📚 Year 10 CIE Physical Education: Formula & Theorem Quick Reference Handbook | Year 10 CIE 体育：公式定理速查手册
This handbook brings together the essential equations, biomechanical laws, and key numerical relationships you need for your Year 10 CIE Physical Education course. Each section presents a formula or principle, explains how it works in sport, and provides a worked example so you can apply it confidently in your written paper and practical analysis.
本手册汇集了 Year 10 CIE 体育课程中必考的公式、生物力学定律和关键数量关系。每一小节都会提供一个公式或原理，解释它在运动中的运作方式，并给出计算示例，帮助你自信地应对笔试和实践分析。
1. Speed (Linear Velocity) | 速度（线速度）
Speed is how fast a body moves from one point to another. It is a scalar quantity and is calculated by dividing distance covered by time taken. In sport, measuring speed helps evaluate sprinters, swimmers and cycling performance.
速度是物体从一点移动到另一点的快慢程度，属标量，由位移距离除以所用时间计算。在运动中，测量速度有助于评价短跑、游泳和自行车选手的表现。
- Formula: speed = distance ÷ time
- 公式：速度 = 距离 ÷ 时间
A 100 m sprinter runs the distance in 12.5 s. Speed = 100 ÷ 12.5 = 8 m/s.
一名 100 米短跑运动员用 12.5 秒跑完全程。速度 = 100 ÷ 12.5 = 8 米/秒。
2. Velocity (Linear Velocity with Direction) | 速度（带方向的线速度）
Velocity is speed in a given direction. It is a vector quantity. A change in direction alters velocity even if speed stays the same. In team games, change-of-direction ability is a key athletic quality.
速度是在特定方向上的速率，属矢量。即使速率保持不变，方向改变也会改变速度。在团体项目中，变向能力是关键的运动素质。
- Formula: velocity = displacement ÷ time
- 公式：速度 = 位移 ÷ 时间
A winger runs 40 m north in 5 s. Velocity = 40 ÷ 5 = 8 m/s north. Displacement must be measured in a straight line from start to finish.
一名边锋向北跑 40 米，用时 5 秒。速度 = 40 ÷ 5 = 8 米/秒 向北。位移必须按起点到终点的直线距离测量。
3. Acceleration | 加速度
Acceleration is the rate at which velocity changes over time. It can be positive (speeding up) or negative (deceleration). Sprint starts, braking, and sudden stops all involve high accelerations.
加速度是速度随时间变化的快慢程度。可为正（加速）或负（减速）。短跑起跑、刹车和急停都涉及较大的加速度。
- Formula: acceleration = (final velocity − initial velocity) ÷ time
- 公式：加速度 = (末速度 − 初速度) ÷ 时间
A cyclist increases velocity from 2 m/s to 10 m/s in 4 s. Acceleration = (10 − 2) ÷ 4 = 2 m/s².
一名自行车运动员在 4 秒内将速度从 2 米/秒提升至 10 米/秒。加速度 = (10 − 2) ÷ 4 = 2 米/秒²。
4. Force and Newton's Second Law | 力与牛顿第二定律
The size of a force determines how quickly an object accelerates. According to Newton's second law, force equals mass multiplied by acceleration. Greater force applied to a lighter object produces larger acceleration.
力的大小决定物体加速的快慢。根据牛顿第二定律，力等于质量乘以加速度。对较轻的物体施加更大的力会产生更大的加速度。
- Formula: force = mass × acceleration (F = m × a)
- 公式：力 = 质量 × 加速度 (F = m × a)
A 70 kg rugby player accelerates at 3 m/s². Force = 70 × 3 = 210 N (newtons). This force is generated by leg muscles pushing against the ground.
一名 70 公斤的橄榄球运动员以 3 米/秒² 加速。力 = 70 × 3 = 210 牛。该力由腿部肌肉推蹬地面产生。
5. Momentum | 动量
Momentum is the quantity of motion an object has. It depends on mass and velocity. In contact sports, a heavier and faster player carries more momentum, making them harder to stop or tackle.
动量是物体具有的运动量，取决于质量和速度。在身体接触类项目中，体重更大、速度更快的运动员动量更大，更难被拦停或擒抱。
- Formula: momentum = mass × velocity (p = m × v)
- 公式：动量 = 质量 × 速度 (p = m × v)
A 85 kg footballer moving at 6 m/s has momentum = 85 × 6 = 510 kg·m/s. To halt this player, an equal impulse must be applied in the opposite direction.
一名 85 公斤的足球运动员以 6 米/秒移动，动量 = 85 × 6 = 510 千克·米/秒。要使其停下，必须在相反方向施加等量的冲量。
6. Impulse and Its Relationship to Momentum | 冲量及其与动量的关系
Impulse is the change in momentum caused by a force acting over a period of time. Coaches use the impulse-momentum relationship to teach longer follow-through in striking sports, increasing contact time and thus impulse.
冲量是力在一段时间内作用所导致的动量变化。教练利用冲量-动量关系指导击打类运动中的延长随挥，以增加接触时间从而提高冲量。
- Formula: impulse = force × time = change in momentum
- 公式：冲量 = 力 × 时间 = 动量的变化量
A tennis racket exerts 120 N on the ball for 0.05 s. Impulse = 120 × 0.05 = 6 N·s. This impulse changes the ball's momentum by 6 kg·m/s.
