📚 Year 12 WJEC Physical Education: Formula & Theorem Quick Reference Handbook | Year 12 WJEC 体育：公式定理速查手册
This handbook presents a concise compilation of essential formulas, laws and theorems for the WJEC Year 12 Physical Education specification. Understanding these mathematical and theoretical principles is crucial for analysing movement, designing training programmes and evaluating psychological and skill-based factors in sport. Use this guide for quick revision and to reinforce your grasp of key quantitative relationships.
本手册简洁汇编了 WJEC Year 12 体育课程中必不可少的公式、定律与定理。理解这些数学与理论原理，对于分析运动、设计训练计划以及评估运动中的心理和技能因素至关重要。可用本指南进行快速复习，强化对关键定量关系的掌握。
Every section follows the same logic: the quantitative rule first, then the sporting context in which WJEC examiners expect you to apply it. Numbers alone earn few marks; the marks come from linking a formula to a named sporting example and to the physiological or biomechanical mechanism behind it.
每一节的逻辑相同：先给出定量规则，再说明 WJEC 考官期望你运用的运动情境。只写数字很难得分；分数来自把公式与具体的运动实例以及背后的生理或生物力学机制联系起来。
1. Kinematics – Linear Motion | 运动学 – 线性运动
constant, the expression simplifies to v = d / t, where d is the total distance travelled.
平均速度 (v) 是单位时间内的位移：v = Δs / Δt。当运动方向不变时，该式可简化为 v = d / t，其中 d 为总行驶距离。
Acceleration (a) measures the rate of change of velocity: a = (v – u) / t, where u is initial velocity and v is final velocity.
加速度 (a) 度量速度变化的快慢：a = (v – u) / t，其中 u 为初速度，v 为末速度。
v = u + a t s = u t + ½ a t² v² = u² + 2 a s s = ½ (u + v) t
These four SUVAT equations describe motion with uniform acceleration and are fundamental for analysing sprints, throws and jumps.
这四个 SUVAT 方程描述匀加速运动，是分析短跑、投掷和跳跃动作的基础。
Always distinguish scalars from vectors: distance, speed, mass and time are scalars, while displacement, velocity, acceleration and force are vectors with both magnitude and direction.
务必区分标量与矢量：路程、速率、质量和时间是标量，而位移、速度、加速度和力是既有大小又有方向的矢量。
| Quantity 物理量 | Symbol 符号 | SI Unit 国际单位 |
|---|---|---|
| Displacement 位移 | s | m |
| Velocity 速度 | v | m s⁻¹ |
| Acceleration 加速度 | a | m s⁻² |
| Time 时间 | t | s |
rinter who accelerates from rest to 10 m s⁻¹ in 2 s has an average acceleration of a = (10 – f the drive phase, where force applied against the blocks produces the change in velocity.
一名 100 米短跑运动员在 2 秒内由静止加速到 10 m s⁻¹，其平均加速度为 a = (10 – 0) / 2 = 5 m s⁻²。该数值量化了起跑蹬伸阶段的效果，即运动员对起跑器施力所产生的速度变化。
2. Newton's Laws & Momentum | 牛顿定律与动量
Newton's First Law (Inertia) states that a body remains at rest, or continues in uniform motion in a straight line, unless acted upon by an external force.
牛顿第一定律（惯性定律）指出：物体在不受外力作用时，保持静止或匀速直线运动状态。
Newton' Law links force to the rate of change of momentum: F = m a, or more generally F = Δp / Δt.
牛顿第二定律把力与动量变化率联系起来：F = m a，更一般的形式为 F = Δp / Δt。
Newton's Third Law states that for every action there is an equal and opposite reaction; the ground pushes back on the sprinter with the same magnitude of force that the sprinter applies to it.
牛顿第三定律指出：每一个作用力都有一个大小相等、方向相反的反作用力；地面给短跑运动员的反作用力，与运动员对地面施加的力大小相等。
p = m v F t = Δp = m v – m u m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Momentum (p) is the product of mass and velocity, measured in kg m s⁻¹. Impulse (F t) is the change in momentum and explains why follow-through in racket and throwing events increases contact time and therefore increases the velocity imparted to the ball.
