Year 12 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 体育：公式定理速查手册

📚 Year 12 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 体育：公式定理速查手册

Welcome to your one-stop quick reference for all key formulas and theorems covered in Year 12 CAIE Physical Education (9396). From cardiovascular dynamics to projectile motion, these equations will help you master the biomechanical and physiological principles required for the AS-level exam. Each section pairs concise explanations with worked examples, ensuring you can quickly recall and apply the right formula under exam conditions.

欢迎来到 Year 12 CAIE 体育（9396）关键公式定理一站式速查手册。从心血管动力学到抛射体运动，这些方程将助你掌握 AS 考试所需的生物力学和生理学原理。每个公式均配有简要说明与示例，帮助你在考场上快速回忆并准确应用。

1. Cardiac Dynamics | 心脏动力学

Cardiac output (Q) is the volume of blood pumped by each ventricle per minute. It is the product of heart rate (HR) and stroke volume (SV).

心输出量（Q）是每个心室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

A resting value for an untrained individual is typically around 5 L/min (70 bpm × 70 ml). During maximal exercise, Q can exceed 20 L/min in elite athletes. Mean arterial blood pressure (BP) can also be expressed through total peripheral resistance (TPR):

未受训者安静时心输出量通常约为 5 L/min（70 bpm × 70 ml）。精英运动员在极限运动时心输出量可超过 20 L/min。平均动脉血压（BP）也可用总外周阻力（TPR）表示：

BP = Q × TPR

The classic estimation of maximum heart rate is:

经典的最大心率估算公式为：

HRmax ≈ 220 − age (years)

This value is used to set training intensity zones.

该值可用于设定训练强度区间。

2. Respiratory Dynamics | 呼吸动力学

Minute ventilation (VE) measures the total volume of air moved in and out of the lungs per minute. It depends on tidal volume (TV) and breathing frequency (f).

每分钟通气量（VE）衡量每分钟进出肺部的空气总量，由潮气量（TV）和呼吸频率（f）决定。

VE = TV × f

At rest, TV ~0.5 L and f ~12 breaths/min give VE ~6 L/min. During heavy exercise, TV can increase to 3 L and f to 40 breaths/min, pushing VE above 120 L/min. Alveolar ventilation (VA) accounts for dead space (DS) and is more directly related to gas exchange:

安静时，潮气量约 0.5 L，呼吸频率约 12 次/分，VE 约为 6 L/min。剧烈运动时潮气量可增至 3 L，呼吸频率达 40 次/分，VE 可超过 120 L/min。肺泡通气量（VA）扣除了死腔量（DS），与气体交换更直接相关：

VA = (TV − DS) × f

Typical anatomical dead space is about 150 ml in a healthy adult.

健康成人的解剖死腔量通常约为 150 ml。

3. Linear Kinematics | 线运动学

Kinematics describes motion without reference to its causes. The core quantities are speed, velocity and acceleration.

运动学描述运动本身而不涉及力的成因，核心量包括速率、速度和加速度。

Average speed = distance travelled / time taken

Average velocity (v) = displacement (Δs) / time (Δt)

Speed is a scalar, while velocity is a vector. Acceleration is the rate of change of velocity:

速率是标量，速度是矢量。加速度是速度的变化率：

a = (v − u) / t

where u = initial velocity, v = final velocity, t = time.

其中 u = 初速度，v = 末速度，t = 时间。

4. Linear Kinetics (Newton’s Laws & Momentum) | 线动力学（牛顿定律与动量）

Newton’s Second Law links net force to acceleration:

牛顿第二定律将合外力与加速度联系起来：

F = m × a

Momentum (p) is the product of mass and velocity:

动量（p）是质量与速度的乘积：

p = m × v

The impulse-momentum relationship states that the change in momentum equals the applied impulse:

冲量-动量关系表明，动量的变化等于施加的冲量：

Impulse = F × t = Δp = m(v − u)

This principle explains how increasing the time of force application (e.g., a follow-through) can alter an object’s final velocity.

