📚 AQA A-level Physical Education Year 12: Formula and Principles Quick Reference | AQA 体育 A-level Year 12 公式定理速查手册
This quick-reference handbook brings together all the essential equations, biomechanical principles and physiological thresholds that appear across the AQA A-level Physical Education Year 12 specification. Each section presents the key relationship in plain English, offers central formulas in a clear format and explains how learners can apply them to sporting examples, from sprint starts to fluid drag on a football. Whether you are revising for a progress test or building flashcards, use this guide to check your understanding of SUVAT, impulse, levers, cardiac output, energy-system power and more.
本速查手册汇集AQA A-level体育学科Year 12课程中所有核心方程、生物力学定理和生理学临界值。每个小节先用简明英语呈现关键关系，再给出中心公式，并解释如何将其应用到冲刺起跑、足球流体阻力等运动实例中。无论你是为阶段测验复习还是制作记忆卡片，都可以用这本手册随时检查自己对SUVAT、冲量、杠杆、心输出量、能量系统功率等概念的理解。
1. Linear Motion Equations (SUVAT) | 直线运动方程 (SUVAT)
city (v), acceleration (a) and time (t). These equations allow sport scientists to calculate sprint times, jump take-off velocities and braking distances. In a 100 m race, when acceleration is constant during the drive phase, the athlete's velocity at the end of that phase can be predicted with v = u + a t.
五个匀加速直线运动方程描述了位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)之间的关系。体育科学工作者利用它们计算冲刺时间、跳跃起跳速度和制动距离。在百米赛跑的驱动阶段，若加速度恒定，该阶段末的速度可以用 v = u + a t 进行预测。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
s = v t — ½ a t²
n uses acceleration due to gravity, g = 9.8 m/s², taken as negative if upward is positive.
使用前务必规定正方向。竖直运动涉及重力加速度 g = 9.8 m/s²，若取向上为正方向则 g 取负值。
2. Newton's Laws of Motion | 牛顿运动定律
Newton' n external resultant force. The second law gives the exact relationship F = m a, linking force, mass and acceleration. The third law declares that every action has an equal and opposite reaction. A swimmer pushing water backwards is propelled forwards by the reaction force from the water.
牛顿三定律支配着力如何产生运动。第一定律（惯性）指出，除非受外力作用，物体将保持静止或匀速直线运动。第二定律给出定量关系 F = m a，将力、质量和加速度联系起来。第三定律说明每个作用力都有一个大小相等、方向相反的反作用力。游泳者向后推水，水则向前产生反作用力推动身体前进。
F = m a
When multiple forces act on an athlete, a free-body diagram shows the resultant. Net force determines the direction and magnitude of acceleration.
当多个力作用在运动员身上时，自由体受力图可以显示合力。合力决定了加速度的方向和大小。
3. Momentum and Impulse | 动量与冲量
Momentum (p) is the product of a body's mass and velocity. Impulse is the product of force and the time for which it acts, and equals the change in momentum. In a rugby tackle, a short impulse time produces a large force, increasing the chance of bringing the opponent to ground quickly. Lowering peak force is possible by increasing contact time, which is the principle behind cushioned landing mats.
动量(p)是物体的质量与速度之积。冲量是力与力作用时间的乘积，且等于动量的变化量。在橄榄球擒抱中，冲量作用时间越短产生的力越大，越容易将对方扑倒。通过延长接触时间可以降低峰值力，这正是缓冲垫的使用原理。
p = m v
Impulse = F × t = Δ p = m (v — u)
The area under a force–time graph represents impulse. In a vertical jump, the impulse generated during the push-off phase determines take-off velocity and thus jump height.
力-时间曲线下面积代表冲量。在纵跳中，蹬伸阶段产生的冲量决定起跳速度，进而影响跳跃高度。
4. Projectile Motion | 抛体运动
An object released into the air and subject only to gravity and air resistance follows a parabolic path. The horizontal component of velocity remains constant (ignoring air resistance), while the vertical component changes uniformly under gravity. A javelin thrower's release angle, speed and height of release all affect the range. The greatest horizontal range in a vacuum is achieved at a 45° release angle, though the optimum angle is usually lower for real sports projectiles due to height of release and air resistance.
被抛向空中并仅受重力和空气阻力作用的物体沿抛物线运动。水平速度分量保持不变（忽略空气阻力），垂直分量则在重力作用下均匀变化。标枪运动员的出手角度、出手速度和出手高度都会影响投掷距离。真空中45°出手角可获得最大水平距离，但在实际运动中由于出手高度和空气阻力，最优角度通常略小。
Horizontal displacement: x = u cosθ × t
Vertical displacement: y = u sinθ × t — ½ g t²
Range (level ground): R = u² sin 2θ / g
Air resistance, lift forces and the Magnus effect alter the flight of balls in sports such as football, tennis and golf; these are considered in section 8 on fluid dynamics.
空气阻力、升力和马格努斯效应会改变足球、网球、高尔夫球等的飞行轨迹，相关内容见第8节流体动力学。
5. Forces and Free Body Diagrams | 力与受力分析图
force, friction, air resistance and muscular force are the main forces acting on a performer. A free body diagram isolates the athlete or object, drawing all external forces as arrows. The resultant force determines acceleration. On a banked velodrome track, the horizontal component of the reaction force provides the centripetal force needed for circular motion.
在生物力学中，力是有大小和方向的矢量。体重(W = m g)、地面反作用力、摩擦力、空气阻力和肌力是作用在运动员身上的主要力。自由体受力图将运动员或物体隔离，并用箭头标出所有外力。合力决定加速度。在倾斜的室内自行车赛道上，反作用力的水平分量提供了圆周运动所需的向心力。
Weight: W = m g
Friction: Ff ≤ μ R
When drawing free body diagrams, label each force clearly and ensure the length of arrows represents relative magnitude. Analyse the vertical and horizontal components separately to find the net force.
