📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick-reference guide compiles the essential formulas and theorems from the Year 12 OCR Physical Education biomechanics syllabus. Having these equations at your fingertips will accelerate your analysis of human movement, force interactions, and sporting techniques.
本速查手册汇集了 Year 12 OCR 体育课程生物力学部分的核心公式与定理。熟记这些方程，能让你更快地分析人体运动、力的相互作用和运动技术。
1. Newton’s First Law (Inertia) | 牛顿第一定律（惯性）
A body will remain at rest or continue to move with constant velocity unless acted upon by a net external force. In sport, a stationary football needs a kick to move, and a sprinter’s body continues moving forward unless friction, air resistance or a braking force is applied.
物体将保持静止或匀速直线运动状态，除非受到净外力作用。在运动中，静止的足球需要被踢才会移动，短跑运动员的身体在不受摩擦、空气阻力或制动力的作用下会继续向前。
2. Newton’s Second Law (F = ma) | 牛顿第二定律（F = ma）
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law explains why a lighter cricket ball accelerates more rapidly than a shot when the same force is applied, and why athletes in weight categories tend to produce greater accelerations.
F = m a
其中 F 为合力（单位 N），m 为质量（kg），a 为加速度（m/s²）。物体的加速度与所受合外力成正比，与质量成反比。这一原理解释了为何施加相同力时，板球比铅球加速更快，以及为何同体重级别的运动员更容易产生大加速度。
3. Newton’s Third Law (Action–Reaction) | 牛顿第三定律（作用力与反作用力）
For every action force there is an equal and opposite reaction force. When a swimmer pushes water backwards, the water pushes the swimmer forwards. The forces act on different bodies, so they do not cancel out. In sprinting, the foot drives against the blocks, and the blocks propel the athlete forwards.
每一个作用力都有一个大小相等、方向相反的反作用力。游泳运动员向后推水时，水将他向前推进。由于这两个力作用在不同物体上，它们不会互相抵消。短跑中，脚蹬起跑器，起跑器将运动员向前送出。
4. Linear Kinematic Equations | 线性运动学方程
These equations describe uniformly accelerated linear motion without the need for multiple force measurements. They are used to calculate velocity, displacement or time in sprint starts, long jump take-offs and other rectilinear movements.
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
这里的 u 是初速度，v 是末速度，a 是恒定加速度，t 是时间，s 是位移。这四个方程用于匀加速直线运动，帮助计算短跑起跑、跳远踏跳等直线运动的位移、速度或时间。
5. Momentum and Impulse | 动量与冲量
Momentum is the product of mass and velocity. Impulse is the change in momentum caused by a force applied over a time interval. In rugby tackles, a longer collision time reduces the peak force, decreasing injury risk. Coaches use impulse to break a fall or improve release speed in throwing events.
p = m v
Impulse = F Δt = Δp = m v − m u
动量 p 是质量与速度的乘积，冲量等于力与其作用时间的乘积，也等于动量的变化。在橄榄球扑搂中，延长碰撞时间能降低峰值力，减少受伤风险。投掷项目中，教练利用冲量原理增加器械出手速度。
6. Levers and Torque | 杠杆与力矩
A lever system consists of an effort force, a load and a fulcrum. Torque (moment of force) causes rotation and depends on both force magnitude and its perpendicular distance from the pivot. The body’s limbs act as lever systems, where muscles provide effort to overcome resistance.
τ = F d
Mechanical advantage = effort arm / resistance arm
力矩 τ 等于力 F 乘以力到支点的垂直距离 d。机械效益等于动力臂与阻力臂之比。人体四肢构成杠杆，肌肉收缩提供动力以克服阻力。例如，肘关节弯曲时肱二头肌施加力，前臂作为杠杆。
7. Angular Motion | 角运动
In rotation, angular velocity (ω) measures the rate of change of angular displacement. Centripetal force keeps an object moving in a circle, acting towards the centre. These concepts are vital for analysing discus throws, hammer turns and gymnastic swings.
