📚 AS CCEA Physical Education Formula & Theorem Quick Reference | AS CCEA 体育：公式定理速查手册
This quick reference guide compiles the most important formulae, principles and theorems you need for the CCEA AS Physical Education specification. Each section provides a concise explanation of the key relationship, followed by the standard equation. Use this resource to reinforce your understanding of biomechanics, exercise physiology and movement analysis.
本速查手册汇集了 CCEA AS 体育课程中最关键的公式、原理和定理。每个小节先简要说明核心关系，再给出标准方程。利用这份资料，你可以巩固对生物力学、运动生理学和动作分析的理解，为考试做高效准备。
1. Linear Motion Equations (SUVAT) | 线性运动方程
For an object moving with constant acceleration in a straight line, the four SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). These are fundamental to analysing sprints, throws and any linear movement in sport.
对于以恒定加速度沿直线运动的物体，四个 SUVAT 方程将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。它们是分析短跑、投掷以及运动中任何直线运动的基础。
The first equation directly gives the final velocity after a period of acceleration.
第一个方程直接给出经过一段加速时间后的末速度。
v = u + at
The second equation relates final velocity squared to initial velocity squared, displacement and acceleration without involving time.
第二个方程将末速度平方与初速度平方、位移和加速度联系起来，不含时间变量。
v² = u² + 2as
The third equation gives the displacement when initial velocity and time are known, including the effect of acceleration.
第三个方程在已知初速度和时间的情况下给出位移，并计入了加速度的作用。
s = ut + ½at²
The fourth equation calculates displacement from the average of the initial and final velocities over time.
第四个方程通过初末速度的平均值和时间来计算位移。
s = ½(u + v)t
Remember to choose the equation that uses the variables you already know. All quantities are in SI units: metres (m), seconds (s), metres per second (m/s) and metres per second squared (m/s²).
记住，选择包含已知变量的那个方程。所有量均采用国际单位：米 (m)、秒 (s)、米/秒 (m/s) 和米/秒² (m/s²)。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s three laws explain how forces cause motion and are central to understanding sports technique, from starting a sprint to applying a tackle.
牛顿三定律解释了力如何引起运动，是从起跑到实施擒抱等体育技术分析的核心。
The First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by an external resultant force. This explains why a footballer needs a force to change direction.
第一定律（惯性）：除非受到外力的合力作用，否则物体将保持静止或匀速直线运动。这解释了为什么足球运动员需要施加力才能改变方向。
The Second Law gives the quantitative link between force, mass and acceleration.
第二定律定量地给出了力、质量与加速度之间的关系。
F = m a
Where F is resultant force (N), m is mass (kg) and a is acceleration (m/s²). A heavier athlete requires more force to achieve the same acceleration.
其中 F 为合力 (N)，m 为质量 (kg)，a 为加速度 (m/s²)。较重的运动员需要更大的力才能获得相同的加速度。
The Third Law: For every action there is an equal and opposite reaction. When a sprinter pushes back against the blocks, the blocks push forward on the athlete.
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。短跑运动员向后蹬起跑器，起跑器则向前推运动员。
F₁₂ = -F₂₁
The forces act on different bodies, so they do not cancel out.
这对力作用在不同物体上，因此不会相互抵消。
3. Momentum and Impulse | 动量与冲量
Momentum measures the ‘quantity of motion’ of an athlete or object, while impulse quantifies the effect of a force applied over time.
动量衡量运动员或物体的“运动量”，而冲量则量化一段时间内力作用所产生的效果。
Momentum is the product of mass and velocity.
动量是质量与速度的乘积。
p = m v
Impulse is the product of the force applied and the time for which it acts, and it equals the change in momentum.
冲量是作用力与作用时间的乘积，并且等于动量的变化量。
Impulse = F t = Δp = m v – m u
In a tackle, increasing the time of impact reduces the average force, which helps explain the use of follow-through or crumple zones in protective equipment.
在擒抱动作中，延长撞击时间可以减小平均力，这解释了为什么会在护具中利用随挥动作或缓冲区域。
The principle of conservation of momentum states that in the absence of external forces, total momentum before a collision equals total momentum after the collision.
