📚 Year 13 SQA Physical Education: Formula & Theorem Quick Reference Handbook | SQA 体育公式定理速查手册
This quick reference handbook compiles the essential formulas, principles, and theorems you need for the Year 13 SQA Physical Education course. It covers biomechanics, exercise physiology, and skill acquisition to help you analyse performance and answer exam questions with confidence.
本速查手册汇编了 SQA 高年级体育课程所需的核心公式、原理和定理，涵盖生物力学、运动生理学和技能学习等领域，帮助你自信地分析运动表现并作答考试题目。
1. Linear Kinematics | 线性运动学
Linear kinematics describes motion in a straight line. The fundamental quantities are displacement, velocity, and acceleration.
线性运动学描述直线运动，基本物理量包括位移、速度与加速度。
Average speed is the rate of covering distance: speed = distance / time. Velocity, being a vector, uses displacement instead: v = s / t.
平均速率是距离对时间的变化率：速率 = 路程 / 时间。速度是矢量，用位移代替：v = s / t。
Acceleration measures how quickly velocity changes: a = (v − u) / t, where u is initial velocity and v is final velocity.
加速度衡量速度变化的快慢：a = (v − u) / t，其中 u 为初速度，v 为末速度。
The equations of uniformly accelerated motion are vital for projectile and sprint analysis:
匀变速直线运动公式对抛体与短跑分析至关重要：
v = u + at
s = ut + ½ at²
v² = u² + 2as
Here s represents displacement in a straight line, and t is the time interval.
其中 s 代表直线位移，t 为时间间隔。
2. Newton's Laws & Force | 牛顿定律与力
Newton's three laws form the foundation of all force analysis in sport.
牛顿运动三定律构成了体育中所有受力分析的基础。
First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a net external force. This explains why a hockey puck glides until friction or a stick acts upon it.
第一定律（惯性）： 物体在不受净外力作用时保持静止或匀速直线运动，这解释了冰球在没有摩擦或球杆击打时会持续滑行。
Second Law (F = ma): The acceleration of an object is directly proportional to the net force and inversely proportional to its mass: F = m × a. A sprinter drives out of the blocks with a large force to achieve high acceleration.
第二定律（F = ma）： 物体的加速度与所受净外力成正比，与其质量成反比：F = m × a。短跑运动员用巨大蹬力获得高加速度。
Third Law (Action–Reaction): For every action force there is an equal and opposite reaction force. When a swimmer pushes water backward, the water pushes the swimmer forward.
第三定律（作用与反作用）： 每一个作用力都有一个大小相等、方向相反的反作用力。游泳者向后推水时，水向前推游泳者。
Weight is the force due to gravity: W = m × g m/s² on Earth. This weight influences friction, ground reaction force, and jumping height.
重力是万有引力引起的力：W = m × g，地球上 g = 9.81 m/s²。体重影响着摩擦力、地面反作用力和跳跃高度。
3. Momentum and Impulse | 动量与冲量
Momentum (p) is the product of an object's mass and its velocity: p = m × v. It is a vector quantity measured in kg·m/s. A rugby player with greater mass and speed carries larger momentum and is harder to stop.
动量（p）是物体质量与速度的乘积：p = m × v，为矢量，单位 kg·m/s。质量大且速度快的橄榄球运动员动量更大，更难被拦截。
The impulse–momentum theorem states that impulse equals change in momentum. Impulse is defined as the product of force and the time over which it acts: Impulse = F × Δt = Δp = m(v − u).
冲量—动量定理指出，冲量等于动量的变化。冲量定义为力与其作用时间的乘积：冲量 = F × Δt = Δp = m(v − u)。
In sport, increasing the time of force application (follow-through in kicking) maximises the change in momentum, resulting in a faster ball. Absorbing a landing over a longer time by bending the knees reduces the average force experienced, preventing injury.
在运动中，延长力作用时间（如踢球的随球动作）可使动量变化最大化，使球速更快。屈膝落地延长缓冲时间可减小平均受力，预防受伤。
4. Projectile Motion | 抛体运动
A projectile is any object released into the air and subject only to gravity and air resistance. The motion is analysed by resolving velocity into horizontal and vertical components.
