IGCSE CCEA PE: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 体育：公式定理速查手册

📚 IGCSE CCEA PE: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 体育：公式定理速查手册

This quick reference handbook brings together the essential formulas and theorems required for the IGCSE CCEA Physical Education specification. From calculating training zones to analysing lever systems and projectile motion, each concept is presented with clear explanations and worked examples in both English and Chinese. Use this guide to reinforce your understanding and to have every vital equation at your fingertips before the exam.

这本速查手册汇集了 IGCSE CCEA 体育课程所考查的核心公式与定理。从训练区间计算到杠杆系统和抛体运动分析，每个概念都配有清晰的中英文说明与示例。请用这本手册巩固你的理解，让你在考前能够随时调用每一个关键方程。

1. Maximum Heart Rate & Target Zones | 最大心率与目标区间

The simplest estimation of maximum heart rate (HRmax) is given by: HRmax = 220 − age. While individual fitness levels can cause variations, this formula provides a practical starting point for setting exercise intensity. For more personalised training zones, the Karvonen method uses heart rate reserve (HRR).

最大心率 (HRmax) 最简的估算公式为：HRmax = 220 − 年龄。尽管个体体能差异会导致变动，该公式仍是设定运动强度的实用起点。卡沃宁(Karvonen)方法利用心率储备(HRR)提供更个性化的训练区间。

The Karvonen formula expresses target heart rate as: Target HR = (HRmax − HRrest) × Intensity + HRrest. Intensity is written as a decimal (e.g., 0.6 for 60%). This method accounts for resting heart rate (HRrest) and is widely used to plan aerobic and anaerobic sessions.

卡沃宁公式将目标心率表示为：目标心率 = (HRmax − HRrest) × 强度 + HRrest。强度以小数表示（如 0.6 代表 60%）。该方法结合了静息心率(HRrest)，广泛用于设计有氧与无氧训练。

Common training zones using the Karvonen method are listed below:

Recovery / Easy: 50–60% HRR
Aerobic / Endurance: 60–70% HRR
Tempo / Threshold: 70–80% HRR
Interval / High intensity: 80–90% HRR
Sprint / Maximum: 90–100% HRR

使用卡沃宁方法的常见训练区间如下：

恢复 / 轻松：50–60% 心率储备
有氧 / 耐力：60–70% 心率储备
节奏 / 临界：70–80% 心率储备
间歇 / 高强度：80–90% 心率储备
冲刺 / 最大：90–100% 心率储备

2. Body Mass Index (BMI) | 身体质量指数

Body Mass Index is a screening tool that relates weight to height: BMI = mass (kg) / height² (m²). It provides a rough indication of underweight, healthy weight, overweight and obesity in adults, although it does not distinguish between muscle and fat mass.

身体质量指数是一项将体重与身高联系起来的筛查工具：BMI = 体重 (kg) / 身高² (m²)。它可以大致指示成年人的体重过轻、健康、超重与肥胖，但无法区分肌肉量与脂肪量。

The World Health Organization classification is as follows:

Underweight: BMI < 18.5
Normal weight: BMI 18.5–24.9
Overweight: BMI 25.0–29.9
Obese Class I: BMI 30.0–34.9
Obese Class II: BMI 35.0–39.9
Obese Class III: BMI ≥ 40.0

世界卫生组织的分类如下：

体重过轻：BMI < 18.5
正常体重：BMI 18.5–24.9
超重：BMI 25.0–29.9
肥胖 I 级：BMI 30.0–34.9
肥胖 II 级：BMI 35.0–39.9
肥胖 III 级：BMI ≥ 40.0

3. Energy & Calorie Calculations | 能量与卡路里计算

Energy content of macronutrients: Carbohydrate = 4 kcal/g, Protein = 4 kcal/g, Fat = 9 kcal/g. Alcohol provides 7 kcal/g but is not considered a nutrient for performance. Total energy intake can be estimated by multiplying grams of each macronutrient by its energy value.

