📚 Year 13 WJEC Physical Education: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 体育：公式定理速查手册
In Year 13 WJEC Physical Education, a solid grasp of the key formulas and biomechanical principles is essential for analysing human movement, optimising performance, and understanding physiological responses. This handbook compiles the must-know equations and laws across biomechanics and exercise physiology, presented with clear definitions and applied examples to support your revision.
在 Year 13 WJEC 体育课程中，扎实掌握关键公式和生物力学原理对于分析人体运动、优化表现以及理解生理反应至关重要。本手册汇编了运动生物力学和运动生理学中必须了解的方程和定律，配以清晰的定义和应用实例，为你的复习提供支持。
1. Linear Velocity and Acceleration | 线速度与加速度
Average velocity (v) is the rate of change of displacement: v = Δs / Δt, where Δs is displacement (m) and Δt is time (s). It is a vector quantity, meaning direction matters. Acceleration (a) is the rate of change of velocity: a = Δv / Δt = (v − u) / t, with u as initial velocity and v as final velocity. A negative acceleration indicates deceleration.
平均速度 (v) 是位移的变化率：v = Δs / Δt，其中 Δs 为位移（米），Δt 为时间（秒）。它是一个矢量，方向至关重要。加速度 (a) 是速度的变化率：a = Δv / Δt = (v − u) / t，u 为初速度，v 为末速度。负加速度表示减速。
In a 100 m sprint, if an athlete covers the distance in 10 s, average velocity is 10 m·s⁻¹, though instantaneous velocity varies. Acceleration during the drive phase can reach 5–6 m·s⁻² before decreasing.
在 100 米短跑中，若运动员 10 秒完成，平均速度为 10 m·s⁻¹，但瞬时速度不断变化。蹬地阶段的加速度可达 5–6 m·s⁻²，随后逐渐降低。
v = s / t a = (v − u) / t
2. Linear Momentum and Impulse | 线动量和冲量
Momentum (p) is the product of mass and velocity: p = m × v. Impulse (J) is the change in momentum caused by a force applied over time: J = F × t = Δp = m(v − u). In sport, increasing the time of force application can reduce peak force and injury risk, as seen in landing techniques.
动量 (p) 是质量与速度的乘积：p = m × v。冲量 (J) 是由力在一段时间内作用而引起的动量变化：J = F × t = Δp = m(v − u)。在运动中，延长力作用时间可以降低峰值力与受伤风险，这在落地技术中常被运用。
When a cricketer catches a ball, they ‘give’ with the hands, extending the impact time, which reduces the force experienced. Similarly, bending the knees on landing lengthens impulse time and decreases the force on joints.
板球运动员接球时，手部会向后缓冲，增加碰撞时间，从而减小所受的力。同样，落地时屈膝可延长冲量作用时间，降低关节受力。
p = m v J = F t = Δp
3. 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 force. Second Law (Acceleration): The acceleration of an object is proportional to the net force and inversely proportional to its mass: F = m a. Third Law (Action-Reaction): For every action force, there is an equal and opposite reaction force.
第一定律（惯性）：没有外力作用时，物体保持静止或匀速直线运动。第二定律（加速度）：物体的加速度与合外力成正比，与其质量成反比：F = m a。第三定律（作用力与反作用力）：对于每一个作用力，总有一个大小相等、方向相反的反作用力。
In sprinting, the athlete pushes backward against the track (action), and the track pushes the athlete forward (reaction). The greater the force exerted, the greater the forward acceleration (F = m a).
在短跑中，运动员向后蹬地（作用力），跑道则向前推动运动员（反作用力）。施加的力越大，向前的加速度就越大（F = m a）。
F = m a
4. Force, Mass, and Weight | 力、质量与重量
Weight (W) is the force due to gravity acting on a mass: W = m × g, where g = 9.81 m·s⁻² on Earth. Mass is a scalar quantity measured in kilograms, while weight is a vector measured in newtons. In biomechanics, the distinction is critical when resolving forces on a free body diagram.
重量 (W) 是重力作用于质量上的力：W = m × g，地表 g = 9.81 m·s⁻²。质量是标量，单位为千克；重量是矢量，单位为牛顿。在生物力学中，绘制自由体受力图时必须区分二者。
An athlete with a mass of 70 kg has a weight of 70 × 9.81 = 686.7 N. On a slope, the component of weight parallel to the slope is W sinθ, which influences downhill acceleration in skiing.