动量守恒定律指出，在没有外力的情况下，碰撞前的总动量等于碰撞后的总动量。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This is used to study collisions between players or between a ball and a bat.
这一原理用于分析球员之间或球与球棒的碰撞。
4. Projectile Motion | 抛体运动
Projectile motion can be analysed by separating the horizontal and vertical components of velocity. Gravity acts only vertically, so horizontal motion remains uniform.
抛体运动可通过分解速度的水平与垂直分量进行分析。重力仅作用于垂直方向，因此水平运动是匀速的。
If a projectile is launched with speed u at an angle θ to the horizontal, the initial horizontal and vertical components are given by:
若抛体以速度 u、与水平方向夹角 θ 发射，则初始水平分量与垂直分量由下式给出：
uₕ = u cosθ, uᵥ = u sinθ
Horizontal displacement is the product of horizontal velocity and time, as there is no horizontal acceleration.
水平位移是水平速度与时间的乘积，因为水平方向上没有加速度。
sₕ = u cosθ × t
Vertical motion uses the constant acceleration equations with a = -g (taking upward as positive):
垂直运动使用匀加速方程，取向上为正，a = -g：
vᵥ = u sinθ – g t
sᵥ = u sinθ × t – ½ g t²
vᵥ² = (u sinθ)² – 2 g sᵥ
Key derivatives include time of flight, maximum height and horizontal range. For a projectile landing at the same level, range R = (u² sin 2θ)/g.
关键物理量包括飞行时间、最大高度和水平射程。对于落在同一水平面的抛体，射程 R = (u² sin 2θ)/g。
5. Angular Motion | 角运动
Almost all sports skills involve rotation around an axis, from a discus throw to a somersault. Angular analogues of the linear equations are valuable for analysing turns and spins.
从掷铁饼到空翻，几乎所有的运动技术都涉及绕轴的转动。线性方程的角运动对应形式对分析转动和旋转很有价值。
Angular velocity measures the rate of change of angular displacement.
角速度衡量角位移的变化率。
ω = θ / t
Angular acceleration describes how quickly angular velocity changes.
角加速度描述角速度变化的快慢。
α = (ω – ω₀) / t
The three angular motion equations, which assume constant angular acceleration, mirror the SUVAT equations:
假设角加速度恒定，下列三个角运动方程与 SUVAT 方程完全对应：
ω = ω₀ + α t
θ = ω₀ t + ½ α t²
ω² = ω₀² + 2 α θ
Moment of inertia (I) measures resistance to angular acceleration, depending on mass and how it is distributed from the axis.
转动惯量 (I) 衡量物体对转动的抵抗能力，取决于物体的质量及其相对于转轴的分布。
I = Σ m r²
Angular momentum remains constant if no external torque acts. This explains why a spinning ice skater speeds up when pulling the arms inward.
若无外力矩作用，角动量守恒。这解释了为什么旋转的滑冰选手收臂后会加速旋转。
L = I ω
Torque causes angular acceleration and obeys an equation analogous to F = m a.
力矩引起角加速度，其关系式类似于 F = m a。
τ = I α
6. Levers and Torque | 杠杆与力矩
The body’s skeletal system functions as a series of levers. Understanding torque and the principle of moments helps analyse movement efficiency and strength requirements.
人体骨骼系统如同一系列杠杆。理解力矩和力矩原理有助于分析动作的效率和力量需求。
Torque (or moment of a force) is the turning effect of a force about a pivot.
力矩是力绕某个转动轴产生的转动效果。
τ = F d
Where d is the perpendicular distance from the pivot to the line of action of the force. In a bicep curl, the biceps muscle exerts a torque about the elbow joint.
其中 d 是从支点到力作用线的垂直距离。在做肱二头肌弯举时，肱二头肌绕肘关节产生一个力矩。
According to the principle of moments, for a lever in equilibrium the sum of clockwise moments equals the sum of anticlockwise moments.
根据力矩原理，处于平衡状态的杠杆，顺时针力矩之和等于逆时针力矩之和。
Σ clockwise moments = Σ anticlockwise moments
Mechanical advantage describes how effectively a lever multiplies the effort force.
机械利益描述杠杆在放大施力方面的有效性。
Mechanical Advantage = Effort Arm / Resistance Arm
In the human body, most third-class levers have a mechanical advantage less than one, meaning they favour speed and range of motion over force.
