📚 Year 9 CAIE Physical Education: Quick-Reference Handbook of Formulas & Principles | Year 9 CAIE 体育：公式定理速查手册
In Year 9 CAIE Physical Education, understanding key formulas and principles is essential for analysing human movement, assessing fitness, and designing training programmes. This quick-reference handbook compiles the most important equations and concepts you need to master, from body composition indices to biomechanics and energy systems. Use it to revise efficiently and apply knowledge confidently in exams and practical scenarios.
在 Year 9 CAIE 体育课程中，理解关键公式和定理对于分析人体运动、评估体能以及设计训练计划至关重要。本速查手册汇总了你需要掌握的最重要方程和概念，从身体成分指标到生物力学和能量系统。用它高效复习，在考试和实际应用中自信运用知识。
1. Body Mass Index (BMI) and Waist-to-Hip Ratio (WHR) | 身体质量指数与腰臀比
The Body Mass Index (BMI) is a widely used screening tool to estimate whether an individual has a healthy body weight relative to their height. It is calculated by dividing body mass in kilograms by the square of height in metres.
身体质量指数 (BMI) 是一种广泛使用的筛查工具，用于估算一个人相对于身高的体重是否健康。它通过体重（千克）除以身高（米）的平方来计算。
BMI = mass (kg) ÷ height² (m²)
BMI = 体重 (kg) ÷ 身高² (m²)
BMI values are interpreted as follows: below 18.5 indicates underweight; 18.5–24.9 is considered normal; 25.0–29.9 suggests overweight; and 30.0 and above points to obesity. However, BMI does not distinguish between muscle and fat mass, so athletes with high muscle mass may be misclassified.
BMI 值的解读如下：低于 18.5 表示体重过轻；18.5–24.9 为正常范围；25.0–29.9 表明超重；30.0 及以上属肥胖。然而，BMI 无法区别肌肉与脂肪质量，因此肌肉发达的运动者可能被误判。
The Waist-to-Hip Ratio (WHR) is another useful measure of body fat distribution. It is determined by dividing the circumference of the waist by that of the hips. A high WHR indicates greater central obesity, which is linked to higher health risks.
腰臀比 (WHR) 是另一项衡量体脂分布的有效指标。它由腰围除以臀围得出。较高的腰臀比表明中心性肥胖更明显，这与更高的健康风险相关联。
WHR = waist circumference (cm) ÷ hip circumference (cm)
WHR = 腰围 (cm) ÷ 臀围 (cm)
For males, a WHR above 0.90 is considered high risk; for females, the threshold is 0.85. These numbers help assess the risk of cardiovascular diseases and metabolic disorders.
男性 WHR 高于 0.90 视为高风险；女性高风险阈值为 0.85。这些数值有助于评估心血管疾病和代谢紊乱的风险。
2. Maximum Heart Rate and Target Heart Rate Zones | 最大心率与目标心率区间
Maximum heart rate (HRmax) is the highest number of beats per minute your heart can achieve during maximal exercise. A simple estimation method is the formula: HRmax = 220 − age (in years). This gives an approximate value for prescription of exercise intensity.
最大心率 (HRmax) 指在极限运动中每分钟心跳的最高次数。估算最大心率的简易公式为：HRmax = 220 − 年龄（岁）。这为制定运动强度提供了近似值。
HRmax ≈ 220 − age
最大心率 ≈ 220 − 年龄
Target heart rate zones are derived from HRmax to ensure training is safe and effective. For moderate-intensity exercise, the target zone is typically 50–70% of HRmax; for vigorous intensity, 70–85% of HRmax. The formula used is: Target HR = HRmax × intensity percentage (expressed as a decimal).
目标心率区间根据最大心率制定，以确保训练安全有效。中等强度运动的靶心率通常为最大心率的 50–70%；高强度则为 70–85%。使用的公式为：目标心率 = 最大心率 × 强度百分比（以小数表示）。
Target HR = HRmax × Intensity
目标心率 = 最大心率 × 强度
For example, a 15-year-old has an estimated HRmax of 205 bpm. To exercise at 60% intensity, the target heart rate would be 205 × 0.60 = 123 bpm. Monitoring heart rate helps adjust workout effort.