网球拍对球施加 120 牛的力，持续 0.05 秒。冲量 = 120 × 0.05 = 6 牛·秒。该冲量使球的动量改变 6 千克·米/秒。
7. Levers and Mechanical Advantage | 杠杆与机械利益
Muscles and bones work as lever systems. Mechanical advantage tells us whether the lever is built for speed and range of motion or for force production. Understanding lever class helps analyse movement efficiency in the body.
肌肉和骨骼组成杠杆系统。机械利益可以说明该杠杆是为了速度和运动幅度而设计，还是为了产生力量。理解杠杆类别有助于分析人体运动的效率。
- Formula: mechanical advantage = effort arm ÷ resistance arm
- 公式：机械利益 = 力臂 ÷ 阻力臂
In the elbow joint (third-class lever), the biceps effort arm is about 5 cm while the resistance arm to the hand may be 35 cm. Mechanical advantage = 5 ÷ 35 = 0.14, which favours speed of movement at the hand.
在肘关节（第三类杠杆）中，肱二头肌的力臂约 5 厘米，而到手部的阻力臂可能为 35 厘米。机械利益 = 5 ÷ 35 = 0.14，这有利于手部的运动速度。
8. Centre of Mass and Stability | 质心与稳定性
The centre of mass is the point where the body's mass is equally distributed in all directions. Stability increases when the centre of mass is low and the base of support is wide. This principle is vital in wrestling, rugby scrimmages and gymnastics landings.
质心是身体质量在所有方向上均匀分布的点。质心越低、支撑面越宽，稳定性就越高。该原理在摔跤、橄榄球 scrum 和体操落地中至关重要。
Key concept: stability ∝ 1 / (height of centre of mass) and stability ∝ base width
关键概念：稳定性与质心高度成反比，与支撑面宽度成正比
A gymnast landing with feet shoulder-width apart and knees flexed lowers the centre of mass and broadens the base, maximising stability.
体操运动员双脚分开与肩同宽、屈膝落地，可降低质心并扩大支撑面，从而最大程度提高稳定性。
9. Projectile Motion: Parabolic Trajectory | 抛体运动：抛物线轨迹
A projectile launched into the air follows a parabolic path determined by its initial velocity, angle of release and height of release. The optimal release angle for maximum range on a flat field is 45°, ignoring air resistance.
抛入空中的物体遵循抛物线路径，由初速度、出手角度和出手高度决定。忽略空气阻力时，平坦场地上获得最大远度的最佳出手角度为 45°。
Factors affecting trajectory: release speed (most influential), release angle, release height
影响轨迹的因素：出手速度（影响最大）、出手角度、出手高度
A shot putter releases the shot at 40° and 2.2 m height. Because the release point is above the landing point, the optimum angle is slightly less than 45°, typically around 38°–42°.
铅球运动员在 40° 角和 2.2 米高度出手。由于出手点高于落地点，最佳角度略低于 45°，通常在 38°–42° 左右。
10. Bernoulli's Principle and Lift Forces | 伯努利原理与升力
Bernoulli's principle states that faster airflow over a surface creates lower pressure. This principle helps explain lift on a discus, the swing of a football, and the Magnus effect in spinning balls.
伯努利原理指出，表面上方气流越快，压力越低。该原理可解释铁饼的升力、足球的弧线以及旋转球的马格努斯效应。
Corollary: lift force ∝ velocity² and ∝ surface area; faster moving fluid = lower pressure
推论：升力与速度的平方和表面积成正比；流体流速越快，压力越低
A spinning football moving through the air experiences a pressure difference: low pressure on one side (faster airflow) and high pressure on the opposite, creating a sideways force that bends the ball.
旋转的足球在空中运行时，一侧气流较快、压力较低，另一侧压力较高，从而产生侧向力使球弯曲飞行。
11. Energy and Work Done | 能量与做功
Work is done when a force moves an object in the direction of the force. In sport, muscular work is converted into kinetic energy. Power is the rate at which work is done — a crucial factor for explosive events.
当力使物体沿力的方向移动时，即做了功。运动中，肌肉做功转化为动能。功率是做功的速率，是爆发力项目的关键因素。
- Work done = force × distance moved in direction of force
- 做功 = 力 × 沿力方向的移动距离
- Power = work done ÷ time
- 功率 = 做功 ÷ 时间
A weightlifter lifts 1500 N a vertical distance of 2 m in 1.5 s. Work = 1500 × 2 = 3000 J. Power = 3000 ÷ 1.5 = 2000 W.
一名举重运动员在 1.5 秒内将 1500 牛的重量垂直提起 2 米。做功 = 1500 × 2 = 3000 焦耳。功率 = 3000 ÷ 1.5 = 2000 瓦。
12. Torque in Rotational Movements | 旋转运动中的力矩
Torque causes rotation around an axis. In the body, muscles create torque at joints to produce angular motion. The amount of torque depends on the force applied and the perpendicular distance from the joint axis.
力矩引起绕轴的旋转。在人体中，肌肉在关节处产生力矩以实现角运动。力矩的大小取决于施加的力以及到关节轴的垂直距离。
- Formula: torque = force × perpendicular distance from pivot (moment arm)
- 公式：力矩 = 力 × 到支点的垂直距离（力臂）
When kicking a ball, the quadriceps extend the knee. If the patellar tendon exerts 800 N at a perpendicular distance of 0.05 m from the knee axis, torque = 800 × 0.05 = 40 N·m.
踢球时，股四头肌伸展膝关节。若髌腱在距离膝关节轴 0.05 米处施加 800 牛的力，力矩 = 800 × 0.05 = 40 牛·米。
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