GCSE WJEC PE: Quick Reference Handbook of Formulas and Theorems | GCSE WJEC 体育：公式定理速查手册

📚 GCSE WJEC PE: Quick Reference Handbook of Formulas and Theorems | GCSE WJEC 体育：公式定理速查手册

This handbook brings together every essential formula, law and theorem you need for the GCSE WJEC Physical Education exam. From health‑related fitness equations to the biomechanical principles underpinning movement, the content is organised into clear sections with English–Chinese paired explanations to support quick revision and deep understanding.

本手册汇总了 GCSE WJEC 体育考试所需的全部关键公式、定律和定理。从健康体能方程到支撑人体运动的生物力学原理，内容按清晰的小节编排，并提供英汉对照解释，助力快速复习与深入理解。

1. Health and Fitness Formulas | 健康与体能公式

BMI = weight (kg) ÷ height² (m²)

Body Mass Index (BMI) is a simple screening tool used to classify adults into weight categories: underweight, healthy weight, overweight or obese.

身体质量指数 (BMI) 是一种简易筛查工具，用于将成人划分体重类别：偏瘦、健康体重、超重或肥胖。

MHR = 220 − age (years)

Maximum Heart Rate (MHR) estimates the highest number of beats your heart can achieve in one minute during all‑out exercise. This formula is widely used despite slight individual variations.

最大心率 (MHR) 估算在全力运动中每分钟心跳的最高次数。尽管存在个体差异，该公式被广泛使用。

Heart Rate Reserve (HRR) = MHR − Resting Heart Rate (RHR)

Heart Rate Reserve reflects the range between resting and maximal heart rate, forming the foundation for the Karvonen method of prescribing training intensity.

心率储备反映静息心率与最大心率之间的范围，是制订训练强度的卡沃宁法的基础。

Target HR = (HRR × %intensity) + RHR

This Karvonen formula calculates a precise heart rate zone for aerobic training, for example 60–80% of HRR for moderate to vigorous effort.

该卡沃宁公式可计算有氧训练的精确心率区间，例如以 HRR 的 60%–80% 进行中等至剧烈运动。

Energy (kcal) = MET × body mass (kg) × time (hours)

One MET equals 3.5 ml O&sub2;/kg/min, representing resting metabolic rate. Multiplying MET value of an activity by body mass and duration estimates energy expenditure.

1 MET 等于 3.5 毫升氧气/千克/分钟，代表静息代谢率。用活动的 MET 值乘以体重和持续时间可估算能量消耗。

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2. Work, Power and Energy | 做功、功率与能量

Work (W) = Force (F) × displacement (d) (Joules, J)

Work is done when a force moves an object in the direction of the force. If displacement is zero or perpendicular to the force, no mechanical work is performed.

当力使物体沿力的方向移动时做功。若位移为零或与力垂直，则不做机械功。

Power (P) = Work (W) ÷ time (t) (Watts, W)

Power measures the rate of doing work or transferring energy. Athletes with high power output can perform the same amount of work in less time.

功率衡量做功或能量转换的速率。高功率输出的运动员能在更短时间内完成相等的功。

Gravitational Potential Energy: E_p = m × g × h

m is mass (kg), g is gravitational field strength (9.8 m/s² on Earth), and h is height gained (m). This energy is stored when an athlete raises their centre of mass.

m 为质量 (kg)，g 为重力场强度 (9.8 m/s²)，h 为提升的高度 (m)。运动员升高重心时便储存了这种能量。

Kinetic Energy: E_k = ½ × m × v²

Where v is velocity. Rapid sprinters and fast‑moving objects possess high kinetic energy.

其中 v 为速度。高速冲刺的短跑运动员和快速移动的物体具有很大的动能。

Efficiency = (useful work output ÷ total energy input) × 100%

Human movement is never 100% efficient; energy is lost as heat. Efficiency helps compare techniques and equipment.

人体运动不可能 100% 高效；部分能量以热的形式散失。效率有助于比较技术和器械。

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3. Newton’s Laws of Motion | 牛顿运动定律

First Law (Inertia): An object stays at rest or moves with constant velocity unless acted on by a net external force.

