📚 Pre-U CIE Physical Education: Formula & Theorem Quick Reference | Pre-U CIE 体育：公式定理速查手册
This handbook presents essential formulas and theorems for the Cambridge Pre-U Physical Education specification. Each section provides the formula in a clear, centred format, followed by a bilingual explanation of its application. Use this resource to reinforce your understanding of biomechanics, physiology, and movement analysis.
这本手册提供了剑桥 Pre-U 体育课程中的核心公式与定理。每个部分以清晰的居中格式展示公式，随后提供中英双语的应用解释。请利用此资源巩固你对运动生物力学、运动生理学和运动分析的理解。
1. Speed and Velocity | 速度与速率
Speed is a scalar quantity measuring how fast an object moves. Velocity is a vector, specifying both magnitude and direction. The basic equation links distance, time, and speed.
速率是标量，衡量物体运动的快慢。速度是矢量，同时指明大小和方向。基本方程将距离、时间与速率联系起来。
v = s / t
where s = displacement (m), t = time taken (s), and v = velocity (m·s⁻¹). For speed, replace displacement with distance travelled. This formula is used to calculate average sprint speed or ball speed in sport.
其中 s = 位移（米），t = 所用时间（秒），v = 速度（米/秒）。对于速率，用路程代替位移。该公式用于计算体育中的平均冲刺速度或球的速度。
2. Acceleration | 加速度
Acceleration describes the rate of change of velocity. It is a key concept in analysing starting speed, changes in direction, and deceleration phases in sport.
加速度描述速度变化的快慢。它是分析运动中起跑速度、方向变化和减速阶段的关键概念。
a = (v − u) / t
where u = initial velocity (m·s⁻¹), v = final velocity (m·s⁻¹), t = time (s), yielding acceleration a in m·s⁻². Negative a indicates deceleration.
其中 u = 初速度（米/秒），v = 末速度（米/秒），t = 时间（秒），得出的加速度 a 单位为米/秒²。负值表示减速。
3. Equations of Uniformly Accelerated Motion | 匀加速运动方程
When acceleration is constant, three classical equations relate displacement, velocity, and time. They are invaluable for analysing projectile motion, sprint starts, or long jump take-off.
当加速度恒定时，有三个经典方程关联位移、速度和时间。它们对于分析抛射体运动、短跑起跑或跳远起跳非常有用。
v = u + at
s = ut + ½ at²
v² = u² + 2as
where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement. These are collectively known as the SUVAT equations.
其中 u = 初速度，v = 末速度，a = 加速度，t = 时间，s = 位移。这些方程统称为 SUVAT 方程。
4. Force and Newton’s Laws of Motion | 力与牛顿运动定律
Newton’s Second Law states that the net force acting on a body equals the product of its mass and acceleration. This principle is fundamental to all movement in sport, from pushing off the blocks to throwing a javelin.
牛顿第二定律指出，作用在物体上的合外力等于其质量与加速度的乘积。这一原理是所有体育运动动作的基础，从起跑器蹬出到投掷标枪。
F = m a
F = net force (N), m = mass (kg), a = acceleration (m·s⁻²). For a given force, smaller mass gives larger acceleration, explaining why lighter athletes can accelerate faster.
F = 合外力（牛顿），m = 质量（千克），a = 加速度（米/秒²）。对于给定的力，较小的质量会产生较大的加速度，这解释了为什么体重较轻的运动员可以更快加速。
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 acting over time. These concepts explain tackling in rugby, follow-through in racket sports, and the effect of impact time on force.
动量是质量与速度的乘积。冲量是由力在一段时间作用引起的动量变化。这些概念解释了英式橄榄球中的擒抱、拍类运动中的随挥动作，以及作用时间对冲击力的影响。
p = m v
Impulse = F t = Δ p = m v – m u
where p = momentum (kg·m·s⁻¹), F = average force (N), t = time of contact (s), Δp = change in momentum. Increasing contact time reduces peak force, important in injury prevention.
其中 p = 动量（千克·米/秒），F = 平均力（牛顿），t = 接触时间（秒），Δp = 动量变化。延长接触时间可降低峰值力，这对预防运动损伤至关重要。
6. Torque and Levers | 力矩与杠杆
Torque (moment of force) is the turning effect produced by a force applied at a distance from an axis of rotation. The human body operates as a system of levers, with joints as fulcrums.
力矩（力的转动效应）是力在离转动轴一定距离处作用时产生的转动效果。人体作为一个杠杆系统运作，关节充当支点。
τ = F d
where τ = torque (N·m), F = force applied (N), d = perpendicular distance from the axis to the line of action of the force (moment arm). Longer muscle moment arms generate larger torques.
其中 τ = 力矩（牛·米），F = 施加的力（牛顿），d = 从轴到力作用线的垂直距离（力臂）。肌肉力臂越长，产生的力矩越大。
7. Work and Energy | 功和能
Work is done when a force moves its point of application. Kinetic energy, gravitational potential energy, and their conservation are central to understanding performance in explosive events like high jump or gymnastics.
当力使其作用点移动时，就做了功。动能、重力势能及其守恒对于理解跳高或体操等爆发性项目的表现至关重要。
W = F s cos θ
KE = ½ m v²
GPE = m g h
where W = work done (J), θ = angle between force and displacement, g = 9.81 m·s⁻², h = height (m). Conservation: KE₁ + GPE₁ = KE₂ + GPE₂ (ignoring air resistance).
其中 W = 做功（焦耳），θ = 力与位移之间的夹角，g = 9.81 米/秒²，h = 高度（米）。守恒公式：KE₁ + GPE₁ = KE₂ + GPE₂（忽略空气阻力）。
8. Power | 功率
Power is the rate of doing work or transferring energy. In sport, power output is a strong predictor of performance in sprints, jumps, and weightlifting.
功率是做功或能量转移的快慢。在体育运动中，功率输出是短跑、跳跃和举重成绩的有力预测指标。
P = W / t = F v
where P = power (W), W = work done (J), t = time (s), F = force (N), v = velocity (m·s⁻¹). The product F v is especially useful for continuous exertion such as rowing or cycling.
其中 P = 功率（瓦特），W = 做功（焦耳），t = 时间（秒），F = 力（牛顿），v = 速度（米/秒）。F v 的乘积对于划船或自行车等持续用力运动特别实用。
9. Angular Motion Quantities | 角运动量
Many sporting movements, such as a golf swing or a skater’s spin, involve rotation. Linear and angular variables are linked through the radius of rotation.
许多运动动作，如高尔夫挥杆或滑冰旋转，都涉及转动。线性变量与角变量通过旋转半径关联。
s = r θ (for θ in radians)
v = r ω
a = r α
where r = radius, θ = angular displacement (rad), ω = angular velocity (rad·s⁻¹), α = angular acceleration (rad·s⁻²). In sports, the speed of the club head or the hand depends on both angular velocity and radius.
其中 r = 半径，θ = 角位移（弧度），ω = 角速度（弧度/秒），α = 角加速度（弧度/秒²）。在体育中，球杆头或手的速度同时取决于角速度和半径。
10. Fluid Forces: Drag and Bernoulli’s Principle | 流体作用力：阻力与伯努利原理
Drag force opposes the motion of a body through a fluid. Bernoulli’s principle explains lift and the Magnus effect in spinning balls. These concepts are crucial for cycling aerodynamics, swimming technique, and ball flight in soccer or tennis.
阻力是物体在流体中运动时阻碍其运动的力。伯努利原理解释了旋转球体的升力和马格努斯效应。这些概念对自行车空气动力学、游泳技术以及足球或网球的球路飞行至关重要。
Fd = ½ Cd ρ A v²
where Fd = drag force (N), Cd = drag coefficient, ρ = fluid density (kg·m⁻³), A = cross-sectional area (m²), v = velocity (m·s⁻¹). Reducing Cd or A (e.g. by tucking in cycling) cuts drag significantly.
其中 Fd = 阻力（牛顿），Cd = 阻力系数，ρ = 流体密度（千克/米³），A = 横截面积（米²），v = 速度（米/秒）。降低 Cd 或 A（例如自行车运动中蜷缩身体）可大幅减小阻力。
11. Training Heart Rate Zones | 训练心率区间
The Karvonen formula is widely used to determine target heart rate zones for training. It personalises intensity based on resting heart rate, improving the precision of endurance prescription.
卡沃宁公式被广泛用于确定训练的目标心率区间。它根据安静心率个性化设定强度，提高了耐力处方的精确性。
HRtarget = HRrest + (HRmax – HRrest) × Intensity
where HRmax is usually estimated as 220 – age. For an aerobic threshold zone (60–70% intensity), the formula adjusts the working heart rate accordingly. It is applied to plan effective cardiovascular conditioning.
其中 HRmax 通常估算为 220 – 年龄。对于有氧阈值区间（60–70% 强度），该公式相应调整工作心率。它被用于规划有效的心血管调节训练。
12. Body Mass Index (BMI) and Energy Expenditure | 身体质量指数 (BMI) 与能量消耗
BMI offers a simple, population-level indicator of body composition. Energy expenditure can be estimated using metabolic equivalent (MET) values, both relevant to nutrition and health in sport.
BMI 提供了一个简单的群体层面身体成分指标。能量消耗可以使用代谢当量 (MET) 值进行估算，两者都与运动营养和健康相关。
BMI = mass (kg) / height² (m²)
Energy expenditure (kcal) = MET × mass (kg) × time (h)
BMI categories: underweight <18.5, normal 18.5–24.9, overweight 25–29.9, obese ≥30. MET values: sleeping ≈ 0.9 MET, walking ≈ 3–5 MET, running ≈ 8–15 MET. These tools help athletes monitor body mass and energy balance.
BMI 分类：体重过轻 <18.5，正常 18.5–24.9，超重 25–29.9，肥胖 ≥30。MET 值：睡眠 ≈ 0.9 MET，步行 ≈ 3–5 MET，跑步 ≈ 8–15 MET。这些工具有助于运动员监测体质量和能量平衡。
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