📚 Year 12 WJEC Physical Education: Formula and Theorem Quick Reference Handbook | Year 12 WJEC 体育：公式定理速查手册
This concise handbook brings together every essential formula, theorem and key quantitative relationship required for the Year 12 WJEC Physical Education course. From biomechanics and motion to training physiology, having these equations at your fingertips will sharpen your exam responses and help you apply scientific principles to sporting examples with confidence.
这本简练的手册汇集了 Year 12 WJEC 体育课程所需的全部重要公式、定理和关键定量关系。从生物力学、运动分析到训练生理学，熟记这些方程能让你在考试作答时更精准，并能自信地将科学原理应用于运动实例中。
1. Kinematics: Speed, Velocity and Acceleration | 运动学：速率、速度与加速度
v = d ÷ t
Average speed (v) is the distance travelled (d) divided by the time taken (t). Speed is a scalar; velocity is a vector and includes direction.
平均速率 (v) 等于移动距离 (d) 除以所用时间 (t)。速率是标量；速度是矢量，包含方向。
a = (v − u) ÷ t
Acceleration (a) is the change in velocity (final v, initial u) per unit time t. Its unit is m/s².
加速度 (a) 是单位时间 t 内速度的变化量（末速 v，初速 u）。单位为米每二次方秒 (m/s²)。
v² = u² + 2as
For uniform acceleration, this equation links initial velocity u, final velocity v, acceleration a and displacement s without time.
对于匀加速直线运动，该方程将初速度 u、末速度 v、加速度 a 和位移 s 联系起来，不涉及时间。
s = ut + ½at²
Displacement s when accelerating from an initial velocity u over time t.
从初速度 u 开始，经过时间 t 加速所得的位移 s。
2. Newton’s Laws of Motion | 牛顿运动定律
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by an external resultant force.
第一定律（惯性）： 任何物体都保持静止或匀速直线运动状态，直到有外力迫使其改变。
Second Law: The acceleration of a body is directly proportional to the net force and inversely proportional to its mass.
第二定律： 物体的加速度与所受合力成正比，与质量成反比。
F = m × a
Force (N) = mass (kg) × acceleration (m/s²). A larger force is needed to accelerate a greater mass at the same rate.
力 (N) = 质量 (kg) × 加速度 (m/s²)。要令更大质量的物体获得相同的加速度，需要更大的力。
Third Law: For every action there is an equal and opposite reaction. When a sprinter pushes against the blocks, the blocks push back with equal force.
第三定律： 每有一个作用力，就有一个大小相等、方向相反的反作用力。短跑运动员蹬起跑器时，起跑器会以相等的力反推运动员。
W = m × g
Weight (W) is the force due to gravity: mass (kg) × gravitational field strength (g ≈ 9.81 N/kg).
重力 (W) 是由引力产生的力：质量 (kg) × 重力场强度 (g 约 9.81 N/kg)。
3. Momentum and Impulse | 动量与冲量
p = m × v
Momentum (p) is the product of mass and velocity. It is a vector quantity measured in kg·m/s.
动量 (p) 是质量与速度的乘积。它是矢量，单位为 kg·m/s。
Impulse = F Δt = Δp
Impulse equals the average force multiplied by the time for which it acts, and equals the change in momentum.
冲量等于平均力乘以力作用的时间，也等于动量的变化量。
Principle of Conservation of Momentum: In a closed system, total momentum before a collision equals total momentum after the collision, assuming no external forces.
动量守恒定律： 在不受外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。
Example: In rugby, the combined momentum of a tackler and ball carrier just after contact equals the sum of their momenta just before contact.
实例：榄球比赛中，擒抱者与持球者在接触后的总动量等于接触前二者动量之和。
4. Work, Energy and Power | 功、能与功率
Work = F × d × cos θ
Work done (J) when a force F moves an object through distance d at angle θ to the direction of motion. When force and displacement are in the same direction, cosθ = 1.
力 F 使物体沿位移方向移动距离 d 时所做的功 (J)，θ 为力与位移的夹角。若力与位移同向，cosθ = 1。
KE = ½mv²
Kinetic energy is the energy possessed by a moving object. Doubling velocity quadruples kinetic energy, explaining why faster collisions carry much greater risk.
