📚 Year 9 CAIE PE: Formula & Theorem Quick Reference Handbook | 九年级 CAIE 体育：公式定理速查手册
This quick reference handbook compiles the essential formulas and theorems that underpin the Year 9 CAIE Physical Education syllabus. Understanding these principles — from how we measure body composition to the mechanics of movement — enables you to analyse performance, design training programmes and explain the science behind sport. Keep this guide handy for revision, homework and practical application.
本速查手册汇编了九年级 CAIE 体育课程所依托的关键公式与定理。从衡量身体成分到运动力学原理，理解这些知识能帮助你分析运动表现、设计训练计划并解释运动背后的科学。复习、做作业和实践应用时，请随时查阅本手册。
1. Body Mass Index (BMI) Formula | 身体质量指数公式
The Body Mass Index (BMI) is a simple screening tool used to categorise individuals as underweight, healthy weight, overweight or obese based on their mass and height. The formula is BMI = body weight (kg) ÷ (height (m))². For example, a person weighing 55 kg with a height of 1.60 m would have a BMI of 55 ÷ (1.60 × 1.60) ≈ 21.5, which falls within the healthy weight range (18.5 – 24.9). It is important to note that BMI does not directly measure body fat and can overestimate adiposity in muscular athletes.
身体质量指数（BMI）是一种简单的筛查工具，根据体重和身高将个体划分为体重过轻、健康体重、超重或肥胖。其公式为 BMI = 体重（千克） ÷ （身高（米））²。例如，一位体重 55 千克、身高 1.60 米的人，BMI 计算为 55 ÷ (1.60 × 1.60) ≈ 21.5，属于健康体重范围（18.5 – 24.9）。需要注意的是，BMI 不能直接测量体脂，且可能高估肌肉发达运动员的肥胖程度。
2. Maximum Heart Rate (MHR) Estimation | 最大心率估算公式
A widely used method to estimate maximum heart rate (MHR) is: MHR = 220 − age (years). For a 14‑year‑old student, the estimated MHR would be 220 − 14 = 206 beats per minute (bpm). This value serves as a reference point to set exercise intensities and training zones. However, individual variations exist, and the formula provides an approximation rather than a precise measurement.
一种广泛使用的最大心率（MHR）估算方法为：MHR = 220 − 年龄（岁）。对于一名 14 岁的学生，估算的最大心率是 220 − 14 = 206 次/分（bpm）。该数值可作为设定运动强度和训练区间的参考基准。但个体差异存在，此公式提供的是一个近似值而非精确测量。
3. Target Heart Rate Zone (Karvonen Formula) | 目标心率区间（卡沃宁公式）
To tailor cardiovascular training, the Karvonen formula calculates target heart rate (THR) using resting heart rate (RHR) and desired intensity: THR = ((MHR − RHR) × intensity%) + RHR. Suppose a 14‑year‑old has an RHR of 70 bpm and wants to train at 70% intensity. MHR is estimated at 206 bpm, so THR = ((206 − 70) × 0.70) + 70 = (136 × 0.70) + 70 = 95.2 + 70 ≈ 165 bpm. Training within this zone optimises aerobic benefits.
为定制心血管训练，卡沃宁公式利用静息心率（RHR）和期望强度计算目标心率（THR）：THR = ((MHR − RHR) × 强度%) + RHR。假设一名 14 岁学生静息心率为 70 bpm，希望以 70% 强度训练。MHR 估算为 206 bpm，那么 THR = ((206 − 70) × 0.70) + 70 = (136 × 0.70) + 70 = 95.2 + 70 ≈ 165 bpm。在此区间内训练可优化有氧收益。
4. Work Done Formula | 做功公式
In biomechanics, work is done when a force moves an object. The formula is Work (J) = Force (N) × distance moved in the direction of the force (m). When force and displacement are in the same direction, the calculation is straightforward. For instance, lifting a 10 kg weight (force ≈ 100 N) through a vertical distance of 0.5 m requires work of 100 × 0.5 = 50 J. This concept explains energy expenditure during resistance training.
在生物力学中，当力移动物体时即做功。公式为：功（焦耳） = 力（牛顿） × 沿力方向移动的距离（米）。当力与位移方向相同时，计算很直接。例如，将 10 千克的重量（力 ≈ 100 牛）垂直抬起 0.5 米，做功为 100 × 0.5 = 50 焦耳。该概念解释了抗阻训练中的能量消耗。
5. Power Output Formula | 功率公式
Power measures the rate of doing work. The formula is Power (W) = Work (J) ÷ time (s). An athlete who performs 500 J of work in 2 seconds produces a power output of 500 ÷ 2 = 250 W, whereas the same work performed in 5 seconds yields only 100 W. Higher power output is crucial in explosive activities such as sprinting and jumping, reflecting both strength and speed.
功率衡量做功的快慢。公式为：功率（瓦） = 功（焦耳） ÷ 时间（秒）。一名运动员在 2 秒内做 500 焦耳的功，其功率输出为 500 ÷ 2 = 250 瓦；而同样的功若在 5 秒内完成，则仅为 100 瓦。更高的功率输出对短跑、跳跃等爆发性活动至关重要，反映了力量与速度的结合。
6. Speed and Velocity | 速度与速率
Average speed is a scalar quantity calculated as Speed (m/s) = total distance (m) ÷ total time (s). Velocity, a vector, includes direction. In sports, sprinters’ average speed over 100 m can be found by dividing distance by time. A runner completing 100 m in 12.5 s has an average speed of 8 m/s. Velocity would additionally specify the direction, e.g. 8 m/s northwards.