宏量营养素的能量含量：碳水化合物 = 4 千卡/克，蛋白质 = 4 千卡/克，脂肪 = 9 千卡/克。酒精提供 7 千卡/克，但不视为有助于运动表现的营养素。总能量摄入可将各营养素的克数乘以其能量值进行估算。

The basic energy balance equation is: Energy Balance = Energy Intake − Total Energy Expenditure. A positive balance leads to weight gain; a negative balance leads to weight loss. Athletes often use this principle to manipulate body composition.

基本的能量平衡方程为：能量平衡 = 能量摄入 − 总能量消耗。正平衡导致体重增加；负平衡导致体重减少。运动员常用这一原理调整身体成分。

4. Lever Systems & Moments | 杠杆系统与力矩

The moment (torque) of a force about a joint is given by: Moment = Force × Perpendicular distance from pivot. In the human body, joints act as fulcra, muscles provide effort, and body segments or external loads provide resistance.

力对于关节的力矩（扭矩）公式为：力矩 = 力 × 支点的垂直距离。在人体中，关节充当支点，肌肉提供力，肢体或外部负荷提供阻力。

The three classes of levers found in sport and movement are:

First class: Fulcrum between effort and resistance. Example: neck extension (atlanto-occipital joint).
Second class: Resistance between fulcrum and effort. Example: calf raise (ankle joint).
Third class: Effort between fulcrum and resistance. Example: biceps curl (elbow joint).

运动与动作中的三类杠杆为：

第一类：支点在力与阻力之间。例如：颈部伸展（寰枕关节）。
第二类：阻力在支点与力之间。例如：提踵（踝关节）。
第三类：力在支点与阻力之间。例如：肱二头肌弯举（肘关节）。

5. Speed, Velocity & Acceleration | 速度、速率与加速度

Speed is a scalar quantity: Speed = Distance / Time. Velocity is a vector: Velocity = Displacement / Time. A sprinter running 100 m in 10 s has a speed of 10 m/s; if running straight, the velocity magnitude is also 10 m/s in that direction.

速率是标量：速率 = 路程 / 时间。速度是矢量：速度 = 位移 / 时间。一名短跑运动员用 10 秒跑完 100 米，其速率为 10 m/s；若沿直线跑动，速度大小也为 10 m/s，方向向前。

Acceleration is the rate of change of velocity: a = (v − u) / t, where v is final velocity, u is initial velocity, and t is time. A negative acceleration indicates deceleration. The area under a velocity–time graph gives the displacement travelled.

加速度是速度的变化率：a = (v − u) / t，其中 v 为末速度，u 为初速度，t 为时间。负加速度表示减速。速度–时间图下的面积代表所经过的位移。

6. Force, Mass & Newton’s Laws | 力、质量与牛顿定律

Newton’s Second Law lies at the heart of biomechanics: F = m × a. The net force acting on an object equals its mass multiplied by its acceleration. For weight, the force due to gravity is: W = m × g, where g ≈ 10 m/s² on Earth.

牛顿第二定律是生物力学的核心：F = m × a。作用在物体上的合力等于其质量与加速度的乘积。对于重力，重量公式为：W = m × g，地球表面 g ≈ 10 m/s²。

Newton’s Third Law states: for every action force there is an equal and opposite reaction force. When a sprinter pushes backward against the blocks, the blocks exert an equal forward force on the sprinter. All three laws must be applied together when analysing sporting movements.

牛顿第三定律指出：每一个作用力都有一个大小相等、方向相反的反作用力。当短跑运动员向后蹬起跑器时，起跑器对运动员施加了一个等大的向前力。分析运动动作时，三条定律必须综合运用。

7. Momentum & Impulse | 动量与冲量

Momentum is the product of mass and velocity: p = m × v. Momentum is conserved in a closed system, meaning the total momentum before a collision equals the total momentum after, provided no external forces act. This principle helps analyse tackle impacts in rugby or racket–ball contacts.

