📚 Year 10 WJEC PE: Formula and Theorem Quick Reference Handbook | Year 10 WJEC 体育：公式定理速查手册
Welcome to your go-to quick reference for the key formulas and theorems you will encounter in Year 10 WJEC Physical Education. This handbook brings together essential calculations from exercise physiology, biomechanics and health-related fitness, helping you to analyse performance, plan training and understand the science behind sport. Keep it handy as you revise for assessments and practical coursework.
欢迎使用这份 Year 10 WJEC 体育常用公式定理速查手册。本手册汇总了运动生理学、生物力学和健康相关体能中的核心计算公式，帮助你分析运动表现、制定训练计划，并理解运动背后的科学原理。在备考和完成实践作业时，记得随时查阅。
1. Maximum Heart Rate and Heart Rate Reserve | 最大心率与心率储备
Maximum heart rate (MHR) is the highest number of beats per minute your heart can achieve during all-out exercise. The most common estimation formula is often written as:
最大心率（MHR）是你在全力运动时心脏每分钟能达到的最高跳动次数。最常用的估算公式通常写作：
MHR = 220 – age
For a 15‑year‑old, this gives an estimated MHR of 205 bpm. Although individual variation exists, this simple calculation is widely used in WJEC specifications to set training intensities.
对于一名 15 岁的学生，估算的最大心率约为 205 次/分。尽管存在个体差异，但是在 WJEC 课程规范中广泛使用这一简单计算来设定训练强度。
Heart rate reserve (HRR) is the difference between your maximum heart rate and your resting heart rate (RHR). It represents the ‘available’ heart rate range that can be used during exercise.
心率储备（HRR）是你的最大心率与静息心率（RHR）之间的差值。它代表了运动期间可供使用的“可用”心率范围。
HRR = MHR – RHR
If an athlete’s RHR is 60 bpm and MHR is 205 bpm, their heart rate reserve is 145 bpm. This figure is crucial for the Karvonen method in the next section.
如果某名运动员的静息心率为 60 次/分，最大心率为 205 次/分，那么其心率储备就是 145 次/分。这一数值对于下一节的卡氏公式至关重要。
2. Karvonen Formula and Target Heart Rate Zones | 卡氏公式和目标心率区间
The Karvonen formula uses heart rate reserve to calculate a target heart rate for a desired training intensity. It is more personalised than the simple percentage of MHR method.
卡氏公式利用心率储备来计算目标训练强度下的靶心率。它比单纯使用最大心率百分比的方法更具个性化。
Target HR = ((MHR – RHR) × % intensity) + RHR
For moderate aerobic training at 65% intensity, a 15‑year‑old with MHR 205 bpm and RHR 60 bpm would calculate: ((205 – 60) × 0.65) + 60 = 154 bpm. This target ensures the athlete works at the correct physiological load.
若进行强度为 65% 的中等有氧训练，一名 15 岁、MHR 205 次/分、RHR 60 次/分的运动员计算如下：((205 – 60) × 0.65) + 60 = 154 次/分。这一靶心率确保运动员以正确的生理负荷进行训练。
WJEC candidates should know the typical training zone percentages for different energy systems. The table below summarises the key zones using both %MHR and %HRR (Karvonen).
WJEC 考生应了解不同能量系统对应的常见训练区间百分比。下表总结了使用 %MHR 和 %HRR（卡氏）的关键区间。
| Training Zone | % MHR | % HRR (Karvonen) | Benefits |
|---|---|---|---|
| Recovery | 50–60% | 40–50% | Active recovery, cool‑down |
| Aerobic (moderate) | 60–70% | 50–60% | Improves cardiovascular base |
| Aerobic (vigorous) | 70–80% | 60–70% | Increases aerobic capacity (VO₂ max) |
| Anaerobic threshold | 80–90% | 70–85% | Lactate tolerance, high‑intensity intervals |
3. Body Mass Index (BMI) | 身体质量指数
Body Mass Index is a screening tool that estimates whether an individual has a healthy body weight for their height. The formula is straightforward and appears in health‑related fitness assessments.
身体质量指数是一种筛查工具，用于估算个体的体重相对身高是否处于健康范围。该公式简单，并出现在健康相关体能评估中。
BMI = weight (kg) ÷ height² (m²)
A student weighing 65 kg and with a height of 1.75 m would have a BMI of 65 ÷ (1.75 × 1.75) = 21.2 kg/m², falling into the ‘healthy weight’ category. WJEC expects you to interpret these values against standard classifications: underweight (below 18.5), healthy weight (18.5–24.9), overweight (25–29.9) and obese (30 and above).
一名体重 65 kg、身高 1.75 m 的学生，BMI = 65 ÷ (1.75 × 1.75) = 21.2 kg/m²，属于“健康体重”类别。WJEC 要求你根据标准分类来解读这些数值：体重不足（低于18.5）、健康体重（18.5–24.9）、超重（25–29.9）和肥胖（30及以上）。
4. Basal Metabolic Rate (BMR) | 基础代谢率
Basal metabolic rate is the amount of energy (calories) the body needs at rest to maintain essential functions such as breathing and circulation. While WJEC does not demand complex BMR calculations, the simplified Mifflin‑St Jeor equations are useful for understanding daily energy needs.
基础代谢率是指身体在静止状态下为维持呼吸、循环等基本功能所需的能量（卡路里）。虽然 WJEC 不要求进行复杂的 BMR 计算，但简化的 Mifflin‑St Jeor 方程有助于理解每日能量需求。
Male BMR ≈ (10 × weight in kg) + (6.25 × height in cm) – (5 × age) + 5
Female BMR ≈ (10 × weight in kg) + (6.25 × height in cm) – (5 × age) – 161
For a 15‑year‑old female who is 165 cm tall and weighs 55 kg, BMR ≈ (10 × 55) + (6.25 × 165) – (5 × 15) – 161 = 550 + 1031.25 – 75 – 161 = 1345.25 kcal/day. This value then multiplies by an activity factor to estimate total energy expenditure, a concept linked to energy balance.
对于一名 15 岁、身高 165 cm、体重 55 kg 的女生，BMR ≈ (10 × 55) + (6.25 × 165) – (5 × 15) – 161 = 550 + 1031.25 – 75 – 161 = 1345.25 千卡/天。该数值再乘以活动因子即可估算总能量消耗，这一概念与能量平衡相关。
5. Speed, Distance and Time | 速度、距离和时间
The relationship between speed, distance and time is fundamental when analysing athletic performance, from sprints to endurance events. The classic formula is:
在分析从短跑到耐力项目的运动表现时，速度、距离和时间的关系是基础。经典公式是：
Speed = Distance ÷ Time
If a footballer sprints 40 metres in 5.0 seconds, their average speed is 40 m ÷ 5.0 s = 8.0 m/s. You can rearrange the formula to find distance (Speed × Time) or time (Distance ÷ Speed). Always check that units are consistent; for example, in cycling, speed is often expressed in kilometres per hour (km/h).
如果一名足球运动员在 5.0 秒内冲刺 40 米，其平均速度为 40 m ÷ 5.0 s = 8.0 m/s。你可以变换公式来求距离（速度 × 时间）或时间（距离 ÷ 速度）。务必检查单位是否一致；例如在自行车运动中，速度通常用千米/小时（km/h）表示。
6. Acceleration and Velocity | 加速度与速度变化率
Acceleration measures how quickly an athlete changes their speed. In WJEC PE, it is often introduced via uniform acceleration equations to examine starts and changes of direction.
加速度衡量运动员改变速度的快慢。在 WJEC 体育中，通常通过匀加速运动方程来分析起跑和变向。
Acceleration (a) = (final velocity – initial velocity) ÷ time
a = (v – u) ÷ t
A sprinter moving from 2 m/s to 9 m/s over 3 seconds has an acceleration of (9 – 2) ÷ 3 = 2.33 m/s². Negative acceleration (deceleration) occurs when the final speed is lower than the initial speed, such as when slowing down after a sprint.
一名短跑运动员在 3 秒内从 2 m/s 提高到 9 m/s，其加速度为 (9 – 2) ÷ 3 = 2.33 m/s²。当末速度低于初速度时（例如冲刺后减速），就会出现负加速度（减速度）。
Although velocity and speed are often used interchangeably in everyday language, in biomechanics velocity has direction. Be prepared to explain how changing direction involves a change in velocity even if speed remains constant.
尽管在日常用语中速度和速率常被混用，但在生物力学中速度具有方向性。要能解释为什么即使速率不变，改变方向也涉及速度的变化。
7. Force and Newton’s Second Law | 力与牛顿第二定律
Newton’s second law of motion links the force applied to an object, its mass and the resulting acceleration. This law is central to understanding how athletes produce and resist forces.
牛顿第二运动定律将施加在物体上的力、物体质量以及所产生的加速度联系起来。这一定律对于理解运动员如何产生和对抗力至关重要。
Force (F) = mass (m) × acceleration (a)
If a rugby player of mass 80 kg accelerates at 3 m/s², the net force required is 80 kg × 3 m/s² = 240 N (newtons). Conversely, a larger force is needed to accelerate a heavier athlete at the same rate, which partly explains why training aims to increase lean body mass and explosive strength.
如果一名体重 80 kg 的橄榄球运动员以 3 m/s² 加速，所需的净力为 80 kg × 3 m/s² = 240 N（牛顿）。反之，要让更重的运动员获得同样的加速度，就需要更大的力，这在一定程度上解释了训练为何旨在增加瘦体重和爆发力。
8. Momentum and Impulse | 动量与冲量
Momentum is the product of a body’s mass and its velocity. In sport, momentum helps explain tackling, throwing and collisions.
动量是物体质量与其速度的乘积。在体育中，动量有助于解释擒抱、投掷和碰撞等现象。
Momentum (p) = mass (m) × velocity (v)
A cricket ball (0.16 kg) travelling at 40 m/s has a momentum of 6.4 kg m/s. The greater an object’s momentum, the harder it is to stop or change its direction.
一个以 40 m/s 飞行的板球（0.16 kg）具有 6.4 kg m/s 的动量。物体的动量越大，就越难使其停止或改变方向。
Impulse is the change in momentum of an object when a force is applied over a period of time. It can be expressed as:
冲量是指力在一段时间内作用于物体时物体动量的变化。可表示为：
Impulse = Force × time = change in momentum
Impulse = Δp = m(v – u)
In a long jump take‑off, the athlete pushes against the board for a short time to generate a large impulse, maximising the change in horizontal velocity. WJEC candidates should relate impulse to techniques like ‘follow‑through’ in striking and throwing movements.
在跳远起跳时，运动员短时间内对踏板施加力以产生大的冲量，从而最大化水平速度的变化。WJEC 考生应将冲量与击球和投掷动作中的“随挥”技术联系起来。
9. Lever Systems and Mechanical Advantage | 杠杆系统与机械优势
Lever systems in the human body consist of a fulcrum (joint), an effort (muscle force) and a resistance (weight of the body part or external load). All levers obey the law of moments.
人体内的杠杆系统由支点（关节）、动力（肌肉力）和阻力（身体部分重量或外部负荷）组成。所有杠杆都遵循力矩定律。
Moment = Force × perpendicular distance from fulcrum
For a balanced lever, the clockwise moment equals the anticlockwise moment. In sport, moments help explain why a small force can overcome a large resistance when the effort arm is long.
对于平衡的杠杆，顺时针力矩等于逆时针力矩。在体育中，力矩有助于解释为什么当力臂较长时，较小的力就能克服较大的阻力。
There are three classes of lever. Mechanical advantage (MA) compares the effort arm to the resistance arm:
杠杆分为三类。机械优势（MA）是力臂与阻力臂的比值：
Mechanical Advantage = Effort arm ÷ Resistance arm
First‑class levers (e.g. head nodding) have the fulcrum between effort and resistance; they can provide either speed or strength advantage depending on arm lengths. Second‑class levers (e.g. calf raise) always have the resistance between fulcrum and effort, giving an MA > 1 (good for strength). Third‑class levers (e.g. biceps curl) have the effort between fulcrum and resistance; their MA is always < 1, favouring speed and range of motion over force – the most common lever type in the body.
第一类杠杆（如点头动作）的支点在动力和阻力之间；根据力臂长度不同，可提供速度或力量优势。第二类杠杆（如提踵）的阻力总在支点和动力之间，机械优势始终大于 1（有利于增力）。第三类杠杆（如肱二头肌弯举）的动力在支点和阻力之间，其机械优势始终小于 1，以速度和活动范围换取力量，这是人体内最常见的杠杆类型。
10. Work, Power and Energy | 功、功率与能量
In biomechanics, work is done when a force moves an object. Power is the rate at which that work is performed, which is critical in explosive athletic events.
在生物力学中，当一个力使物体移动时即做功。功率是做功的速率，对爆发性运动项目至关重要。
Work (W) = Force (F) × distance moved in the direction of the force (d)
Power (P) = Work ÷ time
An alternative expression relates power to force and speed, which is particularly useful in sprinting and cycling analysis:
另一个表达式将功率与力和速度关联起来，在短跑和自行车运动分析中尤为有用：
Power (P) = Force (F) × velocity (v)
If a cyclist pedals with a driving force of 120 N while moving at 10 m/s, the power output is 120 N × 10 m/s = 1200 W. Because energy is the capacity to do work, these formulas also underpin calculations of mechanical efficiency and energy expenditure during sport.
如果自行车运动员以 120 N 的驱动力，在 10 m/s 的速度下骑行，其输出功率为 120 N × 10 m/s = 1200 W。由于能量是做功的能力，这些公式也是计算运动过程中机械效率和能量消耗的基础。
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