📚 SQA Higher PE Formula and Theorem Quick Reference Handbook | SQA 高等体育公式定理速查手册
In SQA Higher Physical Education, a solid command of key formulas and theoretical principles is essential for analysing performance, designing training programmes, and understanding the science behind movement. This quick reference handbook compiles the most commonly used equations and theorems across physiology, biomechanics, and training methodology. Use it to reinforce your understanding and to answer data-based questions with confidence.
在 SQA 高等体育课程中，牢牢掌握关键公式和理论原理对于分析运动表现、设计训练计划以及理解运动背后的科学至关重要。这份速查手册汇集了生理学、生物力学和训练方法学中最常用的公式与定理，帮助你巩固理解，自信应对数据分析类题目。
1. Heart Rate Calculations | 心率计算
Maximum heart rate (MHR) estimates the highest number of beats per minute your heart can safely achieve during maximal exercise. The most widely used field estimate subtracts your age from 220.
最大心率（MHR）估算的是在最大强度运动中心脏每分钟能安全达到的最高搏动次数。最常用的现场估算法是用 220 减去你的年龄。
MHR = 220 − age
To set individual training zones, the Karvonen formula uses heart rate reserve (HRR), which is the difference between MHR and resting heart rate (RHR). The target heart rate for a given intensity is calculated as:
为了设定个人训练区间，卡沃宁公式使用心率储备（HRR），即最大心率与安静心率（RHR）之差。特定强度下的靶心率计算公式为：
Target HR = (HRR × % Intensity) + RHR
For example, a 17‑year‑old performer with a RHR of 60 bpm training at 70% intensity would have MHR = 203 bpm, HRR = 143 bpm, and target HR = (143 × 0.70) + 60 = 160 bpm. This method ensures the exercise intensity is tailored to current fitness levels.
例如，一名 17 岁的运动员安静心率为 60 bpm，以 70% 强度训练时，MHR = 203 bpm，HRR = 143 bpm，靶心率 = (143 × 0.70) + 60 = 160 bpm。该方法确保训练强度与当前体能水平相匹配。
2. Body Composition Indicators | 身体成分指标
Body Mass Index (BMI) provides a simple screening tool for weight categories. It is calculated as mass in kilograms divided by the square of height in metres.
身体质量指数（BMI）是一种简单的体重分级筛查工具，用体重（千克）除以身高（米）的平方计算。
BMI = weight (kg) ÷ height² (m²)
Waist‑to‑hip ratio (WHR) assesses fat distribution and health risk by comparing the circumference of the waist to that of the hips.
腰臀比（WHR）通过比较腰围和臀围来评估脂肪分布与健康风险。
WHR = waist circumference ÷ hip circumference
For a more direct estimate of body fat percentage, skinfold measurements are converted to body density (BD) using equations such as the Jackson‑Pollock formula. The Siri equation then estimates fat percentage:
要更直接地估算体脂率，皮褶厚度测量值可通过 Jackson‑Pollock 等公式转换为身体密度（BD），再用 Siri 方程估算体脂率：
Body Fat % = (495 ÷ BD) − 450
These calculations require precise skinfold calipers and standardised sites, but they allow detailed tracking of body composition changes.
这些计算需要精确的皮褶卡尺和标准化测量位置，但可以详细跟踪身体成分的变化。
3. Predicting One‑Repetition Maximum (1RM) | 预测最大重复次数（1RM）
Estimating maximum strength without a maximal lift reduces injury risk. The Brzycki equation is reliable when fewer than 10 repetitions are performed to failure.
在不进行最大负荷举起的情况下估算最大力量可以降低受伤风险。当力竭次数少于 10 次时，Brzycki 公式较为可靠。
1RM = weight × 36 ÷ (37 − reps)
An alternative formula, commonly used for moderate rep ranges, is the Epley equation:
另一种常用公式 Epley 方程适用于中等次数范围：
1RM = weight × (1 + 0.0333 × reps)
For instance, if an athlete lifts 60 kg for 8 repetitions, Brzycki estimates 1RM ≈ 60 × 36 ÷ 29 = 74.5 kg. These predictions are essential for prescribing resistance‑training loads based on percentage 1RM.
例如，若运动员用 60 kg 完成 8 次，Brzycki 估算 1RM ≈ 60 × 36 ÷ 29 = 74.5 kg。这些预测对于按 1RM 百分比制定抗阻训练负荷至关重要。
4. Linear Motion Equations | 直线运动方程
Kinematic and kinetic quantities describe an object’s movement. Average speed or velocity is displacement over time, and acceleration is the rate of change of velocity.
运动学和动力学量描述物体的运动。平均速度是位移除以时间，加速度是速度的变化率。
v = s ÷ t and a = (v₂ − v₁) ÷ t
Momentum is the product of mass and velocity, while impulse equals the change in momentum and is given by the average force multiplied by the time it acts.
动量是质量与速度的乘积，而冲量等于动量的变化，也等于平均作用力乘以其作用时间。
p = m × v and F × t = Δp
Work is done when a force moves an object through a distance, and power is the rate of doing work, often expressed as force × velocity for constant‑speed movements.
