📚 Pre-U WJEC Physical Education: Formula & Theorem Quick Reference Handbook | Pre-U WJEC 体育：公式定理速查手册
This quick reference handbook compiles essential formulae, equations, and key theorems required for the Pre-U WJEC Physical Education specification. It spans biomechanics, exercise physiology, and skill acquisition, offering a concise bilingual revision aid. All symbols follow standard scientific notation, and each entry is paired in English and Chinese to strengthen comprehension.
本速查手册汇集了 Pre-U WJEC 体育学科所需的核心公式、方程和重要定理，涵盖运动生物力学、运动生理学和技能习得领域，以精简的双语对照帮助复习。所有符号均采用标准科学记法，并逐条提供中英释义，便于理解与运用。
1. Speed and Acceleration | 速度与加速度
Δs is the distance travelled and Δt is the time interval. When direction matters, average velocity uses displacement instead of distance.
平均速率定义为路程随时间的变化率：v = Δs / Δt，其中 Δs 为行进距离，Δt 为时间间隔。当需要考虑方向时，平均速度则采用位移替代距离。
Acceleration measures how quickly velocity changes: a = (vf − vi) / t, where vf is final velocity, vi is initial velocity, and t is the time taken. Negative acceleration indicates deceleration.
加速度衡量速度变化的快慢：a = (vf − vi) / t，其中 vf 为末速度，vi 为初速度，t 为所用时间。加速度为负值表示减速。
2. Momentum and Impulse | 动量与冲量
ector quantity with the same direction as velocity. A change in momentum occurs when a force acts over time.
线动量是质量与速度的乘积：p = m × v。动量是矢量，方向与速度相同。当力在一段时间内作用时，动量会发生改变。
rce applied multiplied by the time for which it acts: J = F × Δt. According to the impulse–momentum relationship, impulse is also equal to the change in momentum: J = Δp = m(vf − vi).
冲量等于作用力乘以作用时间：J = F × Δt。根据冲量–动量定理，冲量也等于动量的变化量：J = Δp = m(vf − vi)。
3. Newton’s Laws of Motion | 牛顿运动定律
Newton’s first law (inertia): an object remains at rest or in uniform motion in a straight line unless acted upon by an external resultant force. No net force means no acceleration.
牛顿第一定律（惯性定律）：物体在不受外部合力作用时，将保持静止或匀速直线运动状态。没有净外力就没有加速度。
ional to its mass: F = m × a. This law links force, mass, and acceleration quantitatively.
牛顿第二定律：物体的加速度与作用在其上的净外力成正比，与其质量成反比：F = m × a。该定律定量描述了力、质量与加速度的关系。
Newton’s third law: for every action there is an equal and opposite reaction. In sports, this is evident when a sprinter pushes against the blocks and the blocks push back with equal force.
牛顿第三定律：每个作用力都有一个大小相等、方向相反的反作用力。在运动中，短跑运动员蹬起跑器时，起跑器同时以相等的力反推运动员。
4. Moments and Levers | 力矩与杠杆
A moment (torque) is the turning effect of a force: τ = F × d⊥, where d⊥ is the perpendicular distance from the pivot to the line of action of the force. The unit is newton-metre (N·m).
力矩（扭矩）是力产生的转动效应：τ = F × d⊥，其中 d⊥ 为支点到力作用线的垂直距离。单位为牛顿·米（N·m）。
hen MA >1, the lever multiplies effort force; when MA <1, it favours speed and range of motion.
杠杆的机械效益（MA）是力臂与阻力臂的比值：MA = 力臂 / 阻力臂。MA >1 时杠杆增力；MA <1 时则有利于速度和运动幅度。
There are three classes of levers. First-class: pivot between effort and load (e.g. see-saw, neck extension). Second-class: load between pivot and effort (e.g. calf raise, wheelbarrow) — always MA >1. Third-class: effort between pivot and load (e.g. biceps curl) — always MA <1 and most common in the human body.
