📚 Year 13 CCEA PE: Formula & Theorem Quick Reference Handbook | Year 13 CCEA 体育：公式定理速查手册
This quick reference handbook brings together the most important formulas and theorems you will encounter in the CCEA Year 13 Physical Education course. Whether you are studying biomechanics, exercise physiology or skill acquisition, a firm grasp of these equations is essential for exam success. Keep this guide close when tackling past papers and practical data analysis.
本速查手册汇集了 CCEA Year 13 体育课程中最关键的公式和定理。无论你学习的是生物力学、运动生理学还是技能获得，牢固掌握这些方程式对于在考试中取得成功至关重要。在练习历年真题和进行实践数据分析时，请将此指南放在手边。
1. Linear Kinematics (SUVAT Equations) | 直线运动学公式
The four SUVAT equations describe uniformly accelerated linear motion. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). You must be able to select the correct equation depending on which variables are known and which one is required.
四个 SUVAT 方程描述了匀加速直线运动。它们将位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)联系在一起。你必须能够根据已知量和未知量来选用正确的方程。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
For example, when calculating the take-off velocity in a long jump where time of flight and acceleration due to gravity (−9.81 m·s⁻²) are known, ‘v = u + at’ becomes the go-to equation. Always remember to define a positive direction before substituting values.
例如，当已知跳远的腾空时间和重力加速度(−9.81 m·s⁻²)来计算起跳速度时，“v = u + at”会成为首选方程。代入数值前，务必先确定正方向。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s three laws form the bedrock of biomechanics. They explain how forces affect the motion of an athlete or an object.
牛顿三大定律构成了生物力学的基石。它们解释了力如何影响运动员或物体的运动。
First Law (Inertia): A body remains at rest or in uniform motion in a straight line unless acted upon by an external resultant force.
第一定律（惯性）： 除非受到外力的合力作用，否则物体将保持静止或匀速直线运动状态。
Second Law (F = ma): The acceleration of a body is directly proportional to the net force acting on it and inversely proportional to its mass.
第二定律（F = ma）： 物体的加速度与作用在其上的净力成正比，与其质量成反比。
F = ma
Third Law (Action–Reaction): For every action force there is an equal and opposite reaction force.
第三定律（作用力与反作用力）： 每一个作用力都有一个大小相等、方向相反的反作用力。
In sprinting, the athlete applies a backward force onto the blocks; the blocks push the athlete forward with an equal force. The second law helps to calculate the force required to accelerate a 75 kg sprinter at 4 m·s⁻²: F = 75 × 4 = 300 N.
在短跑中，运动员对起跑器施加向后的力；起跑器以大小相等的力将运动员向前推。第二定律可用来计算使75公斤的短跑运动员以4 m·s⁻²加速所需的力：F = 75 × 4 = 300 N。
3. Momentum and Impulse | 动量与冲量
Momentum is a measure of the quantity of motion an object possesses. Impulse relates the force applied to the change in momentum over time.
动量是衡量物体运动量的一个指标。冲量则反映了力在一段时间内引起的动量变化。
p = mv
Impulse = FΔt = Δ(mv)
In a tackle in rugby, a player of mass 90 kg moving at 5 m·s⁻¹ has a momentum of 450 kg·m·s⁻¹. By increasing the time over which the tackle occurs (Δt), the force experienced by the tackled player is reduced, which is why modern protective padding and controlled tackling techniques are crucial for injury prevention.
在橄榄球的擒抱动作中，一名质量为90公斤、以5 m·s⁻¹移动的球员拥有450 kg·m·s⁻¹的动量。通过延长擒抱发生的时间(Δt)，被擒抱球员所承受的力就会减小，这也是现代防护垫具和受控擒抱技术对预防受伤至关重要的原因。
4. Work, Energy and Power | 功、能和功率
These three concepts describe the capacity to perform muscular activity and the rate at which that activity is done. Work is the product of force and the distance moved in the direction of the force.
这三个概念描述了进行肌肉活动的能力以及该活动完成的速率。功是力与在力的方向上移动的距离的乘积。
W = Fd
Kinetic Energy = ½mv²
Gravitational Potential Energy = mgh
Power = W / Δt or Power = Fv
A weightlifter raising a 120 kg barbell vertically through 1.8 m does work against gravity: W = mgh = 120 × 9.81 × 1.8 ≈ 2119 J. If the lift takes 1.5 s, the average power output is 2119 / 1.5 ≈ 1413 W, illustrating the explosive nature of the event.