一名质量为 70 公斤的运动员重量为 70 × 9.81 = 686.7 N。在斜坡上，重量平行于坡面的分量为 W sinθ，这影响着滑雪时的下坡加速度。
W = m g
5. Friction and Coefficient of Friction | 摩擦与摩擦系数
The maximum static friction or kinetic friction is given by F = μ R, where μ is the coefficient of friction and R is the normal reaction force. μ depends on the two surfaces in contact. Static friction is usually greater than kinetic friction, aiding movements such as cutting and accelerating.
最大静摩擦力或动摩擦力由 F = μ R 给出，μ 为摩擦系数，R 为正压力。μ 值取决于接触面材质。静摩擦力通常大于动摩擦力，有助于切入和加速等动作。
In athletics, sprinters use spikes to increase μ between the shoe and track, raising the limiting friction force and preventing slip. Gymnasts use chalk to maintain high friction for grip on apparatus.
径赛运动员使用钉鞋增大鞋与跑道的 μ，提升极限摩擦力以防打滑。体操运动员使用镁粉保持对手器械的高摩擦力。
F = μ R
6. Torque and Moment of Force | 力矩与力偶矩
Torque (τ), or moment of a force, is the rotational effect of a force: τ = F × d, where d is the perpendicular distance from the axis of rotation to the line of action of the force. Torque is measured in newton-metres (N·m). It is a vector; clockwise and anti-clockwise torques can be summed around a pivot.
力矩 (τ)，即力的转动效应：τ = F × d，d 为转轴到力作用线的垂直距离。力矩单位为牛顿米 (N·m)。力矩是矢量，可围绕支点把顺时针与逆时针力矩求和。
In a bicep curl, the elbow acts as the pivot. The bicep force creates a torque to overcome the weight of the forearm and dumbbell. Changing the angle alters the perpendicular distance and thus the required muscle force.
在肱二头肌弯举中，肘关节为支点。肱二头肌产生力矩克服前臂和哑铃的重量。角度改变会改变垂直距离，从而影响所需的肌肉力量。
τ = F d
7. Centre of Mass and Stability | 质心与稳定性
The centre of mass (CoM) is the point where the mass of a body appears to be concentrated. Stability depends on CoM height, base of support area, and the line of gravity. An object is stable if the vertical line through its CoM falls within the base of support. Lowering the CoM and widening the base enhance stability.
质心 (CoM) 是物体质量似乎集中于此的点。稳定性取决于质心高度、支撑面面积和重力线。若通过质心的竖直线落在支撑面内，则物体稳定。降低质心、加宽支撑面可增强稳定性。
Wrestlers adopt a low stance with a wide base to prevent being toppled. High jumpers’ CoM can pass underneath the bar while their body clears it, thanks to the Fosbury Flop technique arching the back.
摔跤手采用低姿态和宽支撑面防止被摔倒。背越式跳高时，运动员质心可从横杆下方通过，而身体越过横杆，这得益于背弓技术。
8. Projectile Motion | 抛射体运动
A projectile’s path is determined by launch velocity, angle of release, and height of release. The horizontal component of velocity is constant (ignoring air resistance), while the vertical component changes due to gravity. Key equations include vₕ = v cosθ and vᵥ = v sinθ, with time of flight, maximum height, and range calculable using constant acceleration formulas.
抛射体的轨迹由初速度、出手角度和出手高度决定。忽略空气阻力时，速度的水平分量恒定，而垂直分量因重力改变。关键方程包括 vₕ = v cosθ 和 vᵥ = v sinθ，飞行时间、最大高度和射程可利用匀加速公式计算。
Optimal release angle for a long jumper is around 18–25°, not 45°, because take-off height is above landing height. Shot putters release at approximately 35–40° considering the projection height advantage.
跳远的最佳起跳角度约为 18–25°，而非 45°，因为起跳高度高于落地高度。铅球选手考虑出球高度优势后，出手角度约为 35–40°。
vₕ = v cosθ vᵥ = v sinθ
9. Angular Motion Quantities | 角运动量
Angular velocity (ω) is the rate of change of angular displacement: ω = Δθ / Δt, expressed in rad·s⁻¹. Angular acceleration (α) is α = Δω / Δt. Linear velocity of a point on a rotating body is v = ω r, where r is the radius of rotation. This relationship is vital for understanding bat/club head speed.
