📚 Year 10 CAIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 10 CAIE 体育：公式定理速查手册
This quick reference handbook gathers all the essential formulas, equations and key theorems you need for the Year 10 CAIE Physical Education science components. From basic kinematics to lever systems, heart‑rate calculations and fluid balance, each section presents dual‑language explanations with clear notation. Use it as your on‑the‑go revision companion.
本速查手册汇聚了你学习 Year 10 CAIE 体育科学部分所需的所有核心公式、方程和重要定理。从基础运动学到杠杆系统、心率计算和水平衡，每一节都以中英双语提供清晰说明。把它当作你随时复习的好帮手。
1. Speed, Velocity and Acceleration | 速度、速率与加速度
Speed is the rate of change of distance. Average speed = total distance ÷ total time (v = d ÷ t). Speed is a scalar; adding direction turns it into velocity. Acceleration is the rate of change of velocity: a = (v – u) ÷ t, where u is initial velocity and v is final velocity. In sport, plotting a velocity–time graph allows you to calculate acceleration from the gradient and distance travelled from the area under the line.
速率是距离的变化率。平均速率 = 总距离 ÷ 总时间（v = d ÷ t）。速率是标量；加上方向后就成为速度。加速度是速度的变化率：a = (v – u) ÷ t，其中 u 为初速度，v 为末速度。在运动中，绘制速度–时间图可以从斜率计算加速度，从曲线下面积计算移动距离。
2. Newton's Laws of Motion | 牛顿运动定律
Newton' s acted upon by an external force. Second law (acceleration): force = mass × acceleration, F = m × a. Third law (reaction): every action has an equal and opposite reaction. These laws explain sprint starts, collisions in rugby and the propulsion swimmers generate against water.
牛顿三大定律支配着体育运动中的所有运动。第一定律（惯性）：物体保持静止或匀速直线运动，除非受到外力作用。第二定律（加速度）：力 = 质量 × 加速度，F = m × a。第三定律（反作用）：每一个作用力都有一个大小相等、方向相反的反作用力。这些定律解释了短跑起跑、橄榄球碰撞以及游泳运动员向后推水推进身体的反作用。
3. Force, Mass and Weight | 力、质量与重量
Mass is the amount of matter in a body (kg) and does not change with location. Weight is a force caused by gravity: W = m × g ction) and air resistance also affect sporting performance; streamlined positions reduce drag.
质量是物体所含物质的多少（kg），不随位置改变。重量是由重力引起的力：W = m × g，地球上 g = 9.8 m s⁻²。在生物力学中，重量始终通过重心垂直向下。摩擦力（F_摩擦 = μ × 法向反作用力）和空气阻力同样影响运动表现；流线型姿势可以减小阻力。
4. Momentum and Impulse | 动量与冲量
Linear momentum measures how much motion a body has: p = m × v. Impulse equals the change in momentum and is the product of force and time: Impulse = F × Δt = Δp. A longer force application time—such as following through in a throw or kick—increases impulse and thus the velocity of the object. Coaches use this relationship to improve bat swing and shot‑put delivery.
线动量衡量物体运动的多少：p = m × v。冲量等于动量的变化量，是力与时间的乘积：冲量 = F × Δt = Δp。较长的施力时间——如投掷或踢球的随前动作——能增大冲量，从而提高物体的速度。教练员利用这一关系改进挥棒和推铅球的出手动作。
5. Work, Power and Energy | 功、功率与能量
Work is done when a force moves an object: W = F × d × cos θ (when force is parallel, W = F × d). Power is the rate of doing work: P = W ÷ t. Two important energy forms in sport are kinetic energy (KE = ½ × m × v²) and gravitational potential energy (GPE = m × g × h). Energy transfers between these forms explain why a pole‑vaulter’s run‑up speed converts into height.
当力使物体移动时做功：W = F × d × cos θ（力与位移平行时，W = F × d）。功率是做功的速率：P = W ÷ t。运动中两种重要的能量形式是动能（KE = ½ × m × v²）和重力势能（GPE = m × g × h）。这两种形式之间的能量转换解释了撑竿跳高运动员如何将助跑速度转化为高度。
6. Levers and Moments | 杠杆与力矩
A lever consists of an effort, a load and a fulcrum. The turning effect of a force is called a moment: Moment = Force × perpendicular distance from pivot. For a lever in equilibrium (balanced), effort × effort arm = load × load arm. In the human body, bones act as levers, joints as pivots and muscles supply the effort. Examples include the biceps curl (third‑class lever) and standing on tiptoes (second‑class lever).
杠杆由动力、阻力和支点组成。力的转动效应称为力矩：力矩 = 力 × 力臂（支点的垂直距离）。当杠杆处于平衡状态时，动力 × 动力臂 = 阻力 × 阻力臂。在人体中，骨骼充当杠杆，关节是支点，肌肉提供动力。例子包括肱二头肌弯举（第三类杠杆）和提踵站起（第二类杠杆）。
7. Mechanical Advantage of Levers | 杠杆的机械优势
Mechanical advantage (MA) tells you how much a lever multiplies the effort force: MA = effort arm ÷ load arm. When the effort arm is longer than the load arm (MA > 1), the lever reduces the effort needed—useful in speed levers such as a javelin throw. When MA < 1, the lever increases range and speed of movement at the expense of force, typical of third‑class levers in the body like the elbow during a throw.
