📚 A-Level WJEC Physical Education: Quick Reference Formula and Theorem Handbook | A-Level WJEC 体育：公式定理速查手册
This quick reference handbook collates the essential formulae, theorems and principles required for the WJEC A-Level Physical Education specification. From linear kinetics and angular motion to physiological measures and psychological performance models, mastering these equations is vital for success in both written examinations and practical application. Use this guide as a rapid revision tool to ensure you can recall and apply every relationship with confidence.
本速查手册汇编了 WJEC A-Level 体育课程所需的核心公式、定理与原理。从直线动力学、角运动到生理指标和心理表现模型，掌握这些方程是笔试与应用分析成功的关键。请将本手册作为快速复习工具，确保你能准确回忆并自信运用每一个关系式。
1. Linear Kinematics: SUVAT Equations | 直线运动学：SUVAT 方程
Uniform acceleration governs many sporting actions, from a sprinter leaving the blocks to a cyclist braking. The four SUVAT equations link initial velocity (u), final velocity (v), acceleration (a), time (t) and displacement (s).
匀加速度支配着许多运动动作，从短跑运动员蹬离起跑器到自行车手制动。四个 SUVAT 方程将初速度 (u)、末速度 (v)、加速度 (a)、时间 (t) 和位移 (s) 联系起来。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
These equations only apply when acceleration remains constant. In a 100 m sprint, a velocity–time profile can be split into phases where acceleration is approximately uniform, allowing us to calculate time splits or optimal stride kinematics.
这些方程仅在加速度恒定时成立。在 100 米短跑中，可将速度‑时间曲线划分为加速度近似恒定的阶段，从而计算分段用时或优化步幅运动学。
| Symbol | Meaning | Unit |
|---|---|---|
| u | Initial velocity | m·s⁻¹ |
| v | Final velocity | m·s⁻¹ |
| a | Acceleration | m·s⁻² |
| t | Time | s |
| s | Displacement | m |
2. Projectile Motion | 抛体运动
A projectile released in sport – a shot put, a football, a long jumper's body – follows a parabolic path determined by its initial speed, angle of release and the height difference between launch and landing. Horizontally, velocity remains constant; vertically, gravity produces uniform acceleration.
体育中的抛体——铅球、足球、跳远者身体——遵循由初速度、出手角度和起抛与落地高度差决定的抛物线轨迹。水平方向上速度恒定；竖直方向上重力产生匀加速度。
Horizontal displacement: x = u cosθ × t
Vertical displacement: y = u sinθ × t − ½ gt²
For a projectile launched and landing at the same height, the optimal angle for maximum range is 45°. When the release point is above the landing area, as in shot put, the optimum angle decreases slightly below 45°. Time of flight, maximum height and range can all be derived from the vertical component.
当抛体起抛与落点等高时，最大射程的最佳出手角为 45°。当出手点高于落地区时，如推铅球，最佳角度略低于 45°。飞行时间、最大高度和射程均可从竖直分量推导得出。
Time to apex: t = (u sinθ) / g
Range (level): R = (u² sin 2θ) / g
3. Newton's Laws of Motion | 牛顿运动定律
Newton's three laws underpin every movement in sport. The First Law (inertia) explains why a stationary football stays still until kicked. The Second Law quantifies the acceleration produced by a net force. The Third Law describes action–reaction pairs, essential for understanding ground reaction force in running and jumping.
牛顿三定律是体育中一切动作的基础。第一定律（惯性）解释了静止的足球在受到踢击前为何保持不动。第二定律量化了净力产生的加速度。第三定律描述了作用与反作用力对，对于理解跑、跳中的地面反作用力至关重要。
F = ma
A rugby player tackling receives an impulse from the opponent; the greater the player's mass, the smaller the resulting acceleration for the same force. Coaches use the Second Law to design strength programmes that increase the force an athlete can apply to a given mass, thereby boosting acceleration.
橄榄球中的扑搂动作会给对方一个冲量；对于同样的力，运动员质量越大，产生的加速度越小。教练利用第二定律设计力量训练计划，以增加运动员对给定质量施加的力，从而提高加速度。
4. Momentum and Impulse | 动量与冲量
Linear momentum (p) is the product of mass and velocity. When a force acts over a time interval, it changes momentum; this impulse (Ft) equals the change in momentum. Understanding impulse allows performers to manipulate force and time to control collisions.
