📚 Year 12 CCEA PE: Formulas and Principles Quick Reference Handbook | 体育公式定理速查手册
This handbook compiles the essential equations, laws, and theoretical principles that underpin the Year 12 CCEA Physical Education specification. It is designed as a rapid revision tool covering biomechanics, exercise physiology, and sport psychology, enabling you to locate and apply the correct formula or principle during study and exam practice.
本手册汇编了 CCEA 12 年级体育课程中核心的公式、定律和理论原理。它作为快速复习工具，覆盖生物力学、运动生理学和运动心理学，帮助你在学习和考试练习中迅速查找并运用正确的公式与原理。
1. Kinematic Equations | 运动学基本公式
v = u + at
This linear equation relates final velocity (v) to initial velocity (u), acceleration (a), and time (t). It is fundamental for analysing an athlete’s change in speed during a sprint start or a constant acceleration phase.
该线性方程将末速度 (v) 与初速度 (u)、加速度 (a) 和时间 (t) 联系起来，是分析运动员冲刺起跑或匀加速阶段速度变化的基础。
s = ut + ½at²
This equation calculates displacement (s) when an object undergoes uniform acceleration. It is used to determine the distance covered by a long jumper during the take-off flight phase or the run-up distance of a javelin thrower before release.
此公式用于计算物体在匀加速条件下的位移 (s)，可用来确定跳远运动员起跳飞行阶段的距离或标枪运动员出手前的助跑距离。
v² = u² + 2as
This form eliminates time and directly links velocity, acceleration, and displacement. It is particularly helpful in analysing the braking distance of a cyclist or the stopping distance of a sprinter when decelerating.
该形式消去了时间，直接将速度、加速度和位移联系起来，特别适用于分析自行车手的制动距离或短跑运动员减速时的停止距离。
Average velocity = (u + v) / 2
Only valid for uniform acceleration, this simple average helps estimate overall performance in events with consistent speed changes, such as a swimmer maintaining a steady pace over a length.
该公式仅适用于匀加速情况，这种简单平均值有助于评估速度变化恒定时的整体表现，例如游泳运动员在单程中保持稳定配速。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a net external force. In sport, a football on the penalty spot stays still until kicked, and a sliding hockey puck eventually stops due to friction.
牛顿第一定律（惯性）：物体保持静止或匀速直线运动，除非有净外力作用。在运动中，点球点的足球在踢击前保持静止，而滑行的冰球因摩擦力最终停止。
Newton’s Second Law (Acceleration): The acceleration of an object is directly proportional to the net force and inversely proportional to its mass: F = ma. A shot putter applies a large force to accelerate a heavy shot; the same force applied to a lighter ball would produce greater acceleration.
牛顿第二定律（加速度）：物体的加速度与净外力成正比，与质量成反比: F = ma。铅球运动员施加巨大的力使沉重的铅球加速；同样的力施加在较轻的球上会产生更大的加速度。
Newton’s Third Law (Action-Reaction): For every action, there is an equal and opposite reaction. When a swimmer pushes against the wall, the wall pushes back with equal force, propelling the swimmer forward.
牛顿第三定律（作用与反作用）：每一个作用力都有一个大小相等、方向相反的反作用力。游泳运动员蹬壁时，池壁以相等的力反推，使运动员向前推进。
3. Torque and Levers | 力矩与杠杆
Torque (τ) = Force (F) × Perpendicular distance from axis (d)
Torque is the turning effect produced by a force. In the human body, muscles create torque around joints. For example, the biceps brachii generates torque at the elbow to lift a dumbbell. The formula is applied in calculating the moment arm in a biceps curl or the rotational force in a golf swing.
力矩是力产生的转动效应。在人体中，肌肉围绕关节产生力矩。例如，肱二头肌在肘部产生力矩举起哑铃。该公式可用于计算肱二头肌弯举中的力臂或高尔夫挥杆中的旋转力。
Lever systems are classified by the relative positions of the fulcrum (F), load (L), and effort (E): first class (FLE, e.g., neck extension), second class (ELF, e.g., calf raise), and third class (FEL, e.g., biceps curl). Mechanical advantage = effort arm / load arm.
杠杆系统按支点 (F)、负荷 (L) 和力 (E) 的相对位置分类：第一类 (FLE，如头颈伸展)、第二类 (ELF，如提踵) 和第三类 (FEL，如肱二头肌弯举)。机械效益 = 力臂 / 负载臂。
4. Impulse and Momentum | 冲量与动量
Impulse (J) = Force (F) × Time (Δt)
Impulse measures the effect of a force applied over a period of time. A cricketer ‘giving’ with the hands when catching a fast ball increases the time of contact, reducing the average force and making the catch less painful.
