📚 Year 13 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 13 CAIE 体育：公式定理速查手册
This quick-reference handbook compiles essential formulas and theorems required for the CAIE A2 Physical Education (9696) syllabus. It focuses on the biomechanics and exercise physiology sections, providing you with the equations you need to apply in analysis and problem solving. Use this guide alongside your course notes to reinforce your understanding of linear and angular motion, forces, energy, fluid dynamics, and cardiorespiratory calculations.
本速查手册汇集了 CAIE A2 体育（9696）课程所需的核心公式与定理，重点覆盖运动生物力学和运动生理学部分。你将找到在分析与解题时需要用到的关键方程，包括直线与角运动、力、能量、流体动力学以及心肺计算方法。请结合课堂笔记使用本指南，以强化对这些内容的理解。
1. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law (Inertia): A body will remain at rest or continue to move with constant velocity unless acted upon by an external resultant force.
牛顿第一定律（惯性定律）：物体将保持静止或匀速直线运动状态，除非受到外力的作用。
Newton’s Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This is expressed as F = m a, where F is the resultant force, m is mass, and a is acceleration.
牛顿第二定律：物体的加速度与作用在其上的净外力成正比，与其质量成反比。公式表达为 F = m a，其中 F 为合力，m 为质量，a 为加速度。
F = m a
Newton’s Third Law: For every action, there is an equal and opposite reaction.
牛顿第三定律：作用力与反作用力大小相等、方向相反，并作用在不同物体上。
2. Linear Kinematics | 直线运动学
Linear kinematics describes motion in a straight line. The four key equations (suvat) link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
直线运动学描述沿直线的运动。四个关键方程（suvat）将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。
Equation 1: v = u + a t (no displacement)
方程1：v = u + a t （不含位移）
v = u + a t
Equation 2: s = u t + ½ a t² (no final velocity)
方程2：s = u t + ½ a t² （不含末速度）
s = u t + ½ a t²
Equation 3: s = ½ (u + v) t (no acceleration)
方程3：s = ½ (u + v) t （不含加速度）
s = ½ (u + v) t
Equation 4: v² = u² + 2 a s (no time)
方程4：v² = u² + 2 a s （不含时间）
v² = u² + 2 a s
3. Angular Kinematics | 角运动学
Angular kinematics deals with rotational motion. The equations mirror the linear suvat set, replacing linear quantities with their angular counterparts: angular displacement (θ), initial angular velocity (ω₀), final angular velocity (ω), angular acceleration (α), and time (t).
角运动学处理旋转运动。方程与直线运动的 suvat 相似，只需将直线量替换为对应的角量：角位移 θ、初角速度 ω₀、末角速度 ω、角加速度 α 和时间 t。
Angular equation 1: ω = ω₀ + α t
角运动方程1：ω = ω₀ + α t
ω = ω₀ + α t
Angular equation 2: θ = ω₀ t + ½ α t²
角运动方程2：θ = ω₀ t + ½ α t²
θ = ω₀ t + ½ α t²
Angular equation 3: θ = ½ (ω₀ + ω) t
角运动方程3：θ = ½ (ω₀ + ω) t
θ = ½ (ω₀ + ω) t
Angular equation 4: ω² = ω₀² + 2 α θ
角运动方程4：ω² = ω₀² + 2 α θ
ω² = ω₀² + 2 α θ
4. Momentum and Impulse | 动量与冲量
Linear momentum (p) is the product of an object’s mass and its velocity. It is a vector quantity.
线动量 p 是物体质量与速度的乘积，是一个矢量。
p = m v
Impulse (J) is the change in momentum, equal to the average force multiplied by the time for which it acts.
冲量 J 是动量的变化量，等于平均力与作用时间的乘积。
J = F Δt = Δp = m v – m u
The principle of conservation of momentum states that in a closed system, total momentum before an event (e.g., collision) equals total momentum after the event, provided no external forces act.
动量守恒定律指出，在没有外力作用的系统中，碰撞等事件发生前后的总动量保持不变。
5. Work, Energy and Power | 功、能量和功率
Work (W) is done when a force causes displacement in the direction of the force.
当力使物体在力的方向上产生位移时，力对物体做了功。
W = F d cos θ
Kinetic energy (KE) is the energy possessed by an object due to its motion.
动能 KE 是物体由于运动而具有的能量。
KE = ½ m v²
Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field.
重力势能 GPE 是物体因位于重力场中的某一高度而具有的能量。
GPE = m g h
Power (P) is the rate of doing work or the rate of energy transfer. For constant velocity, it can also be calculated as force times velocity.
功率 P 是做功的速率或能量转换的速率。在恒定速度下，也可用力乘以速度计算。
P = W / t = F v
6. Rotational Dynamics and Levers | 转动动力学与杠杆
Torque (τ, also called moment of force) is the turning effect of a force about a pivot. It depends on the size of the force and the perpendicular distance from the pivot to the line of action.
力矩 τ（亦称为力的力矩）是力绕支点产生的转动效应。它取决于力的大小以及从支点到力作用线的垂直距离。
τ = F d (where d is the moment arm)
Moment of inertia (I) is the rotational equivalent of mass. It depends on both mass and its distribution relative to the axis of rotation.
转动惯量 I 是转动的惯性量度，相当于直线运动中的质量。它取决于质量及其相对于旋转轴的分布。
I = Σ m r²
Newton’s second law for rotation links torque, moment of inertia and angular acceleration.