力使物体移动一段距离便做了功，功率是做功的快慢，通常在恒速运动中用力乘以速度来表示。
W = F × s and P = W ÷ t = F × v
These relationships help analyse sprint acceleration, jumping mechanics, and the power output of an athlete pushing against an external load.
这些关系有助于分析短跑加速、跳跃力学以及运动员推动外部负荷时的功率输出。
5. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law states that a body remains at rest or in uniform motion unless acted upon by a net external force. Second Law quantifies the relationship among force, mass, and acceleration.
牛顿第一定律指出，物体将保持静止或匀速直线运动状态，除非受到净外力作用。第二定律量化了力、质量和加速度之间的关系。
F = m × a
The Third Law explains that every action has an equal and opposite reaction; when a sprinter pushes backward against the blocks, the blocks push the athlete forward.
第三定律说明，每一个作用力都有一个大小相等、方向相反的反作用力；短跑运动员向后蹬起跑器时，起跑器给予运动员一个向前的推力。
Together these laws underpin all linear motion analysis in sport, from the propulsion of a swimmer through water to the collision forces in a rugby tackle.
这些定律共同构成了运动中所有线性运动分析的基础，从游泳运动员在水中前进的动力到橄榄球擒抱中的碰撞力。
6. Levers & Torque | 杠杆与力矩
Levers alter the size and direction of forces. Mechanical advantage (MA) is the ratio of the effort arm to the resistance arm.
杠杆能改变力的大小和方向。机械效益（MA）是力臂与阻力臂的比值。
MA = effort arm ÷ resistance arm
Torque (moment) is the turning effect of a force around a pivot, calculated as force times perpendicular distance.
力矩是力绕支点产生的转动效应，等于力乘以垂直距离。
τ = F × d (perpendicular)
In a balanced state, the sum of clockwise moments equals the sum of anticlockwise moments. Most human movements use third‑class levers (effort between fulcrum and resistance), for example the biceps curl. The calf raise (second‑class) gives a mechanical advantage greater than 1.
在平衡状态下，顺时针力矩之和等于逆时针力矩之和。人体大多数动作使用第三类杠杆（力点在支点和阻力点之间），如肱二头肌弯举。提踵动作（第二类杠杆）则能产生大于 1 的机械效益。
7. Planes and Axes of Movement | 运动平面与轴
Movement is described relative to three anatomical planes, each paired with a perpendicular axis of rotation.
- Sagittal plane divides the body into left and right; movements occur about a front‑to‑back transverse axis (e.g., running hip flexion).
- Frontal plane divides the body into front and back; movements occur about a side‑to‑side sagittal axis (e.g., cartwheel or lateral arm raise).
- Transverse plane divides the body into top and bottom; rotations occur about a vertical axis (e.g., discus spin).
动作是在三个解剖平面中描述的，每个平面都与一个垂直的旋转轴配对。
- 矢状面将身体分为左右两部分，动作围绕前后方向的横轴进行（如跑步时的髋关节屈伸）。
- 冠状面将身体分为前后两部分，动作围绕左右方向的矢状轴进行（如侧手翻或侧平举）。
- 水平面将身体分为上下两部分，旋转动作围绕垂直轴进行（如掷铁饼的旋转）。
Identifying the plane and axis is a core skill in movement analysis and helps coaches design exercises that target specific joint actions.
识别平面与轴是动作分析的一项核心技能，有助于教练设计针对特定关节动作的练习。
8. Energy Expenditure & METs | 能量消耗与代谢当量
A metabolic equivalent (MET) is a convenient unit for estimating energy cost of physical activities. One MET represents the resting oxygen uptake of approximately 3.5 millilitres per kilogram of body mass per minute.
代谢当量（MET）是估算体力活动能量消耗的便捷单位。1 MET 相当于安静时的摄氧量，约为每千克体重每分钟 3.5 毫升。
1 MET = 3.5 ml·kg⁻¹·min⁻¹
Total energy expenditure in kilocalories can be approximated by multiplying the MET value of an activity by body mass in kilograms and duration in hours.
总能量消耗（千卡）可近似用活动的 MET 值乘以体重（千克）再乘以持续时间（小时）来估算。
kcal ≈ MET × weight (kg) × time (h)
When oxygen consumption is measured directly, every litre of O₂ used releases roughly 5 kcal of energy. This relationship underpins the analysis of aerobic endurance performance and weight management programmes.
直接测量耗氧量时，每升氧大约释放 5 千卡能量。这一关系是有氧耐力表现和体重管理计划分析的基础。
9. Fluid Dynamics in Sport | 运动中的流体力学
Air and water resistance (drag) increases with the square of velocity, a principle that profoundly affects cycling, swimming, and sprinting. Drag force is modelled as:
空气和水的阻力与速度的平方成正比，这一原理对自行车、游泳和短跑影响深远。阻力可表示为：
Fd = ½ ρ v² Cd A
Here ρ is fluid density, v is velocity, Cd the drag coefficient, and A the cross‑sectional area.
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