举重运动员将120公斤的杠铃垂直上举1.8米，克服重力做功：W = mgh = 120 × 9.81 × 1.8 ≈ 2119 J。如果上举用时1.5秒，则平均输出功率为2119 / 1.5 ≈ 1413 W，这表明了此项运动的爆发力特征。
5. Projectile Motion | 抛体运动
Any object released into the air becomes a projectile. In the absence of air resistance, the horizontal and vertical components of motion are independent. The horizontal velocity remains constant, while the vertical motion is governed by gravity.
任何被抛射到空中的物体都成为抛体。在没有空气阻力的情况下，运动的水平和竖直分量相互独立。水平速度保持恒定，竖直运动则受重力支配。
Horizontal: sₕ = u cos θ × t
Vertical: sᵥ = u sin θ × t − ½gt²
vᵥ = u sin θ − gt
The optimal angle of release depends on the relative heights of release and landing. For a shot put, the release angle is typically below 45° because the release point is higher than the landing point; the athlete therefore requires a larger horizontal component to maximise range.
最佳出手角度取决于出手点与落地点的相对高度。在推铅球中，出手角度通常小于45°，因为出手点高于落地点；因此运动员需要更大的水平分量以最大化射程。
6. Angular Kinematics | 角运动学
Angular kinematics describes rotation. Quantities such as angular displacement, angular velocity and angular acceleration all have linear counterparts and appear in the analysis of rotational skills like a golf swing or a spin in ice skating.
角运动学描述旋转运动。角位移、角速度和角加速度等量都有对应的线性量，它们出现在高尔夫挥杆或冰上旋转等旋转技能的分析中。
ω = Δθ / Δt
α = Δω / Δt
v = rω
aₜ = rα
An ice skater who pulls her arms in during a spin reduces her moment of inertia, causing her angular velocity (ω) to increase. The relationship v = rω explains why the linear velocity of the club head is crucial for maximising a golf drive: a longer club radius (r) and faster angular velocity both help increase club head speed.
冰上旋转时，运动员收起手臂会减小转动惯量，从而导致角速度(ω)增加。v = rω 的关系解释了一号木杆杆头线速度对高尔夫开球至关重要的原因：更长的杆身半径(r)和更快的角速度都有助于提高杆头速度。
7. Moment of Inertia and Angular Momentum | 转动惯量与角动量
Moment of inertia (I) is the resistance of a body to change in its rotational state. It depends on mass and how that mass is distributed relative to the axis of rotation. Angular momentum (L) is conserved in a closed system.
转动惯量(I)是物体抵抗其转动状态改变的度量。它取决于质量以及质量相对于转轴的分布。角动量(L)在封闭系统中是守恒的。
I = Σmr²
L = Iω
Conservation: I₁ω₁ = I₂ω₂
In a dive with a somersault and twist, the athlete changes body shape to manipulate I. In the tucked position, I is small, so ω is large, allowing fast rotation. Opening out into the layout position increases I, reducing ω and giving the diver time to prepare for a clean entry into the water.
在包含翻腾和转体的跳水动作中，运动员通过改变身体姿态来调控转动惯量(I)。在团身姿态时，I较小，因此ω较大，可实现快速旋转。展开成直体姿态会增加I，减小ω，从而让跳水者有时间为干净入水做准备。
8. Levers and Mechanical Advantage | 杠杆与机械优势
Levers in the human body consist of a bone acting as the lever arm, a joint as the fulcrum, and muscles providing the effort. Mechanical advantage (MA) expresses the ratio of the load to the effort.
人体内的杠杆由骨骼充当杠杆臂、关节作为支点、肌肉提供动力。机械优势(MA)表示负荷与动力的比值。
MA = Load / Effort
MA = Effort arm length / Load arm length
A third-class lever is most common in the body (e.g., the biceps curl with the elbow as fulcrum). Here MA is less than 1, meaning a large effort is needed to move a small load. This arrangement favours speed and range of movement over force production, which suits many sporting actions.
第三类杠杆在人体中最常见（例如，以肘关节为支点的肱二头肌弯举）。此时机械优势小于1，意味着移动一个较小的负荷需要较大的动力。这种结构有利于产生速度和动作幅度，而非力量输出，这正适合许多运动动作。
9. Cardiovascular Parameters | 心血管参数
Cardiac output (Q) is the volume of blood pumped by the heart per minute. It is the product of stroke volume (SV) and heart rate (HR). Understanding these relationships is fundamental to exercise physiology.
心输出量(Q)是心脏每分钟泵出的血液量。它是每搏输出量(SV)与心率(HR)的乘积。理解这些关系是运动生理学的基础。
Q = SV × HR
A trained endurance athlete might have a resting stroke volume of 100 mL·beat⁻¹ and a resting heart rate of 45 bpm, yielding a resting Q of 4.5 L·min⁻¹. During maximal exercise, SV may rise to 160 mL·beat⁻¹ and HR to 195 bpm, giving a maximal Q of 31.2 L·min⁻¹, demonstrating the huge range of the cardiovascular system.
一名训练有素的耐力运动员静息每搏输出量可能为100 mL·beat⁻¹，静息心率为45 bpm，那么静息心输出量为4.5 L·min⁻¹。在最大强度运动中，每搏输出量可能升至160 mL·beat⁻¹，心率升至195 bpm，此时最大心输出量达31.2 L·min⁻¹，显示了心血管系统的巨大变化范围。
10. VO₂ max and Aerobic Capacity | 最大摄氧量与有氧能力
VO₂ max is the maximum volume of oxygen the body can take in, transport and utilise per minute. It is the gold standard measure of aerobic endurance and is expressed in absolute terms (L·min⁻¹) or relative to body mass (mL·kg⁻¹·min⁻¹).
最大摄氧量(VO₂ max)是机体每分钟能够摄入、运输和利用的最大氧量。它是有氧耐力的金标准指标，可以用绝对值(L·min⁻¹)或相对体重值(mL·kg⁻¹·min⁻¹)表示。
VO₂ = (Vₑ × (FₑO₂ − FₑO₂)) / T
In a laboratory, VO₂ max is measured via open-circuit spirometry. While the full formula is complex, the Fick principle underpins the concept:
VO₂ = Q × (a−v)O₂ difference
This shows that VO₂ max depends on both central (cardiac output) and peripheral (oxygen extraction) factors. Improvements in training result from adaptations in both Q and the arterio‑venous oxygen difference.
这表明VO₂ max同时取决于中枢因素（心输出量）和外周因素（氧摄取）。训练带来的提升源于心输出量和动静脉氧差的适应。
11. The Karvonen Formula | 卡氏公式
The Karvonen formula is used to calculate target heart rate zones for training based on heart rate reserve (HRR). It is more personalised than a simple percentage of HR max because it accounts for resting heart rate.
卡氏公式用于根据心率储备(HRR)来计算训练靶心率区间。它比简单地采用最大心率的百分比更具个性化，因为它考虑了静息心率。
Target HR = ((HRmax − HRrest) × % intensity) + HRrest
If an 18‑year‑old athlete has HRmax = 202 bpm and HRrest = 60 bpm, and wishes to train at 70 % intensity, the calculation becomes: ((202 − 60) × 0.7) + 60 = (142 × 0.7) + 60 = 99.4 + 60 = 159.4 bpm. This target sits within the aerobic training zone and helps ensure the session develops cardiovascular fitness safely.
若一名18岁的运动员最大心率(HRmax)为202 bpm，静息心率(HRrest)为60 bpm，并希望以70%强度进行训练，则计算如下：((202 − 60) × 0.7) + 60 = (142 × 0.7) + 60 = 99.4 + 60 = 159.4 bpm。该靶心率处于有氧训练区间内，有助于确保训练安全地发展心血管适能。
12. Energy Expenditure and METs | 能量消耗与代谢当量
The metabolic equivalent (MET) is a convenient unit for expressing the energy cost of physical activities. 1 MET is defined as the resting metabolic rate, approximately 3.5 mL O₂·kg⁻¹·min⁻¹.
代谢当量(MET)是用于表示体力活动能量消耗的便捷单位。1 MET被定义为静息代谢率，约为3.5 mL O₂·kg⁻¹·min⁻¹。
Energy expenditure (kcal·min⁻¹) = METs × body mass (kg) × 0.0175
An 80 kg rugby player jogging at 8 METs expends roughly 80 × 8 × 0.0175 = 11.2 kcal per minute. This formula is widely used to estimate total energy expenditure and to plan nutritional strategies for athletes in weight‑classified sports.
一名80公斤的橄榄球运动员以8 METs慢跑，消耗大约80 × 8 × 0.0175 = 11.2千卡/分钟。该公式被广泛用于估算总能量消耗，并为参加体重分级项目的运动员制定营养策略。
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