冲量测量力在一段时间内施加的效果。板球手接快速球时向后“收手”会延长接触时间，降低平均冲击力，使接球更不易受伤。
Change in momentum = m × (v – u) = Impulse
The impulse-momentum relationship states that the change in an object’s momentum equals the impulse applied. This explains why follow-through in a tennis serve or a hockey push-pass increases the final velocity of the ball by extending force application time.
冲量-动量关系指出，物体动量的变化等于所施加的冲量。这解释了为什么网球发球或曲棍球推传中的随挥动作能通过延长施力时间来增加球的末速度。
5. Work, Energy, and Power | 功、能、功率
Work done (W) = Force (F) × Displacement (s) × cosθ
In biomechanics, work is calculated when a force causes displacement. When the force is parallel to movement (θ = 0), work is simply F×s. A weightlifter does positive work when lifting the barbell and negative work when lowering it under control.
在生物力学中，力引起位移时计算功。当力与运动方向平行时 (θ = 0)，功为 F×s。举重运动员举起杠铃时做正功，控制杠铃下降时做负功。
Kinetic energy (KE) = ½mv²
The energy of motion; doubling an athlete’s speed quadruples their kinetic energy. This principle is critical in sports such as rugby tackling, where larger mass and velocity increase impact energy.
动能即运动能量；运动员速度加倍，动能增至四倍。此原理在橄榄球擒抱等运动中至关重要，质量越大、速度越快，冲击能量越大。
Gravitational potential energy (GPE) = mgh
Where h is the height above a reference point. A pole vaulter converts kinetic energy into GPE to clear the bar.
其中 h 为参考点上方高度。撑竿跳高运动员将动能转化为重力势能以越过横杆。
Power (P) = Work done / time taken = Force × velocity
Power is the rate of doing work. Explosive athletes, like sprinters and jumpers, require high power outputs to accelerate rapidly. The formula P = Fv highlights that maintaining force output at high speed leads to greater power.
功率是做功的速率。爆发力型运动员（如短跑和跳跃运动员）需要高功率输出来快速加速。公式 P = Fv 表明，在高速下保持力输出会带来更大功率。
6. Cardiovascular Formulas | 心血管公式
Maximum heart rate (HRmax) = 220 – age (years)
This widely used estimation predicts the highest heart rate an individual can achieve. A 17-year-old footballer has a predicted HRmax of 203 beats per minute (bpm). This value forms the basis for calculating training zones.
这一广泛使用的估算方法预测个体能达到的最高心率。一名17岁足球运动员的预测最大心率为203次/分。该数值是计算训练区间的基础。
Cardiac output (Q) = Stroke volume (SV) × Heart rate (HR)
Q is the volume of blood pumped by the heart per minute (L/min). Endurance training increases both SV and maximal Q, enhancing oxygen delivery to working muscles. Typical resting Q is around 5 L/min, rising to 20–35 L/min in elite endurance athletes.
心输出量 Q 是心脏每分钟泵出的血量 (升/分)。耐力训练可提高每搏输出量和最大心输出量，增强对工作肌肉的氧气供应。静息心输出量通常约 5 升/分，精英耐力运动员可升至 20–35 升/分。
7. Respiratory Formulas | 呼吸系统公式
Minute ventilation (VE) = Tidal volume (TV) × Breathing frequency (f)
VE is the volume of air moved into and out of the lungs per minute. During intense exercise, both TV and f increase dramatically. A resting TV of 0.5 L at 12 breaths/min gives a VE of 6 L/min, which can rise beyond 150 L/min in elite athletes.
每分钟通气量 VE 是每分钟进出肺部的空气量。剧烈运动时，潮气量 TV 和呼吸频率 f 都会急剧增加。静息时潮气量 0.5 升、呼吸 12 次/分对应 VE 为 6 升/分，精英运动员可超过 150 升/分。
VO₂max = Q × (a-vO₂ diff)
The Fick principle explains that maximal oxygen uptake is the product of cardiac output and the arteriovenous oxygen difference. The a-vO₂ diff reflects how much oxygen muscles extract from the blood; training widens this difference, improving aerobic capacity.
菲克原理说明最大摄氧量是心输出量与动静脉氧差的乘积。a-vO₂ 差反映肌肉从血液中提取氧的能力；训练可增大此差值，提高有氧能力。
8. Body Composition | 身体成分
Body Mass Index (BMI) = Weight (kg) / Height² (m)
BMI is a simple index for classifying underweight, normal weight, overweight, and obesity. While useful at a population level, it does not distinguish between muscle and fat mass, so an athlete with high muscle mass may be misclassified as overweight.