动能是运动物体具有的能量。速度加倍，动能变为四倍，这解释了为什么高速碰撞风险极大。
ΔPE = mgh
Change in gravitational potential energy is mass × gravity × vertical height change h. This energy converts to KE during a downhill sprint or a dive.
重力势能的变化量等于质量 × 重力加速度 × 垂直高度变化 h。在俯冲或下坡冲刺时，这部分能量转化为动能。
P = W ÷ t = F × v
Power (watts) is the rate of doing work or the product of force and velocity. A high power output is essential for explosive movements like a vertical jump or a javelin throw.
功率 (W) 是做功的速率，也等于力与速度的乘积。纵跳或掷标枪等爆发性动作需要高功率输出。
5. Friction and Air Resistance | 摩擦与空气阻力
f ≤ μs N (static); f = μk N (kinetic)
Friction force f depends on the coefficient of friction μ and the normal reaction force N. Static friction (μs) is usually larger than kinetic friction (μk). Proper footwear increases μ, improving grip.
摩擦力 f 取决于摩擦系数 μ 和法向反作用力 N。静摩擦系数 (μs) 通常大于动摩擦系数 (μk)。合适的运动鞋能增大 μ，提升抓地力。
Fdrag = ½ ρ v² Cd A
Air resistance (drag) increases with fluid density ρ, velocity squared v², drag coefficient Cd and cross-sectional area A. Streamlined positions and smooth clothing reduce Cd and A.
空气阻力（拖拽力）随流体密度 ρ、速度的平方 v²、阻力系数 Cd 和迎风横截面积 A 的增大而增大。流线型姿势和光滑服装可以减小 Cd 和 A。
6. Angular Motion | 角运动
ω = Δθ ÷ Δt
Angular velocity (ω) is the rate of change of angular displacement θ, measured in rad/s. A skater spinning faster has a larger ω.
角速度 (ω) 是角位移 θ 的变化率，单位为 rad/s。旋转中的溜冰选手角速度越大，转得越快。
v = r ω
Linear velocity v of a point on a rotating body is the product of radius r and angular velocity ω. A longer tennis racket arm generates a higher v at the racket head for the same ω.
旋转体上某点的线速度 v 等于旋转半径 r 与角速度 ω 的乘积。在相同 ω 下，较长的臂展能使网球拍头获得更高的线速度。
Fc = m ω² r = mv² ÷ r
Centripetal force is the inward force required to keep a body moving in a circle. It is provided by friction (turning a bike) or by the athlete’s grip (hammer throw).
向心力是维持物体做圆周运动所需的指向圆心的力。它可由摩擦力（自行车转弯）或运动员的握力（链球投掷）提供。
7. Projectile Motion and Fluid Effects | 抛射体运动与流体效应
A projectile’s path is determined by its initial velocity, angle of release and height of release. The horizontal and vertical components are independent.
抛射体的轨迹由初速度、出手角度和出手高度决定。水平分运动与竖直分运动彼此独立。
vx = v cosθ, vy = v sinθ
Horizontal velocity vx and vertical velocity vy components separate the motion. Gravity only affects the vertical component.
水平分速度 vx = v cosθ，竖直分速度 vy = v sinθ。重力仅影响竖直分量。
Bernoulli’s Principle: Faster fluid flow creates lower pressure. An aerofoil or a spinning ball (Magnus effect) experiences lift because of pressure differences.
伯努利原理： 流速越大的地方压强越小。机翼或旋转的球体（马格努斯效应）因压力差而产生升力。
Magnus Effect: A spinning ball drags air, creating high pressure on one side and low pressure on the other, causing it to swerve. This explains a curling free kick in football or a top-spin shot in tennis.
马格努斯效应： 旋转的球带动空气，使其一侧气压高、另一侧气压低，从而产生弧线运动。足球中的香蕉球或网球的上旋球皆源于此。
8. Centre of Mass and Stability | 重心与稳定性
The centre of mass (CoM) is the point where the body’s mass is evenly distributed. It can lie outside the body, e.g. during a Fosbury flop high jump.
重心 (CoM) 是身体质量均匀分布的中心点。它可以位于身体外部，例如背越式跳高过杆时。
Stability increases when: the CoM is low, the line of gravity falls within the base of support, and the base of support is wide. A low defensive stance in basketball improves balance.
以下情况稳定性增强： 重心低、重力作用线落在支撑面内、支撑面宽大。篮球防守时的低姿态能提升平衡。
τ = F × d⊥
Torque (moment of force) equals force multiplied by the perpendicular distance from the pivot. It causes angular acceleration; toppling occurs when the torque from weight shifts the line of gravity outside the base.
力矩等于力乘以力到转动轴的垂直距离。它产生角加速度；当重力产生的力矩使重力作用线移出支撑面时，就会发生倾倒。
9. Heart Rate and Target Zone Formulas | 心率与目标区间公式
HRmax = 220 − age
Estimated maximum heart rate in beats per minute (bpm). This is a population-average estimate and individual variation exists.
估算的最大心率 (bpm)。这是人群平均值，存在个体差异。
Karvonen Formula: Target HR = (HRmax − HRrest) × % intensity + HRrest
This formula uses heart rate reserve to set personalised training zones. For a 60–80% intensity range, calculate both boundaries.
卡氏公式利用心率储备来设定个性化训练区间。对于 60%–80% 强度范围，应分别计算上下限。
Borg Scale (RPE): Though not an equation, the 6–20 scale correlates with heart rate (RPE × 10 ≈ HR). It provides a self-reported measure of exercise intensity.
博格自觉疲劳量表 (RPE)： 虽非公式，6–20 级的量表与心率大致相关（RPE × 10 ≈ 心率），提供了主观的运动强度评价。
10. Respiratory Exchange Ratio and Energy Expenditure | 呼吸交换率与能量消耗
RER = V&775;CO2 ÷ V&775;O2
Respiratory Exchange Ratio is the ratio of carbon dioxide produced to oxygen consumed. RER close to 0.7 indicates fat metabolism; 1.0 indicates carbohydrate metabolism. During high-intensity exercise, RER can exceed 1.0 due to buffering.
呼吸交换率 (RER) 是二氧化碳生成量与氧气消耗量的比值。RER 接近 0.7 表示脂肪供能为主；1.0 表示碳水化合物供能为主。高强度运动时，因缓冲作用 RER 可超过 1.0。
Energy Expenditure (kcal) = V&775;O2 (L/min) × time (min) × caloric equivalent (kcal/L)
Caloric equivalent depends on RER. At RER=0.85, roughly 4.86 kcal per litre of O2. Direct measurement of O2 uptake gives total energy cost.
能量当量取决于 RER。RER=0.85 时，每升氧气约产生 4.86 kcal 能量。通过直接测量摄氧量可得出总能量消耗。
11. Body Composition and Useful Indices | 身体成分与实用指标
BMI = mass (kg) ÷ height² (m²)
Body Mass Index provides a simple categorisation of weight status. It does not distinguish between muscle and fat.
身体质量指数 (BMI) 提供简单的体重分级，但不能区分肌肉与脂肪。
Lean Body Mass = Total Mass − Fat Mass
Lean body mass includes muscle, bone, water and organs. Skinfold measurements or bioelectrical impedance can estimate fat mass.
瘦体重 = 总体重 − 脂肪重量。瘦体重包括肌肉、骨骼、水分和器官。皮褶厚度或生物电阻抗法可估算脂肪重量。
Cardiac Output (Q&775;) = Stroke Volume × Heart Rate
Cardiac output (L/min) reflects the volume of blood pumped by the heart per minute. Endurance training increases stroke volume, allowing a lower resting heart rate for the same Q&775;.
心输出量 (L/min) 反映心脏每分钟泵出的血量。耐力训练可增大每搏输出量，从而在相同心输出量下静息心率更低。
12. Mechanical Advantage and Levers | 机械优势与杠杆
Mechanical Advantage = Effort Arm ÷ Resistance Arm
MA describes the efficiency of a lever system. In the human body, most levers have MA < 1, favouring speed and range of motion over force. The calf raise is an exception: the ball of the foot acts as the fulcrum, producing MA > 1.
机械优势 (MA) 描述杠杆系统的效率。人体内多数杠杆 MA < 1，以速度和运动范围为优势，而非力量。提踵是例外：脚掌为支点，产生 MA > 1。
Three classes of levers exist: first class (e.g. neck extension), second class (calf raise) and third class (bicep curl). The body relies heavily on third-class levers for rapid, wide movements.
杠杆分为三类：第一类（如颈后伸）、第二类（提踵）和第三类（肱二头肌弯举）。人体大量依赖第三类杠杆来实现快速、大幅度的动作。
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