📚 GCSE CCEA Physical Education: Formula & Theorem Quick Reference Handbook | GCSE CCEA 体育：公式定理速查手册
This handbook brings together the key formulas, principles, and biomechanical theorems you need for the GCSE CCEA Physical Education exam. It covers training thresholds, body composition, energy expenditure, and sports mechanics — all in one place for rapid revision.
本手册汇集了 GCSE CCEA 体育考试所需的关键公式、原理与生物力学定理，涵盖训练阈值、身体成分、能量消耗以及运动力学，帮助你一站式快速复习。
1. Maximum Heart Rate & Training Zones | 最大心率与训练区
The most common estimate of maximum heart rate is the simple age-based formula: HRmax = 220 − age (years). It provides a rough value with an error of ±10–12 bpm.
最常用的最大心率估算是基于年龄的简单公式：HRmax = 220 − 年龄（岁）。该值误差约为 ±10–12 次/分。
HRmax = 220 − age
A more accurate formula often used in sports science is: HRmax = 206.9 − (0.67 × age).
运动科学中常用的更精确公式为：HRmax = 206.9 − (0.67 × 年龄)。
HRmax = 206.9 − (0.67 × age)
Training zones are then calculated as percentages of HRmax:
训练区按 HRmax 的百分比划分：
| Training Zone | % HRmax | Typical Use |
|---|---|---|
| Moderate / Aerobic | 60–70% | Fat burning, base endurance |
| Vigorous / Anaerobic threshold | 80–90% | Lactate tolerance, high intensity |
中文对照：
| 训练区 | 最大心率百分比 | 典型用途 |
|---|---|---|
| 中等强度 / 有氧区 | 60–70% | 燃脂、基础耐力 |
| 高强度 / 无氧阈区 | 80–90% | 乳酸耐受、高强度训练 |
2. Karvonen Formula & Target Heart Rate | 卡沃宁公式与目标心率
The Karvonen formula uses heart rate reserve (HRR) to calculate a more personalised target heart rate (THR). HRR = HRmax − resting heart rate (RHR).
卡沃宁公式使用心率储备（HRR）计算更个性化的目标心率（THR）。HRR = 最大心率 − 静息心率（RHR）。
THR = RHR + (HRR × intensity%)
For a 16-year-old with RHR = 60 bpm, training at 70% intensity: HRmax = 220 − 16 = 204 bpm; HRR = 204 − 60 = 144; THR = 60 + (144 × 0.7) = 160.8 ≈ 161 bpm.
例如，一名 16 岁、RHR = 60 次/分的学生以 70% 强度训练：HRmax = 220 − 16 = 204 次/分；HRR = 204 − 60 = 144；THR = 60 + (144 × 0.7) = 160.8 ≈ 161 次/分。
This formula is especially useful when prescribing individualised training loads.
该公式在制定个性化训练负荷时尤其有用。
3. Borg Rating of Perceived Exertion (RPE) | 博格自觉用力评分
The Borg RPE scale ranges from 6 to 20, where 6 means no exertion and 20 means maximal exertion. It can be linked to heart rate: RPE × 10 ≈ HR (bpm).
博格 RPE 量表范围为 6–20，6 表示毫不费力，20 表示最大用力。它可与心率关联：RPE × 10 ≈ HR（次/分）。
HR ≈ RPE × 10
A perceived effort of 12 suggests a heart rate near 120 bpm. The scale is valuable when heart rate monitors are unavailable.
自觉用力为 12 表明心率约为 120 次/分。在没有心率监测器时，该量表非常有价值。
4. Body Mass Index (BMI) | 身体质量指数
BMI is a quick screening tool for body composition. It is calculated as mass (kg) divided by the square of height (m).
BMI 是体成分的快速筛查工具，计算方法为体重（kg）除以身高（m）的平方。
BMI = weight (kg) ÷ height² (m²)
Classification ranges (adults): underweight < 18.5, normal 18.5–24.9, overweight 25–29.9, obese ≥ 30. For children, age- and sex-specific centile charts are used.
分类范围（成人）：体重过轻 < 18.5，正常 18.5–24.9，超重 25–29.9，肥胖 ≥ 30。儿童使用年龄与性别特定的百分位图表。
5. Basal Metabolic Rate (BMR) & Energy Expenditure | 基础代谢率与能量消耗
BMR can be estimated via the Mifflin-St Jeor equations, but for GCSE CCEA the focus is often on MET-based energy cost.
