📚 Year 13 WJEC Physical Education: Formula & Theorem Quick Reference | WJEC 体育公式定理速查手册
This quick reference guide compiles the essential formulas, laws, and theorems required for Year 13 WJEC Physical Education. Use it to reinforce your understanding of biomechanics, exercise physiology, and sport psychology. Each formula is accompanied by a concise explanation to help you apply it effectively in exam questions.
这本速查手册汇总了 Year 13 WJEC 体育学科必备的公式、定律和定理。用它来巩固你对生物力学、运动生理学和运动心理学的理解。每条公式都附有简明解释，帮助你有效应对考试题目。
1. Linear Motion Equations | 直线运动方程
Linear motion equations describe the relationship between displacement, velocity, acceleration, and time for an object moving in a straight line with uniform acceleration. They are fundamental to analysing sprints, throws, and jumps.
直线运动方程描述了做匀加速直线运动的物体位移、速度、加速度和时间之间的关系，是分析短跑、投掷和跳跃的基础。
v = u + at
Where v is final velocity (m/s), u is initial velocity (m/s), a is acceleration (m/s²), and t is time (s).
其中 v 是末速度（米/秒），u 是初速度（米/秒），a 是加速度（米/秒²），t 是时间（秒）。
s = ut + ½ at²
Displacement s (m) when initial velocity, time, and acceleration are known. The term ½ at² accounts for the changing velocity.
位移 s（米）在已知初速度、时间和加速度时计算。½ at² 项表示速度变化带来的额外位移。
v² = u² + 2as
Useful when time is not given. It links final velocity, initial velocity, acceleration, and displacement directly.
当时间未知时使用。它直接将末速度、初速度、加速度和位移联系起来。
2. Newton’s Laws of Motion | 牛顿运动定律
These three laws govern how forces affect motion and are the backbone of biomechanical analysis. You must be able to apply them to sporting examples such as starting a race, hitting a ball, or swimming.
三大定律支配着力如何影响运动，是生物力学分析的支柱。你必须能将它们应用于起跑、击球或游泳等运动实例。
First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.
第一定律（惯性定律）： 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。
Second Law: The acceleration of an object is directly proportional to the net force and inversely proportional to its mass: F = ma.
第二定律： 物体的加速度与净外力成正比，与质量成反比：F = ma。
Third Law: For every action, there is an equal and opposite reaction.
第三定律： 作用力与反作用力大小相等、方向相反。
3. Force, Mass, and Acceleration | 力、质量与加速度
Expanding on the Second Law, the formula F = ma can be rearranged to calculate any variable. In sport, this explains why lighter athletes can accelerate faster and why greater force production leads to more powerful movements.
扩展第二定律，公式 F = ma 可变形计算任意变量。在体育中，这解释了为什么体重较轻的运动员能更快加速，以及更大的力量输出如何产生更有力的动作。
F = m × a
Force is measured in newtons (N), mass in kilograms (kg), acceleration in metres per second squared (m/s²).
力的单位是牛顿 (N)，质量单位是千克 (kg)，加速度单位是米/秒² (m/s²)。
Weight is a specific force: W = mg, where g is gravitational field strength (9.8 m/s² on Earth).
重力是一种特殊的力：W = mg，其中 g 是重力场强度（地球表面为 9.8 m/s²）。
4. Momentum and Impulse | 动量与冲量
Momentum describes the quantity of motion an object has. Impulse is the change in momentum caused by a force acting over a period of time. Both are vector quantities.
动量描述物体的运动量。冲量是力在一段时间内作用造成的动量变化。两者均为矢量。
p = mv
Momentum (p) is the product of mass (m) and velocity (v), measured in kg m/s. A rugby player of 90 kg running at 8 m/s has a momentum of 720 kg m/s.
动量 (p) 是质量 (m) 与速度 (v) 的乘积，单位是千克·米/秒。一名90公斤的橄榄球运动员以8米/秒速度奔跑，动量为720千克·米/秒。
Impulse = F × Δt = Δp = mv – mu
Impulse equals the area under a force–time graph. In a long jump take-off, the athlete applies a large force over a short contact time to maximise the change in velocity.
冲量等于力-时间图下的面积。在跳远起跳时，运动员在短暂接触时间内施加大力，以最大化速度变化。
5. Projectile Motion | 抛体运动
Projectile motion analysis separates the horizontal and vertical components of velocity. A body launched into the air follows a parabolic path determined by initial speed, angle of release, and height of release. Air resistance is often ignored in basic calculations.