A football on the pitch will not move until kicked; a skater glides at nearly constant velocity on smooth ice because net force is close to zero.

足球在草地上静止直到被踢；溜冰者在光滑冰面上几乎匀速滑行，因为合外力接近于零。

Second Law: F = m × a

The acceleration of an object is directly proportional to the net force applied and inversely proportional to its mass. Stronger forces produce larger accelerations; heavier athletes require more force to change velocity.

物体的加速度与所受合外力成正比，与质量成反比。较大的力产生较大的加速度；体重较大的运动员需要更大的力来改变速度。

Third Law: For every action there is an equal and opposite reaction.

When a swimmer pushes water backwards, the water pushes the swimmer forwards. Sprinters drive down and back against the blocks – the ground reaction force propels them forward.

游泳者向后推水，水便向前推游泳者。短跑运动员向后下方蹬起跑器——地面反作用力推动其向前。

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4. Forces and Free Body Diagrams | 力与受力图

Weight: W = m × g

Weight acts vertically downwards from the centre of mass. It must not be confused with mass, which is measured in kilograms.

重力竖直向下作用于重心。务必与以千克计量的质量区分开来。

Normal reaction force acts perpendicular to the supporting surface. Friction opposes relative motion. Air resistance (drag) acts opposite to the direction of motion and increases with speed squared and frontal area. Lift is an upward force generated by pressure differences, governed by Bernoulli’s principle.

法向反作用力垂直于支撑面。摩擦力阻碍相对运动。空气阻力（拖曳力）与运动方向相反，且随速度平方和迎风面积增大。升力是由压力差产生的向上力，遵循伯努利原理。

In a free body diagram, forces are drawn as arrows whose lengths represent magnitude. When all forces balance, the object is in equilibrium; when unbalanced, acceleration occurs according to Newton’s second law.

在受力图中，力用箭头表示，长度代表大小。所有力平衡时物体处于平衡态；不平衡时根据牛顿第二定律产生加速度。

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5. Linear Motion and Kinematics | 直线运动与运动学

Speed = distance ÷ time (m/s)

Speed is a scalar quantity; velocity is a vector and includes direction. Acceleration = change in velocity ÷ time taken (m/s²).

速率是标量；速度是矢量，包含方向。加速度 = 速度变化量 ÷ 时间 (m/s²)。

SUVAT equations (valid when acceleration is constant):

v = u + at
s = ut + ½ at²
v² = u² + 2as
s = ½ (u + v) t

Symbols: u = initial velocity, v = final velocity, a = constant acceleration, t = time, s = displacement. These allow calculation of take‑off velocity, braking distance or dive height.

符号：u = 初速度，v = 末速度，a = 恒定加速度，t = 时间，s = 位移。利用这些方程可计算起跳速度、制动距离或跳水高度。

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6. Projectile Motion and Fluid Mechanics | 抛体运动与流体力学

Projectiles follow a curved (parabolic) path. The horizontal and vertical components are independent: v_x = u cos θ, v_y = u sin θ. Gravity only affects the vertical component, while horizontal motion is constant if air resistance is negligible.

抛体沿抛物线轨迹运动。水平和竖直分量彼此独立：v_x = u cos θ，v_y = u sin θ。重力仅影响竖直分量，若空气阻力可忽略则水平运动恒定。

Factors affecting range include release velocity, release angle (optimal 45° in a vacuum) and height of release. In sport, the actual best angle is often lower because of air resistance and body configuration.

影响射程的因素包括出手速度、出手角度（真空中最佳为 45°）和出手高度。在运动中因空气阻力和身体姿态，实际最佳角度往往更小。

Bernoulli’s Principle: faster fluid flow → lower pressure.

An aerofoil or spinning ball creates a pressure differential, generating lift or the bending trajectory of a spinning football (Magnus effect).

翼型或旋转球体产生压力差，形成升力或旋转足球的弧线轨迹（马格努斯效应）。

The magnitude of air resistance depends on speed, cross‑sectional area, surface roughness and fluid density. Cyclists adopt a streamlined position to minimise drag and conserve energy.

