📚 Year 11 CIE Physical Education: Formula and Theorem Quick Reference | CIE 体育公式定理速查手册
This quick reference handbook compiles the essential formulas, equations and biomechanical principles required for the Year 11 CIE Physical Education syllabus. Each entry is presented with a concise explanation in English followed by its Chinese equivalent, helping bilingual learners master the quantitative and theoretical aspects of sports science.
本速查手册汇编了 Year 11 CIE 体育课程所需的关键公式、方程和生物力学原理。每个条目先提供英文简明解释，再给出对应的中文释义，帮助双语学习者掌握运动科学中的定量与理论内容。
1. Speed, Velocity and Acceleration | 速率、速度与加速度
speed = d / t
Speed (s) is a scalar quantity measured as distance travelled per unit time. The standard unit is metres per second (m/s). It gives no information about direction.
速率是标量，表示单位时间内通过的距离。标准单位是米每秒 (m/s)，不包含方向信息。
v = s / t
Velocity (v) is a vector; it equals displacement (s) divided by time (t) and includes both magnitude and direction. A change in velocity can result from a change in speed or direction.
速度是矢量，等于位移 (s) 除以时间 (t)，同时具有大小和方向。速率或方向的改变都会导致速度变化。
a = (v – u) / t
Acceleration (a) is the rate of change of velocity. u is initial velocity, v is final velocity and t is the time taken. Measured in m/s².
加速度 (a) 表示速度的变化率。u 为初速度，v 为末速度，t 为所用时间，单位为 m/s²。
v² = u² + 2as
This equation links final velocity, initial velocity, acceleration and displacement (s) without directly using time. It is useful for analysing sprints or throwing events when time is not given.
该公式将末速度、初速度、加速度和位移 (s) 联系起来而不直接使用时间，在短跑或投掷分析中时间未知时非常有用。
2. Newton's Laws of Motion | 牛顿运动定律
A solid grasp of Newton's three laws is essential for explaining starting, stopping and changing direction in sport.
掌握牛顿三定律对于解释运动中起跑、停止和变向至关重要。
F = ma
Newton's Second Law: the net force (F) acting on an object equals its mass (m) multiplied by its acceleration (a). A larger force produces a greater acceleration for the same mass.
牛顿第二定律：作用于物体的合外力 (F) 等于质量 (m) 与加速度 (a) 的乘积。质量一定时，力越大加速度越大。
W = mg
strength (g ≈ 9.8 m/s² on Earth). Weight differs from mass, which is the amount of matter.
重力 (W) 是由引力产生的力，等于质量 (m) 乘以引力场强度 (g，地球上约 9.8 m/s²)。重量不同于质量。
3. Momentum and Impulse | 动量与冲量
Momentum helps explain collisions, tackles and the follow-through in striking skills.
动量有助于解释碰撞、擒抱及击球技术中的随挥动作。
p = mv
Linear momentum (p) is the product of mass (m) and velocity (v). A heavier or faster athlete has greater momentum and is harder to stop.
线动量 (p) 是质量 (m) 与速度 (v) 的乘积。体重更大或速度更快的运动员动量更大，更难被阻挡。
Impulse = FΔt = Δp
Impulse is the force (F) applied over a time interval (Δt) and equals the change in momentum (Δp). Increasing contact time (e.g. follow-through) reduces the peak force and improves control.
冲量是力 (F) 在时间间隔 (Δt) 内的作用量，等于动量的变化量 (Δp)。延长接触时间 (如随挥) 可减小峰值力并提升控制。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
In a closed system, total momentum before collision equals total momentum after collision. This principle of conservation of momentum governs impacts in rugby tackles and cue sports.
在封闭系统中，碰撞前的总动量等于碰撞后的总动量。动量守恒定律支配着橄榄球擒抱和台球中的碰撞。
4. Work, Energy and Power | 功、能与功率
These concepts quantify the energy cost and intensity of human movement.