抛体运动分析将速度分解为水平和垂直分量。抛射到空中的物体遵循抛物线轨迹，由初速度、出手角度和出手高度决定。基础计算通常忽略空气阻力。
Horizontal velocity: Vₓ = V cos θ
Vertical velocity: Vᵧ = V sin θ
Where V is the resultant velocity and θ is the angle of projection. A shot putt released at 12 m/s and 35° has Vₓ = 9.83 m/s and Vᵧ = 6.88 m/s.
其中 V 是合速度，θ 是投射角。以12米/秒、35°推出的铅球，水平分速度为9.83米/秒，垂直分速度为6.88米/秒。
Time of flight = (2 V sin θ) / g
Range = (V² sin 2θ) / g
Maximum range is achieved at a 45° angle when release and landing heights are equal. The optimal angle decreases if the release point is above the landing point, as in shot put.
当出手高度与落地高度相同时，45°可获得最大射程。若出手点高于落地点（如铅球），最佳角度会减小。
6. Angular Motion | 角运动
Rotational movements – such as a gymnast’s somersault or a diver’s twist – are described by angular displacement, angular velocity, and angular acceleration. The formulas mirror linear motion but use radians.
旋转运动——如体操翻腾或跳水转体——由角位移、角速度和角加速度描述。这些公式与直线运动类似，但使用弧度制。
ω = Δθ / Δt
Angular velocity (ω) is the rate of change of angular displacement (θ), measured in rad/s.
角速度 (ω) 是角位移 (θ) 的变化率，单位是弧度/秒。
α = Δω / Δt
Angular acceleration (α) in rad/s².
角加速度 (α) 单位是弧度/秒²。
ω₂² = ω₁² + 2αθ
Equivalent to v² = u² + 2as for rotation.
旋转运动中等同于 v² = u² + 2as 的公式。
7. Levers and Torque | 杠杆与力矩
A lever is a rigid bar rotating around a fulcrum. In the human body, bones act as levers, joints are fulcrums, and muscles provide effort. Torque (moment of force) determines rotational effect.
杠杆是绕支点转动的刚性杆。在人体中，骨为杠杆，关节为支点，肌肉提供动力。力矩决定转动效果。
Torque (T) = F × d
Where d is the perpendicular distance from the line of force to the axis of rotation (moment arm), measured in metres. Torque is expressed in newton-metres (Nm).
其中 d 是力作用线到旋转轴的垂直距离（力臂），单位为米。力矩单位为牛·米 (Nm)。
Levers are classified into three orders. In a third-class lever (most common in the body, e.g., elbow flexion with biceps), effort lies between the fulcrum and the resistance. This arrangement favours speed and range of motion over force.
杠杆分为三级。第三类杠杆（身体中最常见，如肱二头肌屈肘）动力点在支点和阻力点之间，有利于速度和运动幅度，但力量较小。
8. Fluid Mechanics: Drag and Lift | 流体力学：阻力与升力
Fluid forces affect performance in swimming, cycling, and ball flight. Drag opposes motion through a fluid; lift acts perpendicular to flow. Both depend on velocity, surface area, fluid density, and a coefficient.
流体力影响游泳、自行车和球类飞行中的表现。阻力阻碍流体中的运动；升力垂直于流动方向。二者均取决于速度、表面积、流体密度和相关系数。
Drag force (Fd) = ½ Cd ρ A v²
Cd is the drag coefficient, ρ is fluid density (air ≈ 1.2 kg/m³, water ≈ 1000 kg/m³), A is cross-sectional area, v is velocity. The squared velocity means drag increases dramatically at higher speeds.
Cd 为阻力系数，ρ 为流体密度（空气约 1.2 kg/m³，水约 1000 kg/m³），A 为迎风横截面积，v 为速度。速度的平方意味着高速时阻力急剧增大。
Lift force (FL) = ½ CL ρ A v²
Lift principles explain why spinning balls curve (Magnus effect). A backspin volleyball creates a pressure difference, causing the ball to dip.
升力原理解释了旋转球为何会弯曲（马格努斯效应）。带上旋的排球产生压力差，使球下坠。
9. Energy Systems and ATP Yield | 能量系统与ATP产量
Understanding energy system contributions and ATP resynthesis is vital for analysing exercise intensity and duration. The three systems are ATP-PC, glycolytic, and aerobic.