平均速率是标量，计算公式为：速率（米/秒） = 总距离（米） ÷ 总时间（秒）。速度是矢量，包含方向。在体育运动中，可通过距离除以时间求出短跑运动员 100 米的平均速率。一名选手 12.5 秒跑完 100 米，平均速率为 8 米/秒。速度则会额外指明方向，如向北 8 米/秒。
7. Acceleration | 加速度
Acceleration describes how quickly velocity changes. The formula is a = (v − u) / t, where v = final velocity (m/s), u = initial velocity (m/s) and t = time (s). A cyclist accelerating from 2 m/s to 8 m/s in 3 seconds experiences an acceleration of (8 − 2) / 3 = 2 m/s². Positive acceleration increases speed; negative acceleration (deceleration) reduces it, often analysed in change‑of‑direction movements.
加速度描述速度变化的快慢。公式为 a = (v − u) / t，其中 v = 末速度（米/秒），u = 初速度（米/秒），t = 时间（秒）。一名自行车手在 3 秒内从 2 米/秒加速到 8 米/秒，其加速度为 (8 − 2) / 3 = 2 米/秒²。正加速度提高速度；负加速度（减速度）降低速度，变向动作中常加以分析。
8. Force (Newton’s Second Law) | 力（牛顿第二定律）
Newton’s Second Law states that Force (N) = mass (kg) × acceleration (m/s²). In sporting contexts, the body’s mass and the acceleration produced by muscles determine the force applied. Kicking a football (mass 0.45 kg) causing an acceleration of 40 m/s² requires a force of 0.45 × 40 = 18 N. Understanding this relationship helps athletes optimise technique to produce greater forces safely.
牛顿第二定律指出：力（牛顿） = 质量（千克） × 加速度（米/秒²）。在运动情境中，身体的质量以及肌肉产生的加速度决定了施加的力。踢一个质量为 0.45 千克的足球并产生 40 米/秒² 的加速度，需要 0.45 × 40 = 18 牛的力。理解这一关系有助于运动员优化技术，安全地产生更大的力。
9. Momentum | 动量
Momentum is the product of mass and velocity: p = m × v, measured in kg·m/s. A rugby player of mass 80 kg running at 5 m/s has a momentum of 400 kg·m/s. In collisions, the principle of conservation of momentum explains why protective equipment (e.g. scrum caps, shin pads) extends impact time, reducing force and risk of injury. Tackling technique also aims to manage momentum transfer.
动量是质量与速度的乘积：p = m × v，单位为千克·米/秒。一名质量 80 千克的橄榄球运动员以 5 米/秒的速度奔跑，其动量为 400 千克·米/秒。碰撞中，动量守恒原理解释了为何防护装备（如橄榄球头盔、护腿板）能延长撞击时间，从而减小力并降低受伤风险。擒抱技术同样旨在控制动量传递。
10. Moment of a Force (Torque) | 力矩（转矩）
The moment (or torque) about a pivot is calculated as Moment (Nm) = Force (N) × perpendicular distance from the pivot (m). On a seesaw, a smaller person can balance a larger person by sitting farther from the pivot, because the product of force and distance equalises. In the body, joints act as pivots; muscles generate moments to move limbs. For example, the biceps produce a moment about the elbow to lift a weight.
绕支点的力矩（或转矩）计算公式为：力矩（牛·米） = 力（牛） × 到支点的垂直距离（米）。在跷跷板上，体轻的人可通过坐得离支点更远来平衡体重较大的人，因为力与距离的乘积相等。在人体中，关节充当支点；肌肉产生力矩以移动肢体。例如，肱二头肌绕肘关节产生力矩来举起重物。
11. Levers and Mechanical Advantage | 杠杆与机械效益
A lever system consists of a fulcrum, effort and load. Mechanical Advantage (MA) = Load / Effort. When a lever is balanced, the principle of moments applies: Effort × effort arm = Load × load arm. In second‑class levers (e.g. the ankle during plantar flexion), the effort arm is longer than the load arm, giving MA > 1, which allows a smaller effort to lift a larger load — ideal for powerful movements like jumping.
杠杆系统由支点、动力和阻力构成。机械效益（MA） = 阻力 / 动力。杠杆平衡时适用力矩原理：动力 × 动力臂 = 阻力 × 阻力臂。在第二类杠杆（如跖屈时的踝关节）中，动力臂长于阻力臂，MA > 1，这使较小的动力即可举起较大的阻力——非常适合跳跃等需要爆发力的动作。
12. Projectile Motion – Range Equation | 抛体运动——射程公式
When an object is projected from ground level and lands at the same height, its horizontal range (ignoring air resistance) is given by Range = (v² × sin(2θ)) / g, where v is the initial velocity (m/s), θ is the launch angle, and g is gravitational acceleration (9.8 m/s²). The maximum range is achieved at θ = 45°, where sin(2θ) = sin 90° = 1. Thus, shot‑putters and javelin throwers aim for angles close to this to optimise distance, though the ideal angle may vary due to release height and aerodynamics.
当物体从地面高度抛出并落于同一高度时，其水平射程（忽略空气阻力）由公式 Range = (v² × sin(2θ)) / g 给出，其中 v 为初速度（米/秒），θ 为抛射角，g 为重力加速度（9.8 米/秒²）。最大射程在 θ = 45° 时取得，此时 sin(2θ) = sin 90° = 1。因此，铅球和标枪运动员力求接近该角度以获得最佳距离，尽管实际最佳角度可能因出手高度和空气动力学而略有不同。
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