动量是质量与速度的乘积：p = m × v。在不受外力的封闭系统中动量守恒，即碰撞前的总动量等于碰撞后的总动量。该原理可帮助分析橄榄球擒抱冲击或球拍–球接触。

Impulse is defined as: Impulse = F × t = Δp (change in momentum). To achieve a large change in momentum, either a large force must be applied or the force must act over an extended time. Cricketers ‘give’ with the ball by moving the hands backward to increase contact time and reduce peak force.

冲量定义为：冲量 = F × t = Δp（动量的变化）。要实现大幅动量变化，要么施加大力，要么延长力的作用时间。板球手接球时手向后移动以增加接触时间、减小峰值力。

8. Resolution of Forces & Parallelogram Law | 力的分解与平行四边形法则

When two forces act at a point, the resultant force can be found using the parallelogram law: draw the two forces as adjacent sides of a parallelogram; the diagonal from the point of application represents the magnitude and direction of the resultant. This is essential in determining net muscle force across a joint.

当两个力作用于一点时，可用平行四边形法则求合力：将两力作为平行四边形的邻边绘出；从作用点引出的对角线即表示合力的大小与方向。这对于确定关节处的净肌肉力至关重要。

For calculations, it is often easier to resolve a single force into perpendicular components: F_x = F cos θ and F_y = F sin θ. On an inclined plane, the component of weight parallel to the slope is mg sin θ, and the component perpendicular is mg cos θ.

计算时，常将单个力分解为互相垂直的分量：F_x = F cos θ、F_y = F sin θ。在斜面上，平行于斜面的重力分量为 mg sin θ，垂直于斜面的分量为 mg cos θ。

9. Projectile Motion & Optimal Release | 抛体运动与最佳出手角度

The horizontal range of a projectile released and landing at the same height is given by: Range = (u² × sin 2θ) / g, where u is release speed, θ is the angle of release, and g is gravitational acceleration. For a given release speed, the maximum range is achieved when θ = 45° (sin 90° = 1).

在出手点与落点高度相同时，抛体的水平射程公式为：射程 = (u² × sin 2θ) / g，其中 u 为出手速度，θ 为出手角度，g 为重力加速度。对于给定出手速度，最大射程在 θ = 45° 时取得（sin 90° = 1）。

In many sports, the release height differs from the landing height. A shot put is released from about 2 m above the ground, so the optimal angle is lower than 45°, typically around 35–40°. The parabolic flight path is also influenced by air resistance and the spin of the object.

在许多运动中，出手高度与落点高度不同。铅球从大约 2 米高处推出，所以最佳角度低于 45°，通常约为 35–40°。飞行轨迹还会受空气阻力和物体旋转的影响。

10. Training Principles: FITT & SPOR | 训练原则：FITT 与 SPOR

The FITT principle provides a framework for designing training programmes: Frequency (how often), Intensity (how hard), Time (how long), and Type (what kind of exercise). Adjusting one or more of these variables allows progressive overload and avoids plateaus.

FITT 原则为设计训练计划提供了一个框架：频率（多久一次）、强度（多大负荷）、时间（持续多久）以及类型（何种运动）。调整其中一个或多个变量可实现循序渐进超负荷，避免平台期。

The broader SPOR or SPORT principles capture key training theorems: Specificity (training match demands), Progression (gradually increasing load), Overload (working beyond the comfort zone), and Reversibility (use it or lose it). Many models also add Tedium (variety) to maintain motivation. Collectively, these theorems guide safe and effective athletic development.

更广泛的 SPOR / SPORT 原则涵盖了关键训练定理：专项性（训练匹配专项需求）、渐进性（逐步增加负荷）、超负荷（在舒适

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IGCSE CCEA PE: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 体育：公式定理速查手册