角速度 (ω) 是角位移的变化率：ω = Δθ / Δt，单位为 rad·s⁻¹。角加速度 (α)：α = Δω / Δt。旋转体上一点的线速度为 v = ω r，r 为旋转半径。这一关系对理解球棒或球杆的杆头速度至关重要。
In golf, a longer club shaft (greater r) generates higher clubhead speed for the same angular velocity, increasing ball velocity. A figure skater spins faster by pulling arms in, reducing r and conserving angular momentum.
在高尔夫中，相同角速度下，较长的杆身（更大的 r）产生更高的杆头速度，从而增大球的初速。花样滑冰运动员收臂旋转，减小 r，在角动量守恒下旋转更快。
ω = Δθ / Δt v = ω r
10. Moment of Inertia and Angular Momentum | 转动惯量与角动量
Moment of inertia (I) is the resistance to angular acceleration: I = Σ m r², dependent on mass and its distribution relative to the axis. Angular momentum (L) is the product of I and ω: L = I ω. In a closed system, angular momentum is conserved; decreasing I results in an increase in ω.
转动惯量 (I) 是对角加速度的惯性阻力：I = Σ m r²，取决于质量及其相对于轴的分布。角动量 (L) 为 I 与 ω 的乘积：L = I ω。在封闭系统中，角动量守恒；减小 I 会导致 ω 增大。
Conservation of angular momentum explains why a diver tucks into a somersault: reducing I by drawing mass closer to the axis significantly increases spin rate, allowing multiple rotations before entry.
角动量守恒解释了跳水运动员为何团身翻腾：通过将质量靠拢转轴减少 I，可大幅提升旋转速度，在入水前完成多周转体。
I = Σ m r² L = I ω
11. Work, Energy, and Power in Sport | 运动中的功、能与功率
Mechanical work done is force multiplied by distance moved in the direction of the force: W = F d cosθ. Kinetic energy (KE) is KE = ½ m v²; gravitational potential energy (GPE) is GPE = m g h. Power is the rate of doing work: P = W / t = F v, often used to assess explosive performance.
力学中的功等于力乘以在力的方向上移动的距离：W = F d cosθ。动能 (KE)：KE = ½ m v²；重力势能 (GPE)：GPE = m g h。功率是做功的快慢：P = W / t = F v，常用来评估爆发性表现。
In a vertical jump, the athlete’s kinetic energy at take-off is converted to potential energy at the peak. Power output during a countermovement jump can exceed 4000 W in elite athletes.
纵跳时，运动员起跳瞬间的动能转化为最高点的势能。精英运动员在反向纵跳中的功率输出可超过 4000 W。
KE = ½ m v² GPE = m g h P = F v
12. Cardiac Output and the Fick Equation | 心输出量与菲克方程
Cardiac output (Q̇) is the volume of blood pumped by the heart per minute: Q̇ = heart rate (HR) × stroke volume (SV). Oxygen uptake (V̇O₂) is determined by the Fick equation: V̇O₂ = Q̇ × a-vO₂ diff, where a-vO₂ diff is the arteriovenous oxygen difference. These formulas underpin the measurement of aerobic fitness.
心输出量 (Q̇) 是心脏每分钟泵出的血量：Q̇ = 心率 (HR) × 每搏输出量 (SV)。摄氧量 (V̇O₂) 由菲克方程确定：V̇O₂ = Q̇ × 动静脉氧差。这些公式是测量有氧适能的基础。
At rest, Q̇ ≈ 5 L·min⁻¹; during maximal exercise it can rise to 20–25 L·min⁻¹ in trained athletes. V̇O₂ max, the maximal oxygen uptake, is a gold standard indicator of endurance capacity and is expressed in mL·kg⁻¹·min⁻¹ or L·min⁻¹.
静息时，Q̇ ≈ 5 L·min⁻¹；在最大强度运动期间，训练有素的运动员可升至 20–25 L·min⁻¹。最大摄氧量 (V̇O₂ max) 是耐力水平的黄金标准，单位用 mL·kg⁻¹·min⁻¹ 或 L·min⁻¹ 表示。
Q̇ = HR × SV V̇O₂ = Q̇ × a-vO₂ diff
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