📚 Year 11 SQA Physical Education: Quick Reference Formula & Theorem Handbook | SQA 体育：公式定理速查手册
This handbook gathers the key formulas, equations, and principles that underpin the Year 11 SQA Physical Education course. From biomechanics to exercise physiology, mastering these numerical and conceptual tools will sharpen your ability to analyse performance, design training programmes, and evaluate factors impacting physical activity.
本手册汇集了支撑 Year 11 SQA 体育课程的关键公式、方程式和原理。从生物力学到运动生理学，掌握这些数字和概念工具将提升您分析运动表现、设计训练计划以及评估影响身体活动因素的能力。
1. Body Composition & Body Mass Index | 身体成分与体重指数
Body Mass Index (BMI) provides a simple estimation of whether an individual has a healthy body weight relative to height. It is widely used in health screening and fitness assessment.
体重指数（BMI）提供了一个简单的估算，判断个人体重相对于身高是否健康。它广泛用于健康筛查和体能评估。
BMI = Mass (kg) ÷ (Height in metres)²
Example: A person weighing 70 kg with a height of 1.75 m has a BMI = 70 ÷ (1.75 × 1.75) ≈ 22.9 kg/m². A value between 18.5 and 24.9 is generally considered healthy.
示例：一个人体重 70 kg，身高 1.75 m，其 BMI = 70 ÷ (1.75 × 1.75) ≈ 22.9 kg/m²。数值介于 18.5 到 24.9 之间通常被认为是健康的。
BMI categories help identify underweight, healthy weight, overweight, and obese ranges, but it does not differentiate between muscle and fat mass, so athletes with high muscle mass may be misclassified.
BMI 分类有助于识别体重过轻、健康体重、超重和肥胖范围，但它不区分肌肉和脂肪质量，因此肌肉量高的运动员可能被误分类。
2. Maximum Heart Rate & Training Zones | 最大心率与训练区域
Heart rate monitoring is fundamental when designing cardiovascular training. The most common estimate of maximum heart rate (HRmax) uses the 220-age formula.
设计心血管训练时，心率监测是基础。最大心率（HRmax）最常用的估算采用 220 – 年龄公式。
Estimated HRmax = 220 − Age (years)
For a 16‑year‑old, estimated HRmax = 220 − 16 = 204 bpm. This estimation has a standard deviation of about ±10 bpm, so individual variation is common.
对于 16 岁青少年，估算 HRmax = 220 − 16 = 204 bpm。此估算的标准差约为 ±10 bpm，因而个体差异很常见。
Training zones are then calculated as percentages of HRmax. Aerobic training typically targets 60–80% HRmax, whereas anaerobic training may use 80–90% HRmax.
随后，训练区域按 HRmax 的百分比计算。有氧训练通常以 60–80% HRmax 为目标，而无氧训练可使用 80–90% HRmax。
3. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
A more personalised method of setting training intensity uses the Karvonen formula, which incorporates resting heart rate (HRrest) and heart rate reserve (HRR).
一种更个性化的设定训练强度的方法是卡沃宁公式，它纳入了静息心率（HRrest）和心率储备（HRR）。
Target Heart Rate = (HRmax − HRrest) × %Intensity + HRrest
Step-by-step: measure HRrest immediately upon waking for several mornings; calculate HRR = HRmax − HRrest; multiply HRR by desired intensity (e.g. 0.70 for 70%); then add HRrest.
步骤：连续几个早晨醒来后立即测量 HRrest；计算 HRR = HRmax − HRrest；将 HRR 乘以目标强度（如 70% 时为 0.70）；最后加上 HRrest。
Example: 16‑year‑old student with HRrest 65 bpm, HRmax 204 bpm; HRR = 139 bpm. 70% target = (139 × 0.70) + 65 = 97.3 + 65 ≈ 162 bpm. This method reflects fitness status because fitter individuals have lower HRrest and thus a wider reserve.
示例：16 岁学生，HRrest 65 bpm，HRmax 204 bpm；HRR = 139 bpm。70% 目标心率 = (139 × 0.70) + 65 = 97.3 + 65 ≈ 162 bpm。此方法能反映体能状态，因为更健康的人静息心率更低，储备范围更宽。
4. Speed, Velocity & Acceleration | 速率、速度与加速度
In biomechanical analysis of movement, distinguishing between scalar and vector quantities is essential. Speed is the rate of change of distance, while velocity is rate of change of displacement (including direction).
在运动的生物力学分析中，区分标量和矢量至关重要。速率是距离的变化率，而速度是位移（含方向）的变化率。
Speed = Distance ÷ Time | Velocity = Displacement ÷ Time
Example: A 100 m sprinter running a straight line covers 100 m in 10 s. Average speed = 100 m / 10 s = 10 m/s; average velocity is also 10 m/s forward because displacement equals distance. In a 400 m lap race, distance = 400 m, but displacement may be 0 m if start and finish coincide – average velocity = 0 m/s.