在人体中，大多数第三类杠杆的机械利益小于 1，意味着它们以牺牲力量为代价，换取了速度和较大的运动范围。
7. Centre of Mass and Stability | 质心与稳定性
The centre of mass (CoM) is the point where all the mass of a body appears to be concentrated. Its position influences balance, stability and the execution of many skills.
质心 (CoM) 是物体质量看似集中的那个点，其位置影响平衡、稳定性和许多技术的发挥。
While there is no single equation, stability is often analysed through the relationship between the CoM, line of gravity and base of support.
虽然没有单一的公式，但通常通过质心、重力作用线和支撑面之间的关系来分析稳定性。
An athlete is more stable when the CoM is low, the base of support is wide and the line of gravity falls within the base. For example, a defensive stance in basketball widens the base, lowers the CoM and improves stability.
当质心较低、支撑面较宽且重力作用线落在支撑面之内时，运动员更稳定。例如，篮球防守姿势加宽支撑面、降低质心，提高了稳定性。
The torque caused by the weight force about the edge of the base determines whether the athlete will topple. The condition for tipping is that the line of gravity moves outside the base.
体重绕支撑边缘产生的力矩决定运动员是否会倾倒。当重力作用线移出支撑面时，将发生倾倒。
In static positions, stability can be enhanced by increasing the moment arm on the side opposite any disturbing force.
在静态姿势中，增大对抗扰动的力臂有助于增强稳定性。
8. Fluid Resistance and Magnus Effect | 流体阻力与马格努斯效应
Objects moving through a fluid (air or water) experience drag forces and, when spinning, a lift force known as the Magnus effect. These principles affect balls, javelins and cyclists.
物体在流体（空气或水）中运动时会受到阻力；当旋转时，还会受到一种称为马格努斯效应的升力。这些原理影响着球、标枪和自行车运动员。
Drag force opposes motion and depends on fluid density (ρ), cross-sectional area (A), velocity (v) and a drag coefficient (C_D).
阻力与运动方向相反，取决于流体密度 (ρ)、横截面积 (A)、速度 (v) 和阻力系数 (C_D)。
F_D ∝ ρ A v²
Streamlining reduces the drag coefficient and the frontal area, while high speeds lead to a sharp increase in drag because of the v² term.
流线型外形可减小阻力系数和迎面面积；而由于存在 v² 项，高速运动时阻力会急剧增加。
The Magnus effect creates a sideways force on a spinning object. Topspin causes a downward force, making a tennis ball dip; backspin creates lift, making a ball float.
马格努斯效应对旋转的物体产生侧向力。上旋产生向下的力，使网球下坠；下旋产生升力，使球飘浮。
F_Magnus ∝ ω × v
The direction of the Magnus force is perpendicular to both the spin axis and the velocity vector, governed by the right-hand rule.
马格努斯力的方向垂直于旋转轴和速度矢量，遵循右手定则。
9. Cardiovascular Calculations | 心血管计算
Monitoring heart rate and cardiac output is essential in exercise physiology. The following formulae help prescribe training intensity and evaluate cardiovascular fitness.
监测心率和心输出量在运动生理学中至关重要。下列公式有助于设定训练强度并评价心血管健康。
Maximum heart rate can be estimated from age, though actual values vary.
最大心率可根据年龄估算，但实际值因人而异。
MHR = 220 – age (years)
The Karvonen formula calculates a target heart rate zone using resting heart rate (RHR) and desired training intensity.
卡氏公式利用安静心率 (RHR) 和期望的训练强度计算靶心率范围。
THR = (MHR – RHR) × Intensity + RHR
For example, an 18‑year‑old with RHR 60 bpm working at 70% intensity would aim for a training heart rate of approximately ((202‑60)×0.7)+60 ≈ 159 bpm.
例如，一名 18 岁、安静心率 60 bpm 的运动员按 70% 强度训练，目标心率约为 ((202‑60)×0.7)+60 ≈ 159 bpm。
Cardiac output is the volume of blood pumped by the heart per minute.
心输出量是心脏每分钟泵出的血量。
Q = SV × HR
Stroke volume (
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