画受力分析图时，要清楚标注每个力，并确保箭头长度代表相对大小。分析时应分别处理竖直和水平分量，以求得合力。
6. Moments and Levers | 力矩与杠杆
A moment is the turning effect of a force about a fulcrum. The principle of moments states that for a lever to be in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. The human body uses three classes of lever: first class (e.g. neck nodding), second class (e.g. standing on tiptoe) and third class (e.g. bicep curl). Most joint systems are third-class levers, favouring speed and range of movement over force production.
力矩是力绕支点产生的转动效应。力矩原理指出，杠杆达转动平衡时，顺时针力矩之和等于逆时针力矩之和。人体利用三类杠杆：第一类（如点头）、第二类（如踮脚）和第三类（如肱二头肌弯举）。多数关节系统为第三类杠杆，以力量为代价换取速度和运动幅度。
Moment = Force × perpendicular distance from fulcrum
M = F × d
esistance arm. A trainer can alter resistance-arm length to manipulate exercise intensity.
机械利益 = 施力臂 / 阻力臂。教练可以通过改变阻力臂长度来调控运动强度。
7. Angular Motion | 角运动
Angular motion occurs when a body or body part rotates about an axis. The quantities angular displacement (θ), angular velocity (ω) and angular acceleration (α) parallel their linear counterparts. In a diving somersault, the tuck position reduces the moment of inertia, causing angular velocity to increase (conservation of angular momentum).
角运动发生在身体或身体环节绕某一轴线转动时。角位移(θ)、角速度(ω)和角加速度(α)分别对应直线运动中的位移、速度和加速度。在跳水空翻中，团身姿势减小转动惯量，使角速度增快（角动量守恒）。
ω = Δθ / t
α = Δω / t
Angular momentum = moment of inertia × angular velocity: L = I ω
Moment of inertia depends on the distribution of mass relative to the axis of rotation. A gymnast can spin faster by pulling arms in tight, reducing I and therefore increasing ω.
转动惯量取决于质量相对于转轴的分布。体操运动员通过将手臂收紧可以加快旋转——减小I，因而增大ω。
8. Fluid Dynamics and Bernoulli's Principle | 流体动力学与伯努利原理
When an object travels through a fluid (air or water), drag forces oppose motion. Bernoulli's principle explains how pressure differences are created by variations in flow velocity: faster flow exerts lower pressure. The Magnus force is the result of pressure differences generated when a spinning ball drags air faster on one side. A topspin tennis ball dips sharply because the pressure side difference creates a downward force.
物体在流体（空气或水）中运动时会受到与运动方向相反的阻力。伯努利原理揭示了流速差异如何引起压力差：流速较快处压力较低。马格努斯力正是由旋转球体一侧气流加速产生压力差而形成的。上旋网球会因压力差产生向下的力而急剧下坠。
Bernoulli's equation: P + ½ ρ v² + ρ g h = constant
Drag force depends on velocity, cross-sectional area, fluid density and the drag coefficient of the shape. Cyclists adopt a streamlined crouch to reduce the frontal area and drag coefficient.
阻力大小取决于速度、横截面积、流体密度和物体的形状阻力系数。自行车运动员采用低伏姿势以减小正面面积和阻力系数。
9. Cardiovascular Formulas | 心血管计算公式
Cardiac output (Q) is the volume of blood pumped by the heart per minute. It is the product of stroke volume (SV) and heart rate (HR). Endurance training increases stroke volume, enabling the heart to pump the same cardiac output at a lower heart rate both at rest and during submaximal exercise. Maximum heart rate is approximated using the formula 220 minus age.
心输出量(Q)指心脏每分钟泵出的血量，是每搏输出量(SV)与心率(HR)的乘积。耐力训练可使每搏输出量增加，从而在安静和亚极量运动时以较低心率维持相同的心输出量。最大心率可用220减年龄的公式估算。
Q = SV × HR
Max HR = 220 — age
Heart rate zones are then calculated as percentages of max HR. Cardiorespiratory endurance can also be monitored through VO₂ max, expressed in ml/kg/min.
心率训练区间随后按最大心率的百分比计算。心肺耐力也可通过最大摄氧量(VO₂ max)进行监控，单位是ml/kg/min。
10. Energy Systems and Power Output | 能量系统与功率输出
Three energy systems provide ATP for muscular contraction: the ATP-PC system (immediate, high power, capacity for 8–10 seconds), the glycolytic system (lactic acid system, high intensity for up to 2 minutes) and the aerobic system (low power, high capacity, dominates beyond 3 minutes). Power output refers to the rate of work done or the product of force and velocity. A vertical jump test can evaluate short-term anaerobic power using the force–velocity relationship.
三个供能系统为肌肉收缩提供ATP：ATP-PC系统（即刻供能，功率高，可供能8–10秒）、糖酵解系统（乳酸系统，高强度运动可持续约2分钟）和有氧系统（功率低但容量大，3分钟后占据主导）。功率输出指作功的速率或力与速度的乘积。纵跳测试可通过力-速度关系评估短期无氧功率。
Power = work / time = force × velocity
Work = force × distance moved in direction of force
Knowledge of energy-system interplay helps coaches design interval training: work-to-rest ratios for phosphocreatine replenishment (e.g. 1:6 for maximal sprints) versus shorter recovery for lactic tolerance work (e.g. 1:3).
理解能量系统的相互作用有助于教练安排间歇训练：旨在补充磷酸肌酸的练习可选用大休息比（如最大冲刺采用1:6），而乳酸耐受训练则需要较短的恢复时间（如1:3）。
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