线动量 (p) 是质量与速度的乘积。当力在一段时间内作用时，它会改变动量；此冲量 (Ft) 等于动量的变化。理解冲量有助于运动员操控力与时间来掌控碰撞。
p = mv
Impulse = Ft = Δp = m(v − u)
In cricket, a fielder catches a fast ball by moving the hands backwards upon impact. This increases the stopping time t, reducing the average force experienced. Similarly, in martial arts, a follow‑through motion extends contact time to maximise impulse delivered to a target.
在板球中，外野手接高速球时随球后收双手，延长了缓冲时间 t，从而减小了所受平均力。同样，在武术中，顺势跟进动作延长接触时间，以最大化对靶传递的冲量。
5. Work, Energy and Power | 功、能量与功率
Mechanical work is done when a force moves an object through a distance. Kinetic and gravitational potential energy describe an athlete's capacity to perform further work. Power indicates the rate at which work is done or energy transferred – a key determinant of explosive performance.
当力使物体在力的方向上移动一段距离时，便做了机械功。动能与重力势能描述了运动员进一步做功的能力。功率表示做功或能量转换的速率——是爆发力表现的关键决定因素。
Work: W = F d cosθ
Kinetic energy: KE = ½ mv²
Gravitational potential energy: GPE = mgh
Power: P = W/t = Fv
A weightlifter raising a barbell increases its GPE; the work done equates to the increase in mechanical energy if efficiency is assumed. In cycling, power output is often expressed as the product of force applied to the pedals and crank velocity. Elite sprinters can generate peak powers exceeding 2000 W during a standing start.
举重运动员举起杠铃会增加其重力势能；若假设效率理想，所做的功等于机械能的增加量。在自行车运动中，输出功率常表示为踏板力与曲柄速度的乘积。优秀短跑选手在站立式起跑中可产生超过 2000 瓦的峰值功率。
6. Angular Motion: Angular Kinematics | 角运动：角运动学
Many body segments rotate about joints, and whole bodies rotate during dives, twists and throws. Angular displacement (θ), angular velocity (ω) and angular acceleration (α) are analogous to their linear counterparts. Linear and angular quantities are connected through the radius of rotation.
许多身体环节围绕关节转动，而整个身体在跳水、旋转和投掷中也会旋转。角位移 (θ)、角速度 (ω) 和角加速度 (α) 与其线量对应类似。线量与角量通过转动半径相联系。
Average angular velocity: ω = Δθ / t
Angular acceleration: α = Δω / t
Linear velocity: v = rω
Centripetal acceleration: a = rω² = v²/r
rω), maximising throwing distance. Centripetal force, directed toward the axis, prevents the hammer from flying off tangentially.
在链球投掷中，运动员使链球以高角速度旋转。有效半径 (r) 越长，出手线速度 (v = rω) 越大，从而最大化投掷距离。指向转轴的向心力防止链球沿切线飞出。
7. Moment of Inertia and Torque | 转动惯量与扭矩
Moment of inertia (I) is the rotational equivalent of mass; it depends on both mass and its distribution relative to the axis of rotation. Torque (T) causes angular acceleration, and angular momentum (L) is conserved when no external torque acts.
转动惯量 (I) 是转动中的质量等效量；它取决于质量及其相对于转轴的分布。扭矩 (T) 引起角加速度；当无外力矩作用时，角动量 (L) 守恒。
Moment of inertia: I = Σ mr²
Torque: T = Iα
Angular momentum: L = Iω
Figure skaters exploit conservation of angular momentum: by pulling arms and free leg close to the body, they reduce I, causing ω to increase dramatically. Divers use the same principle to control spin rate during somersaults and twists. Coaches analyse torque production at joints to improve limb acceleration in kicking and striking.
花样滑冰运动员利用角动量守恒：将手臂和自由腿收紧靠近身体，减小转动惯量 I，使角速度 ω 急剧增加。跳水运动员运用相同原理控制空翻和转体速度。教练分析关节扭矩输出，以改善踢击和击打动作的肢体加速。
8. Levers and Moments | 杠杆与力矩
The human musculoskeletal system operates through levers. A moment (torque) is the product of force and the perpendicular distance from the pivot. Understanding lever classes and mechanical advantage clarifies how force and speed trade-offs occur in human movement.
人体骨骼肌肉系统通过杠杆运作。力矩是力与其到支点垂直距离的乘积。理解杠杆类别与机械利益有助于阐明人体运动中力与速度的权衡。
Moment = Force × Perpendicular distance from pivot
Mechanical Advantage (MA) = Effort arm / Resistance arm
- First-class lever: Pivot between effort and resistance (e.g., head nodding). MA can be >1 or <1.