机械优势（MA）反映杠杆放大力度的倍数：MA = 动力臂 ÷ 阻力臂。当动力臂比阻力臂长（MA > 1）时，杠杆省力——用于速度杠杆如掷标枪。当 MA < 1 时，杠杆以牺牲力为代价增大运动幅度和速度，这类第三类杠杆在人体中很常见，如投掷时的肘关节。
8. Centre of Mass and Stability (Toppling Angle) | 重心与稳定性（倾倒角）
The centre of mass (CoM) is the point where all the body's mass appears to be concentrated. Stability depends on CoM height and base‑of‑support width. The maximum tilt angle before toppling can be estimated using: tan θ = (½ × base width) ÷ height of CoM. A lower CoM and wide base increase θ, making a stance more stable—think of a defensive basketball stance or a sumo wrestler. In gymnastics, a balanced handstand requires the CoM to be directly over the hands.
重心是所有质量看起来集中的点。稳定性取决于重心高度和支撑面宽度。倾倒前的最大倾斜角可用以下公式估算：tan θ = (½ × 支撑面宽度) ÷ 重心高度。更低的重心和更宽的支撑面能增大 θ，使站姿更稳定——想象篮球防守站姿或相扑力士。在体操中，平衡倒立需要重心刚好落在手掌上方。
9. Maximum Heart Rate and Target Heart Rate | 最大心率与靶心率
The simplest predictor of maximum heart rate is: HR_max = 220 – age (years) ariation. Target heart rate for exercise is a specified percentage of HR_max. For moderate‑intensity aerobic work, aim for 60–70% HR_max; for vigorous activity, 70–85% e the athlete is working at the correct intensity to develop aerobic or anaerobic fitness.
最大心率的最简单预估公式是：HR_max = 220 – 年龄（岁）。这只是一个估计值，存在个体差异。运动的靶心率是HR_max的特定百分比。中等强度有氧运动可设定在60–70% HR_max；剧烈活动设定在70–85% HR_max。通常监测训练区间以确保运动员在以正确强度锻炼有氧或无氧能力。
10. Karvonen Formula (Heart Rate Reserve) | 卡沃宁公式（心率储备）
The Karvonen formula provides a more personalised target heart rate by incorporating resting heart rate (RHR). Formula: Target HR = RHR + (HR_max – RHR) × % intensity. For example, a 15‑year‑old with RHR = 70 bpm aiming for 65% ethod better reflects fitness differences and is widely used in training programme design.
卡沃宁公式通过纳入静息心率（RHR）提供更个性化的靶心率。公式：靶心率 = RHR + (HR_max – RHR) × % 强度。例如，一名15岁少年 RHR = 70 bpm，目标强度65%：HR_max = 205 bpm，心率储备 = 135 bpm，因此靶心率 ≈ 70 + 0.65 × 135 = 158 bpm。此方法更能反映体能差异，被广泛用于训练计划设计。
11. Body Mass Index (BMI) | 体重指数
BMI is a simple screening tool for body composition: BMI = body mass (kg) ÷ height² (m²). For adults, BMI classifications are: underweight <18.5, normal 18.5–24.9, overweight 25–29.9, obese ≥30. In younger athletes, BMI‑for‑ ular athlete may be misclassified as overweight; it is just one piece of health profiling.
BMI 是一种简单的身体成分筛查工具：BMI = 体重 (kg) ÷ 身高² (m²)。成人分类标准为：偏瘦 <18.5，正常 18.5–24.9，超重 25–29.9，肥胖 ≥30。对青少年运动员，需使用年龄‑BMI 百分位图。注意 BMI 无法区分肌肉和脂肪，肌肉发达的运动员可能被误判为超重；它只是健康画像的一个组成部分。
12. Sweat Rate Calculation and Fluid Balance | 出汗量计算与水平衡
Athletes can estimate their sweat rate to optimise hydration. A practical formula is: Sweat rate (L/h) = [Pre‑exercise mass (kg) – Post‑exercise mass (kg) + Fluid consumed during exercise (L)] ÷ Exercise duration (h) es 1.2 kg in 90 min and drinks 0.5 L: sweat rate = (1.2 + 0.5) ÷ 1.5 ≈ 1.13 L/h. Replacing ~150% of fluid losses helps accelerate recovery.
运动员可以估算出汗量以优化补液。实用公式：出汗速率 (L/h) = [运动前体重 (kg) – 运动后体重 (kg) + 运动中补液量 (L)] ÷ 运动时长 (h)。为更精确有时会减去排尿量。例如，一名足球运动员 90 分钟内体重减轻 1.2 kg 并喝水 0.5 L：出汗速率 = (1.2 + 0.5) ÷ 1.5 ≈ 1.13 L/h。补充大约 150% 的体液损失有助于加速恢复。
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