杠杆分为三类。第一类：支点在用力点与负荷之间（如跷跷板、颈部后伸）。第二类：负荷在支点与用力点之间（如提踵、独轮手推车），始终 MA >1。第三类：用力点在支点与负荷之间（如肱二头肌弯举），始终 MA <1，且在人体中最为常见。
5. Angular Motion | 角运动
nt: ω = Δθ / Δt. Angular acceleration describes how angular velocity changes: α = Δω / Δt.
角位移（θ）以弧度为单位。角速度是角位移的变化率：ω = Δθ / Δt。角加速度描述角速度变化的快慢：α = Δω / Δt。
ngular quantities are linked: tangential velocity v = r × ω, and tangential acceleration at × α, where r is the radius of rotation. Centripetal acceleration (towards the centre) is ac = v² / r = r ω², and the centripetal force required is Fc = m v² / r = m r ω².
线量与角量相互关联：切向速度 v = r × ω，切向加速度 at = r × α，其中 r 为转动半径。向心加速度（指向中心）为 ac = v² / r = r ω²，所需的向心力为 Fc = m v² / r = m r ω²。
e of external torque, angular momentum is conserved, which explains why a spinning ice skater speeds up when pulling arms in.
角动量是转动惯量与角速度的乘积：L = I × ω。对于质点，I = m r²；对于复杂形体，I = Σ m r²。在没有外力矩的情况下，角动量守恒，这就解释了为何花样滑冰运动员收臂时旋转速度加快。
6. Fluid Mechanics and Air Resistance | 流体力学与空气阻力
ciple helps explain the lift force on an aerofoil and the Magnus effect on spinning balls.
伯努利原理指出，流体流速增加时压强降低：p + ½ ρ v² + ρ g h = 常数。该原理解释了机翼升力以及旋转球体的马格努斯效应。
The drag force opposing motion through a fluid is given by Fd = ½ Cd ρ A v², where Cd is the drag coefficient, ρ is fluid density, A is cross-sectional area, and v is velocity. Reducing frontal area or streamlining reduces drag.
物体在流体中运动时所受的阻力为 Fd = ½ Cd ρ A v²，其中 Cd 为阻力系数，ρ 为流体密度，A 为迎风截面积，v 为速度。减小正面面积或采用流线型可降低阻力。
When drag equals weight, a falling object reaches terminal velocity and no longer accelerates. Cyclists and skiers adopt compact tucks to minimise A and Cd for higher terminal speeds.
当阻力等于重力时，下落的物体达到终极速度，不再加速。自行车运动员和滑雪者采用紧凑的蜷缩姿势，以减小 A 和 Cd，从而提高终极速度。
7. Heart Rate and Training Zones | 心率与训练区间
A traditional estimate of maximum heart rate is HRmax = 220 − age. A more accurate formula for adults is HRmax = 208 − 0.7 × age. Resting heart rate (HRrest) should be measured immediately upon waking.
传统估计最大心率的公式为 HRmax = 220 − 年龄。更为精确的成人公式为 HRmax = 208 − 0.7 × 年龄。安静心率（HRrest）应在刚睡醒时测量。
en formula prescribes target heart rate based on heart rate reserve (HRR): Target HR = (HRmax − HRrest) × % intensity + HRrest. For example, a 70% intensity workout uses 0.70 in the formula.
卡沃宁公式根据心率储备（HRR）计算靶心率：靶心率 = (HRmax − HRrest) × 强度百分比 + HRrest。例如，70% 强度的训练将 0.70 代入公式。
Common training zones are: recovery (50–60% HRR), aerobic endurance (60–75% HRR), lactate threshold (75–85% HRR), and anaerobic power (85–95% HRR). Exercising in the correct zone ensures specific physiological adaptations.
常见训练区间为：恢复区（50–60% HRR）、有氧耐力区（60–75% HRR）、乳酸阈区（75–85% HRR）和无氧功率区（85–95% HRR）。在正确区间内训练，可确保获得特定的生理适应。
8. Body Composition and BMI | 身体成分与体重指数
/ [height (m)]². The classification ranges (adults) are: underweight <18.5, normal 18.5–24.9, overweight 25–29.9, obese ≥30. BMI does not distinguish between fat and lean mass.
