Pre-U CAIE Physical Education: Formula & Theorem Quick Reference | Pre-U CAIE 体育：公式定理速查手册

📚 Pre-U CAIE Physical Education: Formula & Theorem Quick Reference | Pre-U CAIE 体育：公式定理速查手册

This handbook compiles essential formulas, laws, and theorems relevant to the Cambridge Pre-U Physical Education syllabus. It covers biomechanics, physiology, psychology, and statistics to aid quick revision and application in exam contexts. Each entry is presented with a concise mathematical expression followed by dual-language explanations, enabling you to understand the principle in English and its practical meaning in Chinese.

本手册汇编了与剑桥 Pre-U 体育教学大纲密切相关的核心公式、定律和定理，涵盖生物力学、生理学、心理学和统计学，以便在备考中快速查阅和应用。每个条目均配有数学表达式及中英双语解释，帮助你同时掌握学术英文术语与中文实际应用。

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1. Linear Kinematics Equations | 直线运动学方程

Linear kinematics describes motion along a straight line with constant acceleration. The following SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). All quantities are vectors; direction must be assigned consistently.

直线运动学描述匀加速直线运动，以下 SUVAT 方程关联位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)。所有量均为矢量，使用中需统一规定正方向。

v = u + at

This equation calculates final velocity directly from initial velocity, acceleration, and time. It is often used to find the velocity at a given instant during uniformly accelerated motion.

此方程直接由初速度、加速度和时间计算末速度，常用于确定匀加速运动中某一时刻的速度。

s = ut + ½at²

This gives the displacement when initial velocity and acceleration are known. The term ½at² accounts for the extra distance covered due to acceleration.

该式给出已知初速度和加速度时的位移。其中 ½at² 项代表因加速度产生的额外距离。

v² = u² + 2as

This equation links final velocity to displacement without involving time. It is particularly useful when time is not given or required.

此方程将末速度与位移联系起来而不涉及时间，当问题未给出或不要求时间时特别有用。

s = ½(u + v)t

This expression states that displacement equals average velocity multiplied by time. It applies only when acceleration is constant.

此式表明位移等于平均速度乘以时间，仅适用于匀加速度情形。

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2. Angular Kinematics Equations | 角运动学方程

When a body rotates about an axis with constant angular acceleration, the equations for angular motion are analogous to linear SUVAT. Angular displacement (θ) is measured in radians, initial and final angular velocities are ω₁ and ω₂, angular acceleration is α, and time is t.

当刚体绕轴以恒定角加速度转动时，其角运动方程与直线 SUVAT 类似。角位移(θ)的单位为弧度，初、末角速度分别为 ω₁ 和 ω₂，角加速度为 α，时间为 t。

ω₂ = ω₁ + αt

This calculates final angular velocity after a period of constant angular acceleration from a known initial angular velocity.

此式计算从已知初角速度经历恒定角加速度一段时间后的末角速度。

θ = ω₁t + ½αt²

Angular displacement is determined by the initial angular velocity and the contribution of angular acceleration over time.

角位移由初角速度以及角加速度随时间产生的附加转角共同决定。

ω₂² = ω₁² + 2αθ

This relates angular velocity to angular displacement without involving time, useful in rotational kinematics where time is unknown.

该式在不涉及时间的情况下连接角速度与角位移，常用于时间未知的旋转运动学问题。

θ = ½(ω₁ + ω₂)t

The angular displacement equals the average angular velocity multiplied by time, assuming constant angular acceleration.

在匀角加速度下，角位移等于平均角速度乘以时间。

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3. Newton’s Laws & Linear Momentum | 牛顿定律与线性动量

Newton’s three laws of motion underpin all force and momentum calculations in sport. Linear momentum (p) is the product of mass and velocity, and impulse (J) equals the change in momentum.

牛顿运动三定律是体育中所有力和动量计算的基础。线性动量(p)为质量与速度的乘积，冲量(J)等于动量的变化量。

F = ma

Newton’s second law: the net force acting on an object is proportional to its mass and acceleration. It is the fundamental equation for analysing forces in sprinting, jumping and striking.

牛顿第二定律：物体所受合力正比于其质量与加速度。这是分析短跑、跳跃和击打动作中受力的基本方程。

p = mv

Linear momentum quantifies the quantity of motion an object possesses. Heavier or faster-moving athletes have greater momentum, making them harder to stop.

线性动量量化了物体的运动量。更重或移动更快的运动员具有更大的动量，因而更难被阻挡。

Impulse = FΔt = Δp = m(v – u)

Impulse is the product of force and the time for which it acts, and it equals the change in momentum. Coaches apply this to maximize momentum change in striking or throwing by increasing either force or contact time.

冲量是力与其作用时间的乘积，等于动量的变化。教练员通过增大力量或延长接触时间来优化击打或投掷时的动量变化。

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

In the absence of external forces, total linear momentum before a collision equals total momentum after. This conservation law applies to collisions and tackles in rugby or football.

无外力作用时，碰撞前的总动量等于碰撞后的总动量。该守恒定律适用于橄榄球或足球中的碰撞和抢断。

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4. Projectile Motion Formulas | 抛体运动公式

A projectile launched with speed u at an angle θ to the horizontal follows a parabolic path. Air resistance is neglected in these idealised formulas. The horizontal component of velocity (vₓ) remains constant while the vertical component (v_y) changes under gravity (g = 9.81 m·s⁻²).

以初速 u、与水平夹角 θ 抛出的物体遵循抛物线轨迹，以下理想方程忽略空气阻力。水平分速度(vₓ)保持不变，垂直分速度(v_y)在重力(g = 9.81 m·s⁻²)作用下变化。

vₓ = u cosθ

The constant horizontal component of velocity is found from the initial speed and launch angle.

恒定的水平分速度由初速度和抛射角得出。

v_y = u sinθ – gt

The vertical velocity component decreases linearly due to gravity until reaching the peak, then becomes negative as the projectile descends.

垂直分速度因重力线性减小，至最高点达零，随后转为负值表示下落。

Time of flight T = 2u sinθ / g

The total time the projectile stays in the air depends on the initial vertical speed and gravity. Doubling the launch speed at a given angle quadruples the range but only doubles the flight time.

物体在空中的总飞行时间取决于初垂直速度和重力。给定角度下加倍初速可使射程增至四倍，但飞行时间仅加倍。

Maximum height H = u² sin²θ / (2g)

The peak altitude is reached when the vertical velocity becomes zero. It is proportional to the square of the initial vertical speed.

当垂直速度为零时达到最大高度，其与初垂直速度的平方成正比。

Range R = u² sin2θ / g

The horizontal distance travelled before landing is maximised when the launch angle is 45° (sin2θ = 1). Wind and air resistance can significantly alter the actual range.

落地前的水平飞行距离在抛射角为 45° 时最大 (sin2θ = 1)。实际射程会因风和空气阻力而显著改变。

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5. Forces, Impulse & Fluid Dynamics | 力、冲量与流体动力学

In sport, fluid forces such as air resistance and the Magnus effect influence the flight of balls, javelins, and cyclists’ performance. Drag is opposed to motion, while lift is perpendicular to the flow.

在体育运动中，空气阻力、马格努斯效应等流体作用力影响着球、标枪的飞行以及自行车运动员的表现。阻力方向与运动相反，升力垂直于来流。

Drag force F_D = ½ C_D ρ A v²

Drag is proportional to fluid density (ρ), cross-sectional area (A), the square of velocity, and the drag coefficient (C_D), which depends on shape and surface texture. Reducing frontal area is key in cycling and speed skating.

阻力与流体密度(ρ)、迎风面积(A

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