例如，一个 15 岁的人估算最大心率为 205 次/分。若以 60% 强度运动，目标心率则为 205 × 0.60 = 123 次/分。监测心率有助于调整训练力度。
3. Karvonen Formula | 卡沃宁公式
The Karvonen formula calculates target heart rate more precisely by incorporating resting heart rate (HRrest) and heart rate reserve (HRR). HRR is the difference between maximum and resting heart rate, reflecting an individual’s true capacity to increase heart rate during exercise.
卡沃宁公式通过引入静息心率 (HRrest) 和心率储备 (HRR)，更精确地计算目标心率。心率储备是最大心率与静息心率的差值，反映个人在运动中提升心率的真实能力。
HRR = HRmax − HRrest
心率储备 = 最大心率 − 静息心率
The target heart rate is then found by adding a percentage of HRR to resting heart rate: Target HR = HRrest + (HRR × intensity). This method allows for individual differences in resting fitness levels.
然后将一定比例的心率储备加回静息心率，即可得出目标心率：目标心率 = 静息心率 + (心率储备 × 强度)。该方法充分考虑到个人静息体能水平的差异。
Target HR = HRrest + (HRmax − HRrest) × Intensity
目标心率 = 静息心率 + (最大心率 − 静息心率) × 强度
If a person has HRrest of 60 bpm and HRmax of 200 bpm, exercising at 70% intensity gives target HR = 60 + (200 − 60) × 0.70 = 158 bpm. The Karvonen formula is especially useful for designing personalised aerobic training programmes.
若某人静息心率 60 次/分，最大心率 200 次/分，以 70% 强度运动，目标心率 = 60 + (200 − 60) × 0.70 = 158 次/分。卡沃宁公式对于设计个性化的有氧训练计划尤为有用。
4. Speed, Distance, and Time | 速度、距离与时间
Speed is the rate at which a person moves, defined as the distance covered per unit of time. The fundamental relationship is: speed = distance ÷ time. This formula is essential for analysing performance in running, swimming, cycling, and most other sports.
速度是人移动的快慢程度，定义为单位时间内通过的距离。基本关系式为：速度 = 距离 ÷ 时间。这一公式是分析跑步、游泳、自行车比赛及多数运动表现的基础。
v = s / t (speed = distance ÷ time)
v = s / t（速度 = 距离 ÷ 时间）
In SI units, speed is measured in metres per second (m/s), distance in metres (m), and time in seconds (s). Other common units include kilometres per hour (km/h). The triangle method helps rearrange the formula: cover the unknown, and you see the relation: s = v × t, and t = s ÷ v.
国际单位制中，速度以米/秒 (m/s) 表示，距离为米 (m)，时间为秒 (s)。其他常用单位包括千米/小时 (km/h)。三角形法有助于变形公式：遮住未知量，即可得到关系式：s = v × t，t = s ÷ v。
Average speed considers the total distance over total time, even if the speed varies. In many sports, improving average speed is a key training goal. Instantaneous speed can be captured by GPS or timing gates at specific points.
平均速度考虑总距离与总时间，即使速度有变化。在众多运动中，提高平均速度是核心训练目标。瞬时速度可通过 GPS 或特定点的计时门进行测量。
5. Acceleration and Momentum | 加速度与动量
Acceleration describes the rate of change of velocity. It is calculated by dividing the change in velocity by the time taken for that change. The formula is: a = (v − u) / t, where u is initial velocity and v is final velocity.
加速度描述速度变化的快慢，由速度变化量除以变化所用时间得出。公式为：a = (v − u) / t，其中 u 为初速度，v 为末速度。
a = (v − u) ÷ t
加速度 = (末速度 − 初速度) ÷ 时间
In many sporting actions, such as sprint starts or jumping, athletes aim to maximise acceleration. A positive value means speeding up; a negative value (deceleration) means slowing down. Units are metres per second squared (m/s²).
在许多运动动作中，如短跑起跑或跳跃，运动员力求最大化加速度。正值表示加速，负值（减速度）表示减速。加速度单位为米/秒² (m/s²)。
Momentum is a measure of the quantity of motion a body possesses, given by the product of mass and velocity: p = m × v. The greater an object’s mass or velocity, the harder it is to stop. In contact sports like rugby, understanding momentum helps explain the effectiveness of tackles and collisions.