空气阻力的大小取决于速度、迎风面积、表面粗糙度和流体密度。自行车运动员采用流线型姿势以减少阻力、节省能量。

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7. Levers and Torque | 杠杆与力矩

Moment (torque) = Force × perpendicular distance from pivot (d⊥)

The turning effect of a force is called a moment. In the body, muscles provide the effort, joints act as fulcrums, and bones act as levers.

力产生的转动效应称为力矩。在人体中，肌肉提供动力，关节充当支点，骨骼充当杠杆。

Principle of Moments: clockwise moments = anticlockwise moments at equilibrium.

Lever systems are classified into three classes: first class (fulcrum between effort and load, e.g. neck extension), second class (load between fulcrum and effort, e.g. calf raise), and third class (effort between fulcrum and load, e.g. biceps curl). Third‑class levers dominate in the human body and favour speed and range of motion over force.

杠杆系统分为三类：第一类（支点在力与负荷之间，如颈后伸），第二类（负荷在支点与力之间，如提踵），第三类（力在支点与负荷之间，如肱二头肌弯举）。人体以第三类杠杆为主，有利于速度和活动范围，而非力。

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8. Centre of Mass and Stability | 重心与稳定性

The centre of mass is the point where body mass is concentrated. Its position changes with body shape and limb movement.

重心是身体质量集中的点，其位置随体型和肢体运动而改变。

Stability increases when the centre of mass is low, the base of support is wide, and the line of gravity falls within the base. When the line of gravity moves outside the base, the body becomes unstable and topples unless a corrective movement is made.

当重心较低、支撑面较宽且重力线落在支撑面内时，稳定性增强。重力线一旦移出支撑面，身体便不稳定，若不进行纠正将会倾倒。

In wrestling or rugby, players lower their centre of mass and widen their stance to resist being pushed over. Gymnasts raise their arms to adjust centre of mass location during balance routines.

在摔跤或橄榄球中，运动员降低重心并加宽站姿以抵抗被推倒。体操运动员在平衡动作中通过抬臂调整重心位置。

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9. Angular Motion | 角运动

Angular velocity ω = θ ÷ t (rad/s)

Linear velocity of a point on a rotating body: v = r × ω, where r is the radius of rotation. The longer the radius (e.g. a golf club), the greater the linear speed of the distal end for the same angular velocity.

旋转体上一点的线速度：v = r × ω，r 为旋转半径。半径越长（如高尔夫球杆），相同角速度下远端线速度越大。

Centripetal force: F = m v² / r or m ω² r

The centripetal force acts toward the centre of rotation and is required for circular motion, e.g. the grip needed to swing a hammer or the lean of a cyclist on a velodrome curve.

向心力指向旋转中心，是圆周运动所必需的，例如甩链球时的握力，或室内自行车赛弯道时运动员的倾斜。

Angular momentum L = moment of inertia (I) × angular velocity (ω)

In the absence of external torque, angular momentum is conserved. A spinning ice skater pulls arms in, decreasing I, which causes ω to increase dramatically.

无外力矩时，角动量守恒。花样滑冰运动员收拢手臂，减小转动惯量 I，导致角速度 ω 急剧增大。

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10. Impulse and Momentum | 冲量与动量

Momentum p = m × v (kg m/s)

Momentum is a vector quantity. Heavier and faster objects have greater momentum, making them harder to stop.

动量是矢量。质量更大、速度更快的物体动量更大，更难以停下。

Impulse = Force × time = change in momentum (F Δt = Δ m v)

To maximise change in momentum (e.g. speeding up a shot), athletes apply force over as long a time as possible – this is the rationale for follow‑through in striking and throwing events.

为了最大化动量变化（如增加射门速度），运动员尽可能延长施力时间——这就是击打和投掷项目强调随挥动作的原理。

In collisions, total momentum is conserved in a closed system. Catching a fast ball with a ‘giving’ motion increases contact time, reduces average force and softens impact.

在碰撞中，封闭系统内总动量守恒。用“顺势后收”的动作接高速球可以增加接触时间，减小平均力，缓和冲击。

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