体重指数是一种简单的筛查工具：BMI = 体重 (kg) / [身高 (m)]²。成人分类范围为：体重过轻 <18.5，正常 18.5–24.9，超重 25–29.9，肥胖 ≥30。BMI 无法区分脂肪与瘦体重。
Waist-to-hip ratio (WHR) and skinfold measurements provide more direct estimates of body fat distribution. Skinfold equations (e.g. Durnin & Womersley) estimate body density and then % body fat, but are less commonly used in the formula-based handbook.
腰臀比（WHR）和皮褶厚度测量能更直接地反映体脂分布。皮褶方程（如 Durnin & Womersley 公式）可估算体密度与体脂百分比，但在公式速查手册中较少使用。
9. Energy Expenditure and Basal Metabolic Rate | 能量消耗与基础代谢率
R = (10 × W) + (6.25 × H) − (5 × A) − 161, where W is mass in kg, H is height in cm, and A is age in years.
基础代谢率（BMR）可采用 Mifflin-St Jeor 公式估算：男性 BMR（千卡/日）= (10 × W) + (6.25 × H) − (5 × A) + 5；女性 BMR = (10 × W) + (6.25 × H) − (5 × A) − 161，其中 W 为体重（kg），H 为身高（cm），A 为年龄（岁）。
ive ≈ 1.55, very active ≈ 1.9. Energy balance is achieved when energy intake matches TDEE.
每日总能量消耗（TDEE）由 BMR 乘以身体活动水平（PAL）得出。久坐人群 PAL ≈ 1.2，中等活跃 ≈ 1.55，非常活跃 ≈ 1.9。当能量摄入与 TDEE 相等时，达到能量平衡。
dy mass (kg) × time (hours). This is useful for prescribing and monitoring training loads.
1 MET（代谢当量）定义为 3.5 毫升氧气·每千克体重·每分钟。运动期间的能量消耗可估算为：能量（千卡）= MET × 体重（kg） × 时间（小时）。这有助于设定和监测训练负荷。
10. Reaction Time, Response Time and Movement Time | 反应时、响应时与运动时
Reaction time is the processing interval before movement begins; movement time is the duration of the action itself.
响应时是从刺激出现到动作完成的总时间：响应时 = 反应时 + 运动时。反应时是动作开始前信息处理的时间段；运动时是执行动作本身所持续的时间。
Reaction time can be affected by the number of stimulus–response choices, the modality of the stimulus, and the athlete’s level of anticipation. Simple reaction time (one stimulus, one response) is faster than choice reaction time.
反应时受刺激–反应选择数量、刺激类型以及运动员预判水平的影响。简单反应时（单一刺激、单一反应）快于选择反应时。
11. Hick’s Law and Fitts’ Law | 希克定律与菲茨定律
Hick’s law describes the relationship between reaction time and the number of stimulus– (n), where n is the number of choices. As n doubles, RT increases by a constant amount, highlighting the cost of increasing options.
希克定律描述反应时与刺激–反应选项数量之间的关系：RT = a + b log₂(n)，其中 n 为选择数目。随着 n 翻倍，反应时增加一个常量，突显选项增多带来的代价。
nce to the target and W is the target width. The log term, index of difficulty (ID), quantifies the speed–accuracy trade-off: wider targets or shorter distances reduce ID and thus MT.
菲茨定律预测快速指向任务所需的运动时间：MT = a + b log₂(2D / W)，其中 D 为到目标的距离，W 为目标的宽度。对数项称为难度指数（ID），量化了速度–准确性权衡：更宽的目标或更短的距离可降低 ID，从而缩短运动时间。
These laws are widely applied in sports such as cricket batting, goalkeeping reactions, and golf putting, where athletes must choose the appropriate response from multiple options and move accurately to a target under time pressure.
这些定律广泛应用于板球击球、守门员反应和高尔夫推杆等运动情景，运动员必须在时间压力下从多个选择中做出适当反应，并准确移动到目标位置。
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