这些概念用于量化人体运动的能量消耗和强度。
W = Fd
Work done (W) equals force (F) applied in the direction of displacement multiplied by that displacement (d). No work is done if the force is perpendicular to movement.
做功 (W) 等于沿位移方向的力 (F) 乘以位移 (d)。如果力与运动方向垂直，则不做功。
KE = ½ mv²
Kinetic energy (KE) is the energy of motion. Doubling the velocity quadruples the kinetic energy, which explains why faster sprinters possess substantially more energy.
动能 (KE) 是物体由于运动而具有的能量。速度翻倍会使动能增至四倍，这解释了为何快速跑者拥有高得多的动能。
GPE = mgh
Gravitational potential energy (GPE) depends on mass (m), gravity (g) and height (h). It is stored energy that can be converted into kinetic energy during downhill running or diving.
重力势能 (GPE) 取决于质量 (m)、重力加速度 (g) 和高度 (h)。它是在下坡跑或跳水时可转化为动能的储存能量。
P = W / t and P = Fv
Power (P) is the rate of doing work. It can also be expressed as force (F) multiplied by velocity (v) when the force is constant. High power output is critical in explosive events like weightlifting.
功率 (P) 是做功的速率。在力恒定时也可表示为力 (F) 与速度 (v) 的乘积。举重等爆发性项目需要高功率输出。
5. Levers and Mechanical Advantage | 杠杆与机械利益
Lever systems in the body determine force and speed trade-offs. Mechanical advantage (MA) quantifies the efficiency of a lever.
身体中的杠杆系统决定了力与速度之间的权衡。机械利益 (MA) 量化了杠杆的效率。
MA = Effort arm / Resistance arm
Mechanical advantage is the ratio of the distance from the fulcrum to the effort (effort arm) to the distance from the fulcrum to the resistance (resistance arm). An MA greater than 1 means less effort is needed to overcome the resistance, favouring force production.
机械利益是支点到力点的距离（力臂）与支点到阻力点的距离（阻力臂）之比。MA 大于 1 表示用较小的力即可克服阻力，有利于产生大力。
MA = Load / Effort
Alternatively, MA can be calculated as the load (resistance) divided by the effort (force applied). This ratio also indicates the mechanical advantage of a lever.
机械利益也可用负荷 (阻力) 除以施力 (作用力) 来计算。该比值同样表示杠杆的机械优势。
6. Torque and Principle of Moments | 力矩与力矩原理
Torque causes rotation and is fundamental to understanding turning forces in gymnastics, diving and joint movement.
力矩引起转动，是理解体操、跳水和关节运动中旋转力的基础。
Moment = F × d
The moment of a force (torque) is the force (F) multiplied by the perpendicular distance (d) from the pivot. Increasing the lever arm increases the torque, making it easier to rotate an object or body segment.
力矩等于力 (F) 乘以力到转动轴的垂直距离 (d)。增大力臂可增大力矩，使转动更容易。
Σ clockwise moments = Σ anticlockwise moments
For a body in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot. This principle explains balances on a beam or the stability of a wrestler's stance.
处于平衡状态的物体，对于任意支点，顺时针力矩之和等于逆时针力矩之和。该原理可解释平衡木上的平衡或摔跤选手站姿的稳定性。
7. Projectile Motion | 抛射体运动
Projectile principles predict the flight path of a shot put, a soccer ball or a long jumper. Air resistance is usually ignored in CIE PE calculations.
抛射体原理可预测铅球、足球或跳远运动员的飞行轨迹。CIE 体育计算中通常忽略空气阻力。
Time of flight: t = (2v₀ sinθ) / g
The total time a projectile stays in the air depends on the initial velocity (v₀), launch angle (θ) and gravity (g). A larger vertical component extends flight time.