理解能量系统供能和ATP再合成对于分析运动强度和持续时间至关重要。三大系统为ATP-PC、糖酵解和有氧系统。
ATP-PC system: PC + ADP → ATP + Cr
Yields 1 ATP per PC molecule; lasts 3–10 seconds. Used in explosive events like 100 m sprint.
每分子磷酸肌酸产生1个ATP；持续3–10秒。用于100米短跑等爆发性项目。
Anaerobic glycolysis: Glucose → 2 ATP + 2 Lactate
No oxygen required; predominates in 30–90 s high-intensity efforts (e.g., 400 m). Accumulation of lactate contributes to fatigue.
无需氧气；在30–90秒的高强度运动中占主导（如400米）。乳酸堆积导致疲劳。
Aerobic system: Glucose + 6 O₂ → 6 CO₂ + 6 H₂O + ~38 ATP
Also oxidises fatty acids (yielding more ATP but requiring more oxygen). Dominant in endurance events lasting over 2 minutes.
同时氧化脂肪酸（产生更多ATP，但需更多氧）。在持续2分钟以上的耐力项目中占主导。
10. Heart Rate and Training Zones | 心率与训练区域
Heart rate monitoring helps prescribe training intensity. Maximum heart rate (HRmax) can be estimated, and training zones are calculated as percentages of HRmax or heart rate reserve (HRR).
心率监控有助于设定训练强度。最大心率 (HRmax) 可估算，训练区域按 HRmax 或储备心率 (HRR) 的百分比计算。
HRmax = 220 – age
Simplified estimation. A 17-year-old has an approximate HRmax of 203 bpm. Standard deviation of this formula is ±10–12 bpm.
简化估算。17岁运动员的最大心率约为203次/分。该公式的标准差为±10–12次/分。
Karvonen formula: Target HR = Resting HR + (Intensity% × (HRmax – Resting HR))
Using heart rate reserve provides a more individualised training zone. For aerobic training, 60–80% HRR is typical.
使用储备心率法能提供更具个性化的训练区间。有氧训练通常采用60–80% HRR。
11. Body Mass Index and Body Composition | 身体质量指数与体成分
BMI is a simple screening tool, but it does not distinguish between fat mass and lean mass. Nevertheless, it appears in health-related fitness discussions alongside other body composition measures.
BMI是一种简单的筛查工具，但它不能区分脂肪量和瘦体重。尽管如此，在讨论健康相关体适能时，它常与其他体成分测量一起出现。
BMI = Weight (kg) ÷ Height² (m)
For athletes with high muscle mass, BMI may overestimate adiposity. Skinfold calipers or bioelectrical impedance analysis offer alternative estimates of body fat percentage.
对于肌肉量高的运动员，BMI可能会高估肥胖程度。皮褶卡钳或生物电阻抗分析可提供替代的体脂百分比估算。
Body fat % = (495 / Body density) – 450
The Siri equation converts body density (estimated from skinfolds) into body fat percentage. A male footballer might aim for 8–12% body fat; female athletes typically range 16–24%.
Siri公式将体密度（通过皮褶厚度估算）转换为体脂百分比。男性足球运动员可能目标为8–12%体脂；女运动员通常在16–24%之间。
12. Psychological Arousal and Performance | 心理唤醒与表现
Sport psychology theories often express the relationship between arousal, anxiety, and performance through graphs or models. While not a formula in the mathematical sense, the Yerkes-Dodson law can be described as a curvilinear relationship.
运动心理学理论通常用图形或模型表达唤醒、焦虑与表现之间的关系。耶克斯-多德森定律虽非数学公式，但可描述为一种曲线关系。
Performance = f(Arousal) under an inverted-U curve
Low arousal leads to underperformance due to lack of focus; moderate arousal yields optimal performance; excessive arousal causes anxiety and deterioration. The optimal level shifts left for complex tasks and right for simple or gross motor tasks.
低唤醒导致注意力不足，表现不佳；中等唤醒产生最佳表现；过度唤醒引发焦虑，表现下降。复杂任务的最佳唤醒水平偏左，简单或粗大动作任务偏右。
SC = P × I
Stress capacity (SC) is modelled as the product of perceived pressure (P) and individual interpretation (I). This helps explain why two athletes respond differently to the same competitive situation.
压力承受力 (SC) 被模拟为感知压力 (P) 与个体解读 (I) 的乘积。这有助于解释为何两名运动员对相同比赛情境反应不同。
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