示例：一名 100 m 直道短跑运动员用 10 s 跑完 100 m。平均速率 = 100 m / 10 s = 10 m/s；平均速度也是 10 m/s 向前，因为位移等于距离。在 400 m 绕圈赛中，距离 = 400 m，但如果起点终点重合，位移可能为 0 m——平均速度 = 0 m/s。
Acceleration is change in velocity over time. It is central to explosive actions such as sprint starts and jumping.
加速度是速度随时间的变化。它对起跑冲刺和跳跃等爆发性动作至关重要。
Acceleration = (Final Velocity − Initial Velocity) ÷ Time
A positive acceleration means speeding up in the direction of motion; negative (deceleration) means slowing down.
正加速度意味着沿运动方向加速；负加速度（减速度）意味着减速。
5. Force, Mass & Newton’s Second Law | 力、质量与牛顿第二定律
Newton’s Second Law explains how a force causes an object to accelerate. It underpins movement analysis in sports from kicking a ball to pushing a sled.
牛顿第二定律解释了力如何使物体加速。它是从踢球到推雪橇的运动分析的基础。
Force (N) = Mass (kg) × Acceleration (m/s²)
If a footballer imparts an average force of 200 N to a 0.43 kg football over a short time, the resulting acceleration can be calculated: a = F/m = 200/0.43 ≈ 465 m/s² – dramatically higher than the acceleration due to gravity.
如果一个足球运动员在短时间内对 0.43 kg 的足球施加 200 N 的平均力，则产生的加速度可通过 a = F/m = 200/0.43 ≈ 465 m/s² 计算——远高于重力加速度。
In biomechanics, impulse (Force × Time) relates to momentum change, and the same principle is used to explain follow-through techniques.
在生物力学中，冲量（力 × 时间）与动量变化相关，同样的原理用于解释随挥技术。
6. Work, Energy & Power | 功、能量与功率
Understanding work and power helps quantify the intensity of physical tasks. Work is done when a force moves an object in the direction of the force.
理解功和功率有助于量化身体活动的强度。当一个力使物体沿力的方向移动时，就做了功。
Work (J) = Force (N) × Distance (m)
Example: Lifting a 20 kg barbell vertically (weight = 20 × 9.8 = 196 N) over 0.8 m requires work = 196 × 0.8 ≈ 157 J. Power is the rate at which work is performed.
示例：将 20 kg 的杠铃垂直举起（重力 = 20 × 9.8 = 196 N） 0.8 m，所需功 = 196 × 0.8 ≈ 157 J。功率是做功的速率。
Power (W) = Work ÷ Time or Power = Force × Velocity
A high‑power output is critical in explosive sports like sprinting and weightlifting, where athletes need to produce large force quickly.
高功率输出在短跑和举重等爆发力体育项目中至关重要，运动员需要快速产生大力。
7. Lever Systems & Mechanical Advantage | 杠杆系统与机械利益
Musculoskeletal levers are classified into first, second, and third‑class systems based on the relative positions of effort, load, and fulcrum. Most body levers are third‑class, favouring range and speed of movement over force.
肌肉骨骼杠杆依据施力、负荷和支点的相对位置分为一类、二类和三类杠杆。人体大多数杠杆是三类杠杆，以运动范围和速度换取力。
Mechanical advantage (MA) compares the effort arm (distance from fulcrum to effort) to the resistance arm (distance from fulcrum to load).
机械利益（MA）比较施力臂（支点到施力点的距离）与阻力臂（支点到负荷的距离）。
Mechanical Advantage = Effort Arm ÷ Resistance Arm
A MA > 1 means the lever can move larger loads with less effort (e.g. a second‑class lever like standing calf raise). A MA < 1, typical of third‑class levers (e.g. biceps curl), requires more force but gives greater speed and range.
MA > 1 表示杠杆可以用较小的力移动较大的负荷（例如，站立提踵这样的二类杠杆）。MA < 1 是三类杠杆（如肱二头肌弯举）的特征，需要更大的力，但能提供更大的速度和范围。
8. Angular Motion & Angular Velocity | 角运动与角速度
Rotary movements such as throwing, swinging, and spinning are described by angular kinematics. Angular velocity measures how quickly an object rotates.
投掷、挥拍和旋转等转动动作由角运动学描述。角速度衡量物体旋转的快慢。
Angular Velocity (ω) = Angular Displacement ÷ Time
Common units include radians per second (rad/s). In a discus throw, an athlete spins through several radians to generate high angular velocity before release.
常用单位是弧度每秒（rad/s）。在掷铁饼中，运动员旋转数弧度，以在出手前产生高角速度。
Moment of inertia (I) and angular momentum (L = I × ω) explain why tucked rotations in diving or gymnastics speed up – reducing I increases ω if angular momentum is conserved.
惯性矩（I）和角动量（L = I × ω）解释了为什么跳水或体操中的团身旋转会加速——如果角动量守恒，减小 I 会增加 ω。
9. Principles of Training (FITT) | 训练原则 (FITT)
The FITT principle provides a framework for designing and adjusting training sessions. It stands for Frequency, Intensity, Time, and Type.
FITT 原则为设计和调整训练课提供了框架，代表频率、强度、时间和类型。
- Frequency – how often training occurs (e.g. 3–5 times per week for cardiovascular fitness).
频率 – 训练多久一次（例如，心血管体能每周 3–5 次）。 - Intensity – how hard training is (e.g. %HRmax, RPE scale).
强度 – 训练有多难（例如，最大心率百分比、RPE 量表）。 - Time – duration of each session (e.g. 20–60 minutes of aerobic work).
时间 – 每次课的时长（例如，20–60 分钟有氧训练）。 - Type – mode of exercise (e.g. running, cycling, resistance training).
类型 – 运动方式（例如，跑步、骑行、抗阻训练）。
Applying the overload principle through FITT variables ensures progressive improvement while monitoring for overtraining and injury.
通过 FITT 变量应用超负荷原则可以确保持续进步，同时监控过度训练和损伤。
10. Energy Balance & Caloric Expenditure | 能量平衡与热量消耗
Energy balance is determined by comparing caloric intake from food and drink with energy expenditure through resting metabolism, physical activity, and the thermic effect of food.
能量平衡通过比较饮食摄入的热量与静息代谢、身体活动和食物热效应消耗的能量来确定。
Energy Balance = Energy Intake − Energy Expenditure
Estimated daily energy expenditure includes Basal Metabolic Rate (BMR), often estimated by the Mifflin‑St Jeor equation, multiplied by a physical activity level (PAL).
每日能量消耗估算包括基础代谢率（BMR），通常用 Mifflin‑St Jeor 方程估算，再乘以身体活动水平系数（PAL）。
A simple walking estimate: Energy cost ≈ 0.1 kcal per kg per minute of moderate walking. A 60 kg person walking briskly for 30 minutes burns approximately 0.1 × 60 × 30 = 180 kcal.
简单步行估算：能量消耗 ≈ 每分钟中等速度步行每公斤体重 0.1 千卡。一个 60 kg 的人快步走 30 分钟大约消耗 0.1 × 60 × 30 = 180 千卡。
11. Oxygen Uptake & VO₂max | 摄氧量与最大摄氧量
VO₂max represents the maximal volume of oxygen the body can use per minute per kilogram of body mass during intense exercise. It is the gold standard for cardiorespiratory fitness.
VO₂max 代表在剧烈运动中，每分钟每公斤体重身体所能使用的最大氧气体积。它是心肺适能的金标准。
VO₂max = Maximum Cardiac Output × Maximal a‑vO₂ difference
Although direct measurement requires laboratory equipment, field tests such as the Cooper 12‑minute run and the Multi‑Stage Fitness Test (bleep test) provide estimates. A higher VO₂max is associated with better endurance performance.
虽然直接测量需要实验室设备，但如 Cooper 12 分钟跑和多阶段体能测试（折返跑）等场地测试可提供估算。较高的 VO₂max 与更好的耐力表现相关。
Factors affecting VO₂max include genetics, age, sex, and training status; endurance training can improve it by 15–20% in previously untrained individuals.
影响 VO₂max 的因素包括遗传、年龄、性别和训练状态；耐力训练可使先前未受过训练的个体提高 15–20%。
12. Hydration & Sweat Rate | 水合作用与出汗率
Estimating fluid loss during exercise helps athletes personalise hydration strategies and maintain performance.
估算运动中的体液流失有助于运动员制定个性化的补水策略并保持运动表现。
Sweat Rate (L/h) = (Pre‑exercise body mass (kg) − Post‑exercise mass (kg)) ÷ Hours of exercise
Remember to account for fluid consumed during the session: add the drink volume (in litres) to the mass loss. 1 kg mass loss ≈ 1 L fluid loss.
记得要计算运动期间摄入的液体量：将饮水量（升）加到体重丢失中。1 kg 体重下降 ≈ 1 L 体液流失。
Example: A runner starts at 71.0 kg and finishes at 69.5 kg after a 1.5‑hour session, consuming 0.5 L water. Mass loss = 71.0 − 69.5 = 1.5 kg. Add water intake: total sweat ≈ 2.0 L over 1.5 h, giving sweat rate ≈ 1.33 L/h. This guides how much to drink per hour to avoid dehydration.
示例：一名跑步者训练前重 71.0 kg，1.5 小时后训练结束重 69.5 kg，期间饮水 0.5 L。体重丢失 = 71.0 − 69.5 = 1.5 kg。加上饮水量：总出汗量 ≈ 2.0 L，出汗率 ≈ 1.33 L/h。这可以指导每小时补水量，避免脱水。
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