抛射体在空中的总时间取决于初速度 (v₀)、发射角 (θ) 和重力 (g)。垂直分速度越大，飞行时间越长。
Maximum height: h = (v₀² sin²θ) / (2g)
The peak height is reached when the vertical velocity becomes zero. It increases with initial speed and a steeper launch angle.
抛射体达到最高点时垂直速度为零。最大高度随初速度和发射角增大而增加。
Range: R = (v₀² sin 2θ) / g
° (sin 2θ = 1). The same range can be achieved by complementary angles (e.g. 30° and 60°).
水平射程在发射角为 45° 时最大 (sin 2θ = 1)。互补角度 (如 30° 与 60°) 可获得相同的射程。
8. Angular Motion | 角运动
Angular quantities are used to describe spinning, twisting and swinging movements in sport.
角运动量用于描述体育中的旋转、扭转和摆动动作。
ω = Δθ / Δt
Angular velocity (ω) is the rate of change of angular displacement (θ). It is measured in radians per second (rad/s).
角速度 (ω) 是角位移 (θ) 的变化率，单位为弧度每秒 (rad/s)。
α = Δω / Δt
Angular acceleration (α) is the rate of change of angular velocity. It describes how quickly a spinning ice skater's rotation speeds up or slows down.
角加速度 (α) 是角速度的变化率，描述旋转中的花样滑冰选手加速或减速的快慢。
v = rω and a = rα
Linear velocity (v) and acceleration (a) at a point on a rotating body are related to the radius (r) and angular values. A longer tennis racket produces a faster head speed for the same angular velocity.
旋转体上一点的线速度 (v) 和线加速度 (a) 与半径 (r) 及角量相关。在相同角速度下，较长的网球拍能产生更快的拍头速度。
Centripetal acceleration: a = v² / r = ω²r
An object moving in a circular path experiences acceleration directed toward the centre. This centripetal acceleration keeps a hammer thrower's implement on its circular path.
做圆周运动的物体具有指向圆心的加速度。这种向心加速度使链球保持在圆形轨迹上。
Centripetal force: F = mv² / r = mω²r
Centripetal force is the net force causing circular motion. It increases with mass, speed and tighter radius, explaining why cyclists lean heavily on steep banked tracks.
向心力是产生圆周运动的合外力，随质量、速度和半径减小而增大，这解释了自行车运动员在陡峭弯道上需要大幅倾斜的原因。
9. Heart Rate Calculations | 心率计算
Monitoring heart rate (HR) allows athletes to train within specific intensity zones. Several formulas are used in CIE PE to prescribe exercise intensity.
监测心率可以让运动员在特定强度区间训练。CIE 体育中常用以下公式设定运动强度。
HRmax = 220 – age
-year-old, predicted HRmax is 204 bpm. This formula has a standard error of about ±10 bpm.
最简单的心率最大值估算公式。16 岁运动员的预测最大心率约为 204 bpm。该公式存在约 ±10 bpm 的标准误差。
HRmax = 208 – (0.7 × age)
A more accurate age-predicted maximum heart rate formula, especially for young people. It yields slightly different values that often align better with actual measured HRmax.
更为准确的年龄预测最大心率公式，尤其适合青少年，所得数值通常更接近实测 HRmax。
Target HR = ((HRmax – HRrest) × %intensity) + HRrest
The Karvonen formula determines target heart rate by using heart rate reserve (HRmax – HRrest) multiplied by the desired intensity and then adding resting HR. It personalises training zones more effectively than a simple percentage of HRmax.
卡沃宁公式利用储备心率 (HRmax – HRrest) 乘以目标强度百分比后再加上安静心率来计算目标心率，比单纯使用最大心率的百分比更能个性化地设定训练区间。
10. Body Composition and BMI | 身体成分与 BMI
Body mass index (BMI) and body fat percentage are used to assess health risks and suitability for certain sports.
身体质量指数 (BMI) 和体脂率用于评估健康风险及是否适合某些运动项目。
BMI = mass (kg) / (height (m
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