该原理解释了延长受力时间（如随挥动作）如何改变物体的末速度。

5. Angular Motion | 角运动

Angular velocity (ω) measures rotation rate:

角速度（ω）衡量转动快慢：

ω = angular displacement (θ) / time (t)

Linear velocity at a point on a rotating lever is related to angular velocity and the radius of rotation:

旋转杠杆上一点的线速度与角速度和转动半径有关：

v = ω × r

For an object moving in a circle, centripetal acceleration and force are given by:

做圆周运动的物体，其向心加速度和向心力由下式给出：

a = v² / r = ω² r

F = m v² / r = m ω² r

Moment of inertia (I) depends on mass and its distribution from the axis:

转动惯量（I）取决于质量及其到转轴的分布：

I = Σ m r²

Angular momentum (L) is conserved when no external torque acts:

当无外力矩作用时，角动量（L）守恒：

L = I × ω

6. Projectile Motion | 抛射体运动

Assuming negligible air resistance, the horizontal and vertical components of the initial velocity (V) for a projectile launched at angle θ are:

忽略空气阻力时，以初速度 V、角度 θ 发射的抛射体，其水平和竖直分量为：

Vₓ = V cos θ， Vᵧ = V sin θ

Key kinematic equations for a projectile released and landing at the same height:

在同高度抛出与落地的抛射体运动中，关键运动学方程如下：

Time of flight： T = 2 V sin θ / g

Maximum height： H = (V sin θ)² / (2g)

Range： R = V² sin(2θ) / g

where g = acceleration due to gravity (9.81 m/s²). The optimum angle for maximum range in a vacuum is 45°.

其中 g = 重力加速度（9.81 m/s²）。真空中获得最大射程的最佳发射角为 45°。

7. Levers and Moments | 杠杆与力矩

The moment (torque) of a force about a fulcrum is:

力对支点产生的力矩（扭矩）为：

Moment = Force × perpendicular distance from fulcrum

Mechanical advantage (MA) indicates the efficiency of a lever system:

机械优势（MA）反映杠杆系统的效率：

MA = effort arm / resistance arm

When MA > 1, the lever amplifies force (e.g., a plantar flexor acting as a second-class lever). When MA < 1, the lever amplifies speed and range of motion (e.g., a third-class lever like the elbow flexors).

当 MA > 1 时，杠杆省力（如跖屈肌构成的 II 类杠杆）；当 MA < 1 时，杠杆放大速度和运动幅度（如肘屈肌构成的 III 类杠杆）。

8. Training Heart Rates & Body Composition | 训练心率与身体构成

Training zones are often calculated using the Karvonen (heart rate reserve) method:

训练区间常用卡氏（储备心率）法计算：

Target HR = Resting HR + (Intensity × (HRmax − Resting HR))

where intensity is expressed as a decimal (e.g., 0.7 for 70% effort).

其中强度以小数表示（如 70% 为 0.7）。

Body Mass Index (BMI) is a simple indicator of body composition:

身体质量指数（BMI）是简单的身体组成指标：

BMI = mass (kg) / height² (m)

A value of 18.5–24.9 kg/m² is considered healthy for adults. Additionally, waist-to-hip ratio (WHR = waist circumference / hip circumference) is used to assess fat distribution and health risk.

成人健康 BMI 范围为 18.5–24.9 kg/m²。此外，腰臀比（WHR = 腰围 / 臀围）用于评估脂肪分布与健康风险。

9. Reaction Time & Hick’s Law | 反应时间与希克定律

Response time is the sum of reaction time (decision-making phase) and movement time:

应答时间是反应时间（决策阶段）与动作时间的总和：

Response time = Reaction time + Movement time

Hick’s Law describes how reaction time increases with the number of stimulus-response alternatives:

希克定律描述反应时间如何随刺激-反应选项数增加而延长：

RT = a + b log₂(n)

RT = reaction time
n = number of equally probable choices
a, b = empirically derived constants

RT = 反应时间
n = 等概率选项数量
a, b = 经验常数

In sport, a goalkeeper facing a penalty kick has a higher n and thus slower RT compared to a simple sprint start. Coaches can use the single-channel hypothesis and psychological refractory period to understand performance under time pressure.

在体育中，守门员面对点球时 n 更高，因而反应时间比简单的短跑起跑更慢。教练可利用单通道假说和心理不应期来理解时间压力下的表现。

10. Fluid Forces in Sport | 体育中的流体力学

Drag force opposes motion through a fluid. It is approximated by:

阻力是流体中阻碍运动的力，近似由下式计算：

F_d = ½ C_d ρ A v²

C_d = drag coefficient (shape-dependent)
ρ = fluid density
A = cross-sectional area
v = velocity relative to fluid

C_d = 阻力系数（取决于形状）
ρ = 流体密度
A = 横截面积
v = 相对流体速度

The Magnus effect occurs when a spinning object generates a pressure difference. According to Bernoulli’s principle, faster airflow on one side reduces pressure, creating a lift force perpendicular to the direction of motion. This explains the swerve of a spinning football or the topspin dip in tennis.

马格努斯效应因旋转物体产生压力差而出现。根据伯努利原理，物体一侧流速较快导致压力降低，产生垂直于运动方向的升力。这解释了旋转足球的弧线偏转或网球上旋球的下坠。

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Year 12 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 体育：公式定理速查手册