动量 (p) 是质量与速度的乘积，单位为 kg m s⁻¹。冲量 (F t) 等于动量变化，这解释了为什么球拍类与投掷类项目的随挥动作能延长接触时间，从而增大传递给球的速度。
Conversely, in catching a cricket ball the hands give way on impact, lengthening the time of contact so that the force experienced by the hands is reduced.
相反，在板球接球时，双手随球缓冲，延长了接触时间，因此手部所受到的力随之减小。
Conservation of momentum is applied to collisions in contact sports and to the transfer of momentum in rowing, where the oar and water interact with the boat.
动量守恒可应用于对抗性项目中的碰撞，也可用于赛艇中桨、水与艇之间的动量传递分析。
3. Projectile Motion | 抛体运动
A projectile is any object released into the air and moving under the influence of gravity alone. Its horizontal and vertical components of motion are independent of one another.
抛体是指被释放到空中、仅受重力作用的物体。其水平方向与垂直方向的运动相互独立。
uₕ = u cos θ uᵥ = u sin θ t = 2 u sin θ / g R = u² sin 2θ / g H = u² sin²θ / (2 g)
In these equations u is the speed of release, θ is the angle of release, g is the acceleration due to gravity (9.81 m s⁻²), R is the range and H is the maximum height.
式中 u 为出手速度，θ 为出手角度，g 为重力加速度（9.81 m s⁻²），R 为射程，H 为最大高度。
The four factors affecting the trajectory of a projectile are the speed of release, the angle of release, the height of release and the relative height of release.
影响抛体轨迹的四个因素是：出手速度、出手角度、出手高度以及出手高度与落地高度的相对差值。
Theoretically, 45° gives maximum range on level ground, but javelin and shot put throwers use angles closer to 35° because of release height and aerodynamic constraints, while the high jump requires a large angle combined with a curved approach.
理论上，在平地上 45° 可获得最大射程，但标枪与铅球运动员的出手角度更接近 35°，原因是出手高度与空气动力条件的限制；而跳高则需要较大的起跳角度并配合弧线助跑。
Air resistance makes the parabola asymmetric; the descent is steeper than the ascent because drag reduces both the horizontal and vertical velocity components.
空气阻力使抛物线不再对称：由于阻力同时减小水平与垂直速度分量，下降段比上升段更陡。
4. Force, Levers & Mechanical Advantage | 力、杠杆与机械效益
ce about an axis: Moment = Force × perpendicular distance from the pivot, measured in N m.
力矩是力绕轴转动产生的效果：力矩 = 力 × 到支点的垂直距离，单位为 N m。
M = F × d Mechanical Advantage = Effort Arm / Resistance Arm
There are three classes of lever. In a first-class lever the fulcrum lies between the effort and the load; in a second-class lever the load lies between the fulcrum and the effort; in a third-class lever the effort lies between the fulcrum and the load.
杠杆分为三类。第一类杠杆的支点位于用力点与载荷之间；第二类杠杆的载荷位于支点与用力点之间；第三类杠杆的用力点位于支点与载荷之间。
| Class 类别 | Arrangement 排列 | Sporting Example 运动实例 |
|---|---|---|
| First 第一类 | F – L – E | Neck extension at the atlanto-occipital joint 颈部伸展 |
| Second 第二类 | F – E – L | Plantar flexion at the ball of the foot 提踵/蹬地 |
| Third 第三类 | F – L – E | Elbow flexion by the biceps brachii 屈肘 |
The vast majority of levers in the human body are third-class levers, which operate at a mechanical disadvantage but gain speed and range of movement, a trade-off that is essential for fast limb actions in sport.
人体中的绝大多数杠杆属于第三类杠杆，其机械效益较低，但换来了速度与运动幅度的提升；这种取舍对于体育中快速的肢体动作至关重要。
5. Work, Energy, Power & Efficiency | 功、能、功率与效率
s done when a force moves an object through a distance: W = F × d, measured in joules (J).
当力使物体产生位移时即做功：W = F × d，单位为焦耳 (J)。
W = F d KE = ½ m v² GPE = m g h P = W / t = F v Efficiency = (Useful Output / Total Input) × 100%
Kinetic energy depends on the square of velocity, so doubling running speed quadruples the kinetic energy
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