BMI 是划分过轻、正常、超重和肥胖的简便指标。虽然在群体层面有用，但它不区分肌肉与脂肪，因此肌肉量高的运动员可能被误判为超重。
Body fat percentage can be estimated from skinfold measurements using equations such as Durnin and Womersley’s four-site formula, but the core concept remains: % body fat = (fat mass / total body mass) × 100. A healthy range varies by age and sex.
体脂率可通过皮褶厚度测量并使用 Durnin-Womersley 四点法等方程估算，但核心概念仍是：体脂率 = (脂肪质量/总体重) × 100。健康范围因年龄和性别而异。
9. Training Intensity Formulas | 训练强度公式
Target Heart Rate (THR) using Karvonen formula: THR = RHR + (HRmax – RHR) × intensity %
The Karvonen method uses heart rate reserve (HRR = HRmax – RHR) to prescribe exercise intensity. For a 16-year-old with RHR 65 bpm aiming for 70% intensity: THR = 65 + (204 – 65) × 0.70 = 162 bpm. This method is more individualized than a simple percentage of HRmax.
Karvonen 公式使用心率储备 (HRR = 最大心率 – 安静心率) 来制定运动强度。一名 16 岁安静心率 65 次/分的运动员想达到 70% 强度：THR = 65 + (204 – 65) × 0.70 = 162 次/分。该方法比单纯使用最大心率百分比更具个性化。
Training zones: aerobic training usually targets 60–80% HRmax or 50–70% HRR; anaerobic threshold training often corresponds to 80–90% HRmax. These zones guide the specificity of training adaptations.
训练区间：有氧训练通常以 60–80% 最大心率或 50–70% 心率储备为目标；无氧阈训练常对应 80–90% 最大心率。这些区间指导训练适应的特异性。
10. Fluid Mechanics & Magnus Effect | 流体力学与马格努斯效应
Bernoulli’s Principle: P + ½ρv² + ρgh = constant
In a streamlined flow, an increase in fluid velocity results in a decrease in pressure. A discus thrower exploits this by orienting the discus to create lift; faster airflow over the upper surface reduces pressure and generates a lifting force.
在流线型流动中，流速增加导致压力降低。铁饼运动员利用这一原理调整铁饼方向产生升力；上表面更快的气流降低了压力，产生向上的升力。
The Magnus effect explains the curved flight of spinning balls. When a football is kicked with spin, the air on the side rotating in the direction of motion moves faster (low pressure), while the opposite side experiences higher pressure. This pressure imbalance creates a sideways force that bends the ball’s path, observed in free kicks and spinning tennis shots.
马格努斯效应解释了旋转球的弯曲飞行。当足球被踢出旋转时，与运动方向同向旋转的一侧气流更快（低压），另一侧压力较高。这种压力不平衡产生侧向力，使球路弯曲，常见于任意球和旋转网球。
11. Psychological Principles | 心理学原理
The Yerkes-Dodson law (inverted-U theory) describes the relationship between arousal and performance: performance improves with increased arousal up to an optimal point, after which further arousal leads to a decline in performance. The optimal level varies by skill complexity and individual personality – a fine, complex skill like putting requires lower arousal than a gross, powerful activity like a rugby tackle.
耶克斯-多德森定律（倒U理论）描述了唤起与表现的关系：表现随唤起增加而提升，直至最优水平，之后继续增加的唤起导致表现下降。最优水平因技能复杂度和个人个性而异——精细而复杂的技能（如推杆）需要比粗放发力活动（如橄榄球擒抱）更低的唤起水平。
Catastrophe theory adds that when cognitive anxiety is high, exceeding optimal arousal can cause a dramatic, discontinuous drop in performance rather than a gentle decline. Understanding these principles helps athletes use relaxation or psyching-up strategies to regulate arousal.
突变理论补充说，当认知焦虑较高时，超过最优唤起会导致表现急剧而非缓慢的、不连续的下降。理解这些原理有助于运动员通过放松或激励策略调节唤起水平。
12. Quick Reference Summary Table | 公式速查汇总表
| Formula (English) | Equation | 公式 (中文) |
|---|---|---|
| Velocity-time | v = u + at | 末速度 = 初速度 + 加速度 × 时间 |
| Displacement
Published by TutorHao | Year 12 体育 Revision Series | aleveler.com 更多咨询请联系16621398022（同微信） CommentsMore posts |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply