📚 Year 12 Cambridge Physical Education: Quick-Reference Formulas & Theorems | Year 12 剑桥体育：公式定理速查手册
This quick-reference handbook compiles the essential formulas, principles, and theorems required for the Year 12 Cambridge AS Physical Education course. Use it as a ready-made revision tool to strengthen your understanding of biomechanics, exercise physiology, and skill acquisition. Every formula is presented with its practical sporting application to help you connect theory with performance.
这本速查手册汇编了 Year 12 剑桥 AS 体育课程所必需的关键公式、原理和定理。你可以将其作为现成的复习工具，以加深对生物力学、运动生理学和技能获得的理解。每个公式都配有其实际的运动应用，帮助你建立理论与运动表现之间的联系。
1. Newton’s Laws of Motion & Linear Kinematics | 牛顿运动定律与线性运动学
Newton’s First Law (Law of Inertia): An object will remain at rest or continue in uniform straight-line motion unless acted upon by a net external force. In sport, a stationary football stays still until kicked; a skater glides in a straight line until friction or a push changes that state.
牛顿第一定律（惯性定律）：除非受到净外力作用，否则物体将保持静止或匀速直线运动状态。在运动中，静止的足球在被踢之前保持不动；滑冰者沿直线滑行，直到摩擦力或一次蹬冰改变其状态。
F = m a
Newton’s Second Law: The net force acting on an object equals its mass multiplied by its acceleration. This explains why a more muscular sprinter can accelerate a given body mass faster, and why a lighter tennis racket can be swung more quickly through the contact zone.
牛顿第二定律：作用于物体的净力等于其质量乘以加速度。这解释了为什么肌肉发达的短跑运动员能以更大的加速度推动自身，以及为什么较轻的网球拍能在击球区挥动得更快。
Newton’s Third Law: For every action there is an equal and opposite reaction. When a sprinter pushes backwards against the blocks, the blocks push her forwards with equal magnitude. This force couple underpins all ground-reaction forces in running, jumping, and throwing.
牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力。当短跑运动员向后蹬起跑器时，起跑器以相等大小的力向前推动她。这一对力是跑、跳、投掷中所有地面反作用力的基础。
v = u + a t
The first SUVAT equation relates final velocity (v) to initial velocity (u), constant acceleration (a), and time (t). Coaches use this to predict the speed a javelin thrower’s hand reaches during the final acceleration phase.
第一个 SUVAT 方程将末速度 (v) 与初速度 (u)、恒定加速度 (a) 和时间 (t) 联系起来。教练员用此公式来预测标枪运动员在最后加速阶段手部达到的速度。
s = u t + ½ a t²
Displacement (s) under constant acceleration is given by the initial velocity multiplied by time plus one-half of acceleration multiplied by time squared. It helps calculate the distance covered by a bobsleigh from a standing start over a given time.
在恒定加速度下的位移 (s) 等于初速度乘以时间加上加速度乘以时间的平方的一半。它有助于计算雪橇从静止出发在一定时间内行驶的距离。
v² = u² + 2 a s
This equation links final velocity, initial velocity, acceleration, and displacement without involving time. It is useful for estimating the take-off velocity of a long jumper knowing their acceleration distance on the runway.
该方程将末速度、初速度、加速度和位移联系起来而不涉及时间。它可用于根据跳远运动员在跑道上的加速距离来估算其起跳速度。
2. Projectile Motion | 抛物运动
When a body is projected into the air, its trajectory is determined solely by the initial velocity, angle of release, and height of release, assuming air resistance is negligible. The horizontal and vertical components are analysed independently.
身体被抛向空中时，在忽略空气阻力的情况下，其轨迹完全由初速度、出手角度和出手高度决定。水平和垂直分量需独立分析。
vx = u cosθ, vy = u sinθ
The initial horizontal component of velocity (vx) and vertical component (vy) are resolved from the projection velocity (u) at an angle θ to the horizontal. A higher release angle increases vertical lift but reduces horizontal range if too steep.