- Second-class lever: Resistance between pivot and effort (e.g., calf raise). MA >1, favours force production.
- Third-class lever: Effort between pivot and resistance (e.g., biceps curl). MA <1, favours speed and range of motion.
Most human body levers are third class, built for speed at the expense of force. A rugby kicker uses the hip and knee as levers, with muscles producing effort moments that accelerate the foot to high velocity before contacting the ball.
多数人体杠杆为第三类，以牺牲力为代价换取速度。橄榄球踢球手利用髋和膝作为杠杆，肌肉产生施力力矩，使脚在触球前加速至极高速度。
9. Fluid Mechanics: Bernoulli and Magnus | 流体力学：伯努利与马格努斯
When an object moves through air or water, fluid pressure differences can create lift, drag and curved flight paths. Bernoulli's principle states that faster fluid flow produces lower pressure. The Magnus effect explains how spin generates a sideways force on a ball.
当物体在空气或水中运动时，流体压力差可产生升力、阻力和曲线轨迹。伯努利原理指出流速越快处压力越低。马格努斯效应解释了旋转如何使球受侧向力。
Bernoulli: Pressure + ½ ρ v² + ρgh = constant (along a streamline)
Drag force: Fd = ½ Cd ρ A v²
A spinning football experiences a pressure differential: the side moving against the airflow has higher pressure, pushing the ball toward the low‑pressure region. This produces the characteristic 'bend' of a free‑kick. Aerodynamic drag depends on fluid density (ρ), velocity squared, frontal area (A) and the drag coefficient (Cd). Cyclists adopt streamlined positions to reduce both A and Cd.
旋转的足球受到压力差：与气流方向相反旋转的一侧压力较高，将球推向低压区，从而产生任意球特有的弧线。空气阻力取决于流体密度 (ρ)、速度的平方、迎风面积 (A) 和阻力系数 (Cd)。自行车手采用流线型姿势来减小 A 和 Cd。
10. Physiological Formulas | 运动生理学公式
Key physiological measures allow sport scientists to evaluate cardiovascular fitness and pulmonary function. Cardiac output, maximal oxygen uptake and minute ventilation are computed from directly measurable variables. These metrics underpin training prescription and performance prediction.
关键的生理指标使运动科学家能够评估心血管适能与肺部功能。心输出量、最大摄氧量和每分通气量可由直接测量值计算得出。这些指标是训练处方与成绩预测的基础。
Cardiac output: Q = HR × SV
VO₂max = Q × (a−vO₂ difference)
Minute ventilation: VE = TV × f
During a maximal treadmill test, heart rate (HR) plateaus near 220 − age, while stroke volume (SV) reaches its ceiling. The arteriovenous oxygen difference widens as active muscles extract more O₂. Elite endurance athletes achieve high VO₂max values through a combination of genetic endowment and training‑induced central and peripheral adaptations.
在跑台最大测试中，心率 (HR) 趋近于 220 − 年龄，与此同时每搏输出量 (SV) 到达上限。随着活动肌肉提取更多氧，动静脉氧差增大。高水平耐力运动员通过遗传天赋与训练引起的中枢和外周适应，实现高 VO₂max 值。
11. Psychological Theories: Arousal and Performance | 心理技能理论：唤醒与表现
Several theoretical models describe the relationship between arousal (or anxiety) and sport performance. Drive theory, the inverted‑U hypothesis (Yerkes‑Dodson law) and catastrophe theory each offer predictions about optimal arousal and the effects of cognitive and somatic anxiety.
若干理论模型描述了唤醒（或焦虑）与运动表现之间的关系。驱力理论、倒 U 假设（耶克斯‑多德森定律）以及灾难模型各自对最佳唤醒及认知焦虑与躯体焦虑的影响作出预测。
Drive theory: Performance = Habit × Drive (P = H × D)
Drive theory suggests that well‑learned skills are facilitated by high arousal, whereas novices suffer because their dominant response may be incorrect. The inverted‑U model proposes a bell‑shaped relationship with an optimal arousal level that varies by skill complexity and personality. Catastrophe theory extends this by showing that under high cognitive anxiety, performance can plummet dramatically once arousal passes a threshold, with recovery requiring a significant reduction in arousal.
驱力理论认为高唤醒促进熟练技能的表现，而新手则因优势反应可能错误而受干扰。倒 U 模型提出钟形关系，其最佳唤醒水平随技能复杂度和人格而异。灾难模型扩展了此观点，指出在高认知焦虑下，一旦唤醒超过阈值，表现会急剧崩溃，且恢复需要大幅降低唤醒。
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