept is used to understand body positioning and balanced poses.
平衡状态： 物体处于静力或动力平衡时，任意方向上的合力必须为零（ΣF=0）。这一概念用于理解身体姿势和平衡姿态。
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