初速度的水平分量 (vx) 和垂直分量 (vy) 由初始速度 (u) 与水平面的夹角 θ 分解得到。较高的出手角可增加垂直高度，但如果角度太大会减小水平位移。
T = (2 u sinθ) / g
The time of flight (T) depends on the vertical component of velocity and the acceleration due to gravity (g = 9.81 m s⁻²). For a shot-put released from ground level, a larger upward vertical component keeps the shot in the air longer.
飞行时间 (T) 取决于垂直速度分量和重力加速度 (g = 9.81 m s⁻²)。对于从地面推出的铅球，较大的向上垂直分量能使铅球在空中停留更长时间。
H = (u² sin²θ) / 2 g
The maximum height (H) achieved by the projectile’s centre of mass is governed by the square of the vertical component of velocity. In basketball, a higher release creates a higher arc, making it harder for defenders to block.
抛射体质心达到的最大高度 (H) 由垂直速度分量的平方决定。在篮球运动中，较高的出手能产生较高的弧线，使防守队员更难封盖。
R = (u² sin 2θ) / g
Range (R) for a projectile released and landing at the same height is maximised when the release angle is 45°, because sin 2θ reaches its peak of 1. However, in real sports, the optimum angle varies due to release height and air resistance.
当出手点与落地点高度相同时，水平射程 (R) 在出手角为 45° 时最大，因为 sin 2θ 达到最大值1。然而，在实际运动中，由于出手高度和空气阻力，最佳角度会有所不同。
3. Momentum & Impulse | 动量与冲量
p = m v
Linear momentum (p) is the product of an object’s mass and its velocity. A rugby forward with a larger mass moving at the same speed as a back carries greater momentum, making him harder to tackle.
线动量 (p) 是物体质量与其速度的乘积。一名质量更大的橄榄球前锋与后卫以相同速度移动时，他具有更大的动量，因此更难被擒抱。
Impulse = F Δt = Δp
Impulse equals the average force applied multiplied by the time over which it acts, and it equals the change in momentum. Following through a tennis stroke increases contact time, imparting a larger impulse and a faster ball.
冲量等于施加的平均力乘以力作用的时间，也等于动量的变化量。在网球击球中，顺势随挥能增加接触时间，从而传递更大的冲量，使球速更快。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
The law of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision. In a head-on tackle, the combined momentum of the two players just after impact equals the momentum the ball carrier had just before, assuming negligible external forces.
动量守恒定律指出，在孤立系统中，碰撞前后的总动量保持不变。在正面擒抱中，若忽略外力，撞击后两名球员的合动量等于持球者撞击前所具有的动量。
4. Levers & Torque | 杠杆与力矩
Torque (τ) = F × d
Torque (or moment) is the turning effect of a force, calculated as the force (F) multiplied by the perpendicular distance (d) from the pivot to the line of action. Biceps produce torque about the elbow joint to lift a dumbbell, with the elbow acting as the fulcrum.
力矩（或称转矩）是力的转动效应，等于力 (F) 乘以从支点到力作用线的垂直距离 (d)。肱二头肌以肘关节为支点产生力矩来举起哑铃。
Mechanical advantage (MA) = effort arm ÷ resistance arm. In a second-class lever, such as the calf raise (ball of foot as fulcrum, load at ankle, effort through Achilles tendon), the effort arm is longer than the resistance arm, providing MA > 1 and enabling a small effort to lift a large load.
机械效益 (MA) = 力臂 ÷ 阻力臂。在第二类杠杆中，例如提踵（前脚掌为支点，负荷在踝关节，力通过跟腱），力臂长于阻力臂，提供大于1的机械效益，使较小的力可以举起较大的负荷。
The three classes of levers are found throughout the body: first class (e.g., neck extension), second class (calf raise), and third class (bicep curl). Most human levers are third class, favouring speed and range of motion over force production.
人体中可找到三类杠杆：第一类（如颈部伸展）、第二类（提踵）和第三类（肱二头肌弯举）。人体大部分杠杆属于第三类，它们以速度和活动范围为先，而非力量输出。
5. Angular Kinematics & Moment of Inertia | 角运动学与转动惯量
ω = Δθ / Δt
Angular velocity (ω) measures how quickly an object rotates, expressed as the change in angular displacement (Δθ) over time. A gymnast performing a giant swing on the high bar exhibits varying angular velocity, accelerating on the downward phase and decelerating on the way up.