动量是物体运动量的量度，由质量与速度的乘积给出：p = m × v。物体的质量或速度越大，越难使其停下。在橄榄球等对抗运动中，理解动量有助于解释擒抱与冲撞的效果。
p = m × v (momentum = mass × velocity)
动量 = 质量 × 速度
In the absence of external forces, the total momentum of a system is conserved. This principle of conservation of momentum underlies many sporting impacts, such as two players colliding or a bat hitting a ball.
在无外力作用时，系统总动量守恒。动量守恒定律是许多运动碰撞现象的基础，如两名球员相撞，或球棒击球。
6. Force: Newton’s Second Law | 力：牛顿第二定律
Newton’s Second Law of Motion states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: F = m × a. This law explains how changes in motion occur in response to forces in sport.
牛顿第二运动定律指出，作用在物体上的力等于物体质量与其加速度的乘积：F = m × a。这一定律解释了运动中力如何引发运动状态的改变。
F = m × a (Force = mass × acceleration)
力 = 质量 × 加速度
Force is measured in newtons (N). One newton is the force required to give a mass of 1 kg an acceleration of 1 m/s². In sports, large forces are generated during explosive movements like throwing, kicking, or jumping.
力的单位为牛顿 (N)。1 牛顿定义为使 1 千克的物体产生 1 m/s² 加速度所需的力。在投掷、踢球或跳跃等爆发性动作中会产生巨大的力。
The weight of an object is the force due to gravity: W = m × g, where g is the acceleration due to gravity (≈ 9.8 m/s² on Earth). An athlete’s weight influences the force required to jump or accelerate, and it is crucial in sports where weight categories are used, such as boxing.
物体的重量是由重力引起的力：W = m × g，其中 g 为重力加速度（地球上约 9.8 m/s²）。运动员的体重影响着跳跃或加速所需的力，在拳击等划分体重级别的运动中至关重要。
W = m × g (Weight = mass × gravitational field strength)
重量 = 质量 × 重力场强度
7. Work, Power, and Energy | 功、功率与能量
Work is done when a force causes displacement in the direction of the force. It is calculated as work = force × distance moved in the direction of the force: W = F × d. Work is measured in joules (J). For example, lifting a weight through a vertical height does work against gravity.
功是力使物体沿力的方向发生位移时所做的作用。计算公式为：功 = 力 × 沿力方向上移动的距离：W = F × d。功的单位为焦耳 (J)。例如，将重物竖直上举一段高度即是在克服重力做功。
W = F × d (Work = force × distance)
功 = 力 × 距离
Power is the rate at which work is done or energy is transferred. It is given by power = work ÷ time, or P = W / t. Power is measured in watts (W), where 1 W = 1 J/s. An alternative expression useful in sports is P = F × v, linking power directly to force and velocity.
功率是做功或能量转换的速率，由功率 = 功 ÷ 时间，即 P = W / t 给出。功率单位为瓦特 (W)，1 W = 1 J/s。在体育中另一有用表达式为 P = F × v，直接将功率与力和速度联系起来。
P = W ÷ t or P = F × v
功率 = 功 ÷ 时间 或 功率 = 力 × 速度
Explosive athletes such as sprinters and throwers require high power output because they must generate large forces at high speeds. Energy is the capacity to do work; two main forms in sport are kinetic energy (Ek = ½mv²) and gravitational potential energy (Ep = mgh). Understanding energy transfers helps analyse movements like pole vaulting or long jump.
短跑和投掷等爆发性项目运动员需要高功率输出，因为他们必须在高速下产生巨大力量。能量是做功的本领；体育中两种主要形式为动能 (Ek = ½mv²) 和重力势能 (Ep = mgh)。理解能量转换有助于分析撑竿跳高或跳远等动作。
8. Levers and Moments | 杠杆与力矩
A lever is a simple machine consisting of a rigid bar that pivots around a fixed point called a fulcrum. In the human body, bones act as levers, joints serve as fulcrums, and muscles provide the effort force to overcome resistance. The turning effect produced by a force is called the moment of force.
杠杆
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