BMR 可通过 Mifflin-St Jeor 公式估算，但 GCSE CCEA 更侧重基于 MET 的能量消耗计算。
1 MET = 3.5 mL O₂ · kg⁻¹ · min⁻¹
Energy (kcal) = MET × body mass (kg) × duration (h)
能量（千卡）= MET × 体重（kg） × 时间（h）
kcal = MET × kg × hours
For example, a 70 kg athlete cycling at 8 METs for 1.5 hours expends about 8 × 70 × 1.5 = 840 kcal.
例如，70 kg 运动员以 8 METs 骑行 1.5 小时消耗约 8 × 70 × 1.5 = 840 千卡。
6. Speed, Velocity & Acceleration | 速率、速度与加速度
Speed is a scalar quantity; velocity is a vector. The basic equation is distance over time. Acceleration measures the change in velocity.
速率为标量，速度为矢量。基本公式为距离除以时间。加速度衡量速度的变化率。
v = d / t
a = (v − u) / t
Where v = final velocity, u = initial velocity, t = time taken. Units: speed in m/s, acceleration in m/s².
其中 v = 末速度，u = 初速度，t = 所用时间。单位：速度 m/s，加速度 m/s²。
In a 100 m sprint completed in 11 seconds, average speed = 100 ÷ 11 ≈ 9.09 m/s.
在 100 米短跑用时 11 秒时，平均速度 = 100 ÷ 11 ≈ 9.09 m/s。
7. Force, Momentum & Impulse | 力、动量与冲量
Newton’s second law relates force, mass, and acceleration. Momentum is the product of mass and velocity. Impulse equals the change in momentum.
牛顿第二定律将力、质量和加速度联系起来。动量是质量与速度的乘积。冲量等于动量的变化。
F = m a
p = m v
J = F × t = Δp = m v − m u
When a footballer kicks a stationary ball (mass 0.45 kg) and gives it a velocity of 20 m/s, the impulse imparted is 0.45 × 20 = 9 N·s.
足球运动员踢静止的球（质量 0.45 kg）使其速度达到 20 m/s 时，施加的冲量为 0.45 × 20 = 9 N·s。
8. Levers & Moments | 杠杆与力矩
Levers are classified by the relative positions of effort, fulcrum, and load. The principle of moments states that for equilibrium, clockwise moments = anticlockwise moments.
杠杆根据力、支点和负载的位置分类。力矩原理指出，平衡时顺时针力矩等于逆时针力矩。
Moment (N·m) = Force (N) × perpendicular distance from fulcrum (m)
Mechanical Advantage (MA) = Load ÷ Effort
In a first-class lever (e.g., a seesaw), the fulcrum lies between effort and load. In a second-class lever (e.g., the ankle during plantar flexion), the load is between fulcrum and effort. In a third-class lever (e.g., the biceps curl), effort is between fulcrum and load; this is the most common lever in the human body and favours speed and range of motion over force.
第一类杠杆（如跷跷板）支点在力和负载之间。第二类杠杆（如跖屈时的踝关节）负载在支点与力之间。第三类杠杆（如肱二头肌弯举）力在支点与负载之间；这是人体最常见的杠杆，偏重速度与活动范围而非力。
9. Newton’s Laws of Motion | 牛顿运动定律
Newton’s three laws explain the motion of athletes and objects.
牛顿三定律解释了运动员和物体的运动。
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a net external force.
第一定律（惯性）： 物体在不受净外力作用时保持静止或匀速直线运动。
Second Law: Acceleration is directly proportional to net force and inversely proportional to mass (F = m a).
第二定律： 加速度与净外力成正比，与质量成反比（F = m a）。
Third Law: For every action there is an equal and opposite reaction.
第三定律： 每一个作用力都有一个大小相等、方向相反的反作用力。
When a sprinter pushes back against the blocks, the blocks push the sprinter forward with equal force.
短跑运动员向后蹬起跑器时，起跑器以相等的力向前推动运动员。
10. Stability & Centre of Mass | 稳定性与重心
Stability is the ability to resist being moved or toppled. It depends on the base of support, centre of mass height, and line of gravity.
稳定性指抵抗移动或倾倒的能力，取决于支撑面、重心高度和重力线。
Greater stability = larger base + lower centre of mass + line of gravity within base
To increase stability, athletes widen their stance and lower their body. In contact sports, a low centre of mass makes a player harder to push over.
为增加稳定性，运动员加宽站姿并降低身体。在对抗性运动中，低重心使运动员更难被推倒。
The centre of mass can shift outside the body during movements like high jump (Fosbury Flop), where it passes under the bar while the body clears it.
重心在运动中可移出体外，如背越式跳高时身体过杆而重心从杆下通过。
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