角速度 (ω) 衡量物体旋转的快慢，用角位移变化量 (Δθ) 除以时间来表示。体操运动员在单杠上做大回环时表现出不同的角速度，下降阶段加速，上升阶段减速。
α = Δω / Δt
Angular acceleration (α) is the rate of change of angular velocity. Spinning ice skaters increase angular acceleration by pulling their arms closer to the rotational axis, reducing the moment of inertia and causing a rapid spin.
角加速度 (α) 是角速度的变化率。花样滑冰运动员通过将手臂向旋转轴收拢来增加角加速度，从而减小转动惯量并实现快速旋转。
ac = v² / r = ω² r
Centripetal acceleration (ac) points toward the centre of rotation and keeps an object moving in a circle. A cyclist on a velodrome banking experiences centripetal acceleration, and the track’s tilt helps provide the necessary centripetal force.
向心加速度 (ac) 指向旋转中心，使物体保持圆周运动。在场地自行车赛道上骑行的运动员会经历向心加速度，而赛道倾斜有助于提供所需的向心力。
I = Σ m r² and L = I ω
Moment of inertia (I) is the sum of the products of each particle’s mass and the square of its distance from the axis. Angular momentum (L) is the product of I and ω. A figure skater pulling in her arms reduces I, and since angular momentum is conserved (L constant), ω must increase dramatically.
转动惯量 (I) 是各质点的质量与其到转轴距离平方的乘积之和。角动量 (L) 是 I 与 ω 的乘积。花样滑冰运动员收拢手臂时减小了 I，由于角动量守恒（L 不变），ω 必然会急剧增大。
6. Bernoulli’s Principle & Magnus Effect | 伯努利原理与马格努斯效应
Bernoulli’s principle states that an increase in the velocity of a fluid (air or water) occurs simultaneously with a decrease in pressure. When air flows faster over the curved upper surface of a discus, pressure drops, creating an upward lift force that helps it stay airborne longer.
伯努利原理指出，流体（空气或水）流速增加的同时压强会降低。当空气在铁饼弯曲的上表面流速较快时，压强下降，产生向上的升力，帮助铁饼在空中停留更长时间。
The Magnus effect is the lateral force generated by a spinning object moving through a fluid. A clockwise-spinning football experiences higher pressure on one side and lower on the opposite, causing it to curve in flight—this is the physics behind a bending free kick.
马格努斯效应是旋转物体在流体中运动时产生的侧向力。顺时针旋转的足球一侧压力较高，对侧压力较低，导致它在飞行中弯曲——这正是弧线任意球背后的物理原理。
7. Drag & Lift Forces | 阻力与升力
Fd = ½ Cd ρ A v²
Drag force (Fd) depends on the drag coefficient (Cd), fluid density (ρ), cross-sectional area (A), and velocity squared. Cyclists adopt an aerodynamic tuck to reduce A, while time-trial helmets and skinsuits reduce Cd to minimise the retarding force.
阻力 (Fd) 取决于阻力系数 (Cd)、流体密度 (ρ)、迎风面积 (A) 和速度的平方。自行车选手采用空气动力蜷缩姿势来减小 A，而计时赛头盔和连体服则降低 Cd，以尽可能减小制动力。
Lift force is generated when fluid flows asymmetrically around a body, according to Bernoulli’s principle. Formula 1 cars use inverted wings to produce downward lift (downforce), pressing the tyres onto the track for greater grip during cornering.
根据伯努利原理，当流体不对称地流经物体时会产生升力。一级方程式赛车使用倒置翼面来产生向下的升力（下压力），过弯时将轮胎紧压在赛道上以获得更强的抓地力。
8. Work, Energy & Power | 功、能量与功率
Work (W) = F d cosθ
Work is done when a force moves an object through a displacement, and it is greatest when force and displacement are in the same direction (cos 0° = 1). A weightlifter does positive work lifting the barbell; gravity does negative work as the barbell is lowered.
当力使物体产生位移时便做了功；当力与位移方向相同时功最大（cos 0° = 1）。举重运动员举起杠铃时做正功；在杠铃下放时重力做负功。
KE = ½ m v²
Kinetic energy (KE
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