ω = Δθ / Δt
v = r ω
a = v² / r = r ω²
F = m v² / r
角速度 ω 等于角位移除以时间，线速度 v 等于半径 r 乘以角速度。向心加速度 a 等于 v²/r 或 rω²，向心力 F 是使物体维持圆周运动的净力。在掷铁饼、链球旋转和体操大回环中，运动员通过控制半径和角速度来优化线速度。
8. Projectile Motion | 抛体运动
Projectiles follow a parabolic path determined by initial velocity, angle of release and relative height of release. The horizontal component of velocity remains constant if air resistance is ignored, while the vertical component changes uniformly due to gravity.
Horizontal displacement: sₓ = (v cosθ) t
Vertical displacement: sᵧ = (v sinθ) t − ½ g t²
水平位移 sₓ 等于初速度的水平分量乘以时间；垂直位移 sᵧ 受重力加速度 g（9.81 m/s²）影响。最佳抛射角度，当出手高度高于落点时小于 45°，在跳远和篮球投篮中需要依据相对高度进行调整。
9. Fluid Mechanics (Bernoulli’s Principle & Magnus Effect) | 流体力学（伯努利原理与马格努斯效应）
Bernoulli’s principle states that faster fluid flow creates lower pressure. This explains lift on a discus or javelin. The Magnus effect arises when a spinning ball drags air, creating a pressure difference that curves its path. Topspin in tennis causes the ball to dip, while backspin extends flight.
Pressure + ½ ρ v² + ρ g h = constant
Magnus force ∝ spin rate × velocity × cross‑section
伯努利原理表明，流速越快压强越小，这为铁饼和标枪提供了升力。马格努斯效应是由于旋转球体带动周围空气，导致两侧压力差而使轨迹弯曲。网球的上旋使球下坠，下旋则延长飞行距离。
10. Centre of Mass and Stability | 重心与稳定性
The centre of mass (CoM) is the point where the entire mass of a body appears to be concentrated. An athlete is most stable when the line of gravity falls inside the base of support and the CoM is low. Widening the stance and lowering the body are fundamental strategies for improving balance.
Stability ∝ (base of support) × (CoM height)⁻¹
重心是身体全部质量集中的假想点。当重力作用线落在支撑面内且重心较低时，运动员最稳定。加宽站距、降低身体是提高平衡能力的基本手段。体操落地和橄榄球争球都运用了这些原理。
11. Energy and Work | 能量与功
Kinetic energy, gravitational potential energy and mechanical work are conserved in isolated systems. Understanding these transfers helps explain why a pole-vaulter stores elastic energy in the pole, and why a cyclist coasts faster downhill.
KE = ½ m v²
GPE = m g h
W = F d cosθ
Power = W / t
动能 KE 与速度平方成正比，重力势能 GPE 取决于高度。做功 W 是力在位移方向上的分量与位移的乘积。功率是做功的速率。撑竿跳高运动员将动能转化为弹性势能再转化为重力势能，下坡自行车手将重力势能转化为动能。
12. Summary Table of Key Formulas | 关键公式汇总表
The table below provides a one‑glance reference for the most frequently examined formulas in OCR Year 12 PE biomechanics. Use it as a quick check before answering analysis questions.
下表一目了然地汇总了 OCR Year 12 体育生物力学中最常考的公式，用作分析题前的快速核对。
| Formula (English) | 公式（中文） | Application / 应用 |
|---|---|---|
| F = m a | 力 = 质量 × 加速度 | Sprinting start / 短跑起跑 |
| v = u + a t | 末速度 = 初速度 + 加速度 × 时间 | Uniform acceleration / 匀加速运动 |
| Impulse = F Δt = Δp | 冲量 = 力 × 时间 = 动量变化 | Tackle, landing / 扑搂、落地缓冲 |
| τ = F d | 力矩 = 力 × 力臂 | Joint rotation / 关节转动 |
| ω = Δθ / Δt | 角速度 = 角位移 / 时间 | Discus spin / 铁饼旋转 |
| sᵧ = (v sinθ) t − ½ g t² | 垂直位移 = 初速度垂直分量×时间−½ gt² | Shot put height / 铅球飞行高度 |
| KE = ½ m v² | 动能 = ½ × 质量 × 速度² | Sprinting speed / 冲刺速度 |
| W = F d cosθ | 功 = 力 × 位移 × cosθ | Weightlifting / 举重做功 |
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