转动形式的牛顿第二定律将力矩、转动惯量和角加速度联系起来。
τ = I α
Angular momentum (L) is the product of moment of inertia and angular velocity. In the absence of external torque, angular momentum is conserved.
角动量 L 是转动惯量与角速度的乘积。当没有外力矩作用时，角动量守恒。
L = I ω
A lever system amplifies an effort force to overcome a resistance. Mechanical advantage (MA) is the ratio of effort arm to resistance arm.
杠杆系统用于放大施力以克服阻力。机械利益 MA 是力臂与阻力臂的比值。
MA = Effort arm / Resistance arm
7. Projectile Motion | 抛体运动
A projectile’s path is resolved into independent horizontal and vertical components. The horizontal velocity remains constant (ignoring air resistance), while the vertical motion is affected by gravity (g = 9.81 m s⁻²).
抛体的轨迹可分解为独立的水平分量和垂直分量。水平速度保持不变（忽略空气阻力），而垂直运动受重力影响（g = 9.81 m s⁻²）。
Horizontal and initial vertical components for a launch speed u at angle θ:
初速度为 u、发射角为 θ 时的水平和初速度垂直分量：
u_x = u cos θ, u_y = u sin θ
Time of flight (T) to return to the same vertical level:
返回到同一水平高度时的飞行时间 T：
T = 2 u sin θ / g
Maximum height (H) reached:
最大高度 H：
H = u² sin² θ / (2 g)
Range (R) – horizontal distance travelled:
水平射程 R：
R = u² sin(2θ) / g
8. Fluid Mechanics: Bernoulli and Magnus | 流体力学：伯努利与马格努斯效应
Bernoulli’s principle states that in a streamlined fluid flow, an increase in velocity occurs simultaneously with a decrease in pressure. For a horizontal flow, this simplifies to:
伯努利原理指出，在流线型流体中，流速增加的同时压强减小。在水平流动中，可简化为：
P + ½ ρ v² = constant (for horizontal flow)
The full Bernoulli equation includes the gravitational potential term:
完整的伯努利方程包含重力势能项：
P + ½ ρ v² + ρ g h = constant
The Magnus effect explains the curved flight path of a spinning ball. Air pressure differences on opposite sides of a spinning object create a force perpendicular to the direction of motion. Topspin creates a downward force; backspin creates an upward lift.
马格努斯效应解释了旋转球体的弯曲飞行轨迹。旋转物体两侧的气压差产生垂直于运动方向的力。上旋球产生向下的力，下旋球产生向上的升力。
9. Cardiovascular Calculations | 心血管计算
Cardiac output (Q) is the volume of blood pumped by the heart per minute. It is the product of heart rate (HR) and stroke volume (SV).
心输出量 Q 是心脏每分钟泵出的血液量，等于心率 HR 与每搏输出量 SV 的乘积。
Q = HR × SV
Oxygen consumption (VO₂) is the volume of oxygen used by the body per minute. It can be calculated using the Fick equation:
摄氧量 VO₂ 是身体每分钟利用的氧气体积，可用菲克方程计算：
VO₂ = Q × (a-v O₂ difference)
Where a-v O₂ difference is the difference in oxygen content between arterial and mixed venous blood.
其中 a-v O₂ 差为动脉血与混合静脉血之间的氧含量差。
Maximal oxygen uptake (VO₂ max) is a key indicator of aerobic endurance, often expressed in ml kg⁻¹ min⁻¹.
最大摄氧量 VO₂ max 是有氧耐力的关键指标，通常用 ml kg⁻¹ min⁻¹ 表示。
10. Respiratory Calculations and RER | 呼吸计算与呼吸交换率
Minute ventilation (VE) is the volume of air breathed in or out per minute. It depends on breathing frequency (f) and tidal volume (TV).
每分通气量 VE 是每分钟吸入或呼出的气体体积，取决于呼吸频率 f 和潮气量 TV。
VE = f × TV
The respiratory exchange ratio (RER) is the ratio of carbon dioxide produced (VCO₂) to oxygen consumed (VO₂). It indicates the predominant fuel source during exercise (e.g., RER = 0.7 for fat, RER = 1.0 for carbohydrate).
呼吸交换率 RER 是二氧化碳产生量 VCO₂ 与摄氧量 VO₂ 的比值，用于判断运动中的主要能量来源（如 RER = 0.7 表示脂肪供能，RER = 1.0 表示碳水化合物供能）。
RER = VCO₂ / VO₂
11. Friction and Mechanical Efficiency | 摩擦力与机械效率
Friction (F_f) is the force opposing motion between two surfaces in contact. The limiting friction is proportional to the normal reaction force (R) and the coefficient of friction (μ).
摩擦力 F_f 是接触面之间阻碍运动的力。最大静摩擦力与法向反作用力 R 和摩擦系数 μ 成正比。
F_f = μ R
Mechanical efficiency evaluates how effectively the body converts metabolic energy into external mechanical work. Net efficiency (%) is the ratio of mechanical work output to the energy expended above resting level.
机械效率评估身体将代谢能转化为外部机械功的有效程度。净效率（%）是机械功输出与超出静息水平的能量消耗之比。
Net efficiency (%) = (Mechanical work / Energy expenditure above rest) × 100
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