📚 Formulas and Theorems Quick Reference | 公式定理速查手册
Welcome to your Year 10 Cambridge Physical Education quick-reference handbook. This guide distils the essential formulas and scientific principles that underpin human movement, training and biomechanics. Mastering these concepts will sharpen your analytical skills and enhance your practical performance in sport.
欢迎使用Year 10剑桥体育公式定理速查手册。本手册浓缩了支撑人体运动、训练和生物力学的基本公式与科学原理。掌握这些概念将提升你的分析能力，并优化你的运动表现。
1. Speed and Velocity | 速度与速率
Speed is the rate at which an object covers distance, regardless of direction. It is a scalar quantity measured in metres per second (m/s).
速度是物体通过距离的速率，与方向无关，是标量，单位为米每秒 (m/s)。
average speed = total distance / total time v = s / t
Velocity describes both the speed and the direction of motion, making it a vector quantity. For motion in a straight line, velocity can be positive or negative depending on the chosen direction.
速度既描述运动的快慢也描述方向，因此是矢量。在直线运动中，速度根据选定的方向可取正值或负值。
2. Acceleration | 加速度
Acceleration is the rate of change of velocity. It is a vector quantity and in sport it often results from applying a force. The SI unit is metres per second squared (m/s²).
加速度是速度的变化率，为矢量，在运动中通常由施加力产生。国际单位是米每二次方秒 (m/s²)。
a = (v - u) / t
where u is the initial velocity, v is the final velocity and t is the time taken. A positive value indicates speeding up in the direction of motion, while a negative value (deceleration) indicates slowing down.
其中 u 为初速度，v 为末速度，t 为时间。正值表示朝运动方向加速，负值表示减速。
3. Newton's Laws of Motion | 牛顿运动定律
Newton's three laws form the foundation for understanding force and motion in physical activity.
牛顿三大定律是理解体育活动中力与运动的基础。
First Law – Inertia: An object will remain at rest or move with constant velocity unless acted upon by an external resultant force.
第一定律—惯性定律：一切物体总保持匀速直线运动状态或静止状态，直到有外力迫使它改变这种状态为止。
Second Law – Acceleration: The acceleration of an object is directly proportional to the resultant force and inversely proportional to its mass.
第二定律—加速度定律：物体的加速度与所受合外力成正比，与质量成反比。
F = m a
In sport, a sprinter drives more force into the ground to produce greater acceleration. Weight is a specific force: W = m g, where g = 9.8 m/s².
在运动中，短跑运动员向地面施加更大的力以获得更大的加速度。重力是一种特殊的力：W = m g，g 取 9.8 m/s²。
Third Law – Action-Reaction: For every action force there is an equal and opposite reaction force. When a swimmer pushes water backwards, the water pushes the swimmer forwards.
第三定律—作用力与反作用力：两个物体之间的作用力和反作用力总是大小相等，方向相反。游泳运动员向后推水，水则向前推运动员。
4. Moment of Force and Levers | 力矩与杠杆
The moment of a force (torque) measures the turning effect about a pivot. It depends on the force applied and the perpendicular distance from the pivot to the line of action of the force.
力矩衡量力对支点的转动效应，取决于所施加的力以及支点到力作用线的垂直距离。
moment = force × perpendicular distance M = F × d
In a lever system, equilibrium occurs when the clockwise moments equal the anticlockwise moments: effort × effort arm = load × load arm.
在杠杆系统中，当顺时针力矩等于逆时针力矩时达到平衡：动力 × 动力臂 = 阻力 × 阻力臂。
The three classes of lever appear throughout the human body and sport equipment:
三类杠杆在人体和运动器械中普遍存在：
| Class | Fulcrum position | Example in sport |
|---|---|---|
| First class | Between effort and load | Heading a football (neck joint), see-saw |
| Second class | Load between fulcrum and effort | Standing on tiptoes (ankle joint), wheelbarrow |
| Third class | Effort between fulcrum and load | Biceps curl (elbow joint), tennis racket swing |
5. Projectile Motion | 抛射体运动
A projectile is any object released into the air and influenced only by gravity and air resistance. In sport, shot puts, footballs and long jumpers follow a parabolic path.
抛射体指任何被抛入空中且仅受重力和空气阻力影响的物体。在体育中，铅球、足球和跳远运动员都沿抛物线轨迹运动。
The key principle is that horizontal and vertical motions are independent. Horizontal velocity remains constant (ignoring air resistance), while vertical motion is affected by gravity.
关键原理是水平运动和竖直运动相互独立。水平速度保持不变（忽略空气阻力），竖直运动受重力影响。
For a projectile launched with speed u at an angle θ to the horizontal:
对于以初速度 u、与水平方向成 θ 角抛出的物体：
Horizontal range, R = (u² sin 2θ) / g
Maximum height, H = (u² sin² θ) / (2g)
can be calculated using the vertical component u sin θ and the equation s = u t + ½ a t².
其中 g = 9.8 m/s²。在不计空气阻力时，获得最大水平射程的最佳角度为 45°。滞空时间可利用竖直分量 u sin θ 和公式 s = u t + ½ a t² 计算。
6. Work, Energy and Power | 功、能量与功率
When a force moves an object, work is done. Energy is the capacity to do work, and power measures the rate of energy transfer.
力使物体移动时做功。能量是做功的本领，功率衡量能量转移的速率。
Work done = force × distance W = F d
The unit is the joule (J). In sport, two important forms of mechanical energy are kinetic energy (KE) and gravitational potential energy (GPE).
单位为焦耳 (J)。在运动中，两种重要的机械能是动能 (KE) 和重力势能 (GPE)。
KE = ½ m v²
GPE = m g h
and h is height above a reference level. Energy is conserved; a pole-vaulter converts KE into GPE and back.
其中 m 为质量，v 为速度，g = 9.8 m/s²，h 为参考零点以上的高度。能量守恒：撑竿跳高运动员将动能转化为重力势能再转化回来。
Power is calculated as work done divided by time, or force multiplied by velocity.
功率等于做功除以时间，也等于力乘以速度。
P = W / t or P = F v
The unit is the watt (W). Explosive athletes, such as sprinters, develop high power outputs over short periods.
单位为瓦特 (W)。爆发型运动员如短跑选手能在短时间内输出高功率。
7. Centre of Mass and Stability | 质心与稳定性
The centre of mass is the point in an object where its mass is considered to be concentrated. In a uniform human body standing upright, it lies roughly at the navel, but it shifts during movement.
质心是物体质量集中的点。人体直立时质心大约位于肚脐处，但在运动中会移动。
Stability refers to the ability to resist being toppled. Three factors increase stability:
稳定性指抵抗倾倒的能力。三个因素可增强稳定性：
1. A lower centre of mass – bending the knees in a rugby tackle.
1. 降低质心高度——橄榄球擒抱时屈膝。
2. A wider base of support – standing with feet apart in a defensive stance.
2. 增大支撑面积——防守站立时双脚分开。
3. The line of gravity falling within the base of support – shifting weight to keep the vertical line from the centre of mass inside the feet.
3. 重力线落在支撑面内——调整体重使质心的垂线保持在双脚范围内。
The moment the line of gravity moves outside the base, a turning moment is created and stability is lost.
一旦重力线移出支撑面，就会产生力矩，稳定性丧失。
8. Drag Forces and Fluid Resistance | 阻力与流体阻力
Drag is a force that opposes the motion of an object through a fluid (air or water). In sport, reducing drag improves speed and efficiency.
阻力是物体在流体（空气或水）中运动时受到的与其运动方向相反的力。在运动中，减小阻力可提高速度和效率。
Drag depends on several factors: the density of the fluid, the cross-sectional area of the object, the speed squared, and the shape (drag coefficient).
阻力大小取决于多个因素：流体密度、物体的迎风截面积、速度的平方以及形状（阻力系数）。
Fdrag = ½ ρ v² Cd A
A cyclist adopts a streamlined position to reduce frontal area (A) and drag coefficient (Cd). Swimmers wear tight suits to lower skin friction. Because drag increases with v², doubling the speed quadruples the drag force.
自行车运动员采取流线型姿势以减小迎风面积 (A) 和阻力系数 (Cd)。游泳运动员穿紧身泳衣以减少表面摩擦。由于阻力与 v² 成正比，速度加倍时阻力将变为原来的四倍。
9. Momentum and Impulse | 动量与冲量
Momentum is the product of an object's mass and its velocity. It is a vector quantity that describes the 'quantity of motion'.
动量是物体质量与速度的乘积，是描述“运动量”的矢量。
momentum = mass × velocity p = m v
Impulse measures the effect of a force applied over a time interval and equals the change in momentum.
冲量衡量力在一段时间内的作用效果，等于动量的变化量。
impulse = force × time = Δp J = F t = m v - m u
In a collision or when a ball is struck, a larger force applied for a longer time produces a greater change in momentum. This principle explains follow-through in striking sports: extending contact time increases impulse and thus final velocity.
在碰撞或击球时，较大的力作用较长时间可产生更大的动量变化。这一原理解释了击球类运动的随挥动作：延长接触时间可以增加冲量，从而增大末速度。
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