抛体是指被抛出后只受重力和空气阻力作用的物体，其运动通过将速度分解为水平和竖直分量来分析。
For an initial velocity v₀ at an angle θ to the horizontal:
若初速度为 v₀，与水平方向夹角为 θ：
Horizontal component: vₓ = v₀ cos θ
Vertical component: vᵧ = v₀ sin θ
Horizontal displacement assumes constant velocity: x = v₀ cos θ × t. Vertical displacement is governed by y = v₀ sin θ × t − ½ g t², where g = 9.81 m/s².
水平位移看作匀速运动：x = v₀ cos θ × t。竖直位移遵循：y = v₀ sin θ × t − ½ g t²，g = 9.81 m/s²。
The optimum release angle for achieving maximum range on a flat surface is 45° (ignoring air resistance). In many sport situations, such as a basketball free throw or a javelin throw, the optimum angle is adjusted because of the height of release and aerodynamic factors.
忽略空气阻力时，在平面上获得最大射程的最佳出手角度为 45°。在篮球罚球或标枪投掷等实际运动场景中，由于出手高度和空气动力学因素，最佳角度会有所调整。
5. Work, Energy & Power | 功、能与功率
Work (W) is done when a force moves an object in the direction of the force: W = F × d × cos θ and displacement vectors. When force and motion are in the same direction, cos 0° = 1, so W = F × d.
功 (W) 是力使物体沿力的方向发生位移时所做的效果：W = F × d × cos θ，θ 为力与位移的夹角。两者同向时 cos 0° = 1，W = F × d。
Kinetic energy (KE) is the energy of motion: KE = ½ m v². Gravitational potential energy (GPE) is stored due to height: GPE = m g h.
动能 (KE) 是运动具有的能量：KE = ½ m v²。重力势能 (GPE) 是由于高度而储存的能量：GPE = m g h。
Power is the rate of doing work or transferring energy: P = W / t or P = (F × d) / t. Since distance over time is velocity, power can also be expressed as P = F × v for constant force and velocity parallel.
功率是做功或能量转移的快慢：P = W / t 或 P = (F × d) / t。因为距离除以时间即速度，当力与速度同向稳定时功率可表示为 P = F × v。
A weightlifter generating high power can move a barbell faster; a cyclist's power output determines speed against resistance.
举重运动员产生高功率能更快举起杠铃；自行车运动员的功率输出决定了对抗阻力时的速度。
6. Levers and Torque | 杠杆与力矩
A lever amplifies force or speed. It consists of a fulcrum (pivot), an effort force, and a load. The mechanical advantage (MA) is the ratio of effort arm length to load arm length: MA = Effort arm / Load arm.
杠杆用于放大力量或速度，由支点、动力和阻力组成。机械效益 (MA) 是动力臂与阻力臂之比：MA = 动力臂 / 阻力臂。
Torque (τ) is the rotational equivalent of force: τ = F × r where F is the applied force and r is the lever arm (perpendicular distance from pivot to line of force).
力矩 (τ) 是力对转动的作用：τ = F × r，F 为作用力，r 为力臂（支点到力作用线的垂直距离）。
In the body, joints act as fulcrums, muscles provide effort, and bones carry loads. For example, the biceps curl is a third‑class lever (effort between fulcrum and load), favouring speed and range of motion over force production.
人体中关节为支点，肌肉提供动力，骨骼承受负荷。例如肱二头肌弯举是第三类杠杆（动力在支点和阻力之间），它牺牲部分力量来换取较大的运动速度和幅度。
7. Fluid Dynamics & Aerodynamics | 流体动力学与空气动力学
Fluid forces play a key role in sports such as cycling, swimming, and ball games. The drag force opposing motion through a fluid is given by:
流体作用力在自行车、游泳和球类等运动中非常关键。阻碍物体在流体中运动的阻力公式为：
Fdrag = ½ ρ Cd A v²
where ρ (rho) is fluid density, Cd is the drag coefficient, A is the cross‑sectional area, and v is the velocity relative to the fluid.
其中 ρ 为流体密度，Cd 为阻力系数，A 为横截面积，v 为与流体的相对速度。
Notice that drag increases with the square of speed, which explains why even small increases in velocity demand far greater energy output. Streamlining body position and wearing tight suits reduce A and Cd.
注意阻力随速度的平方增加，这解释了为什么速度即使略微提升也需要显著增加能量输出。保持流线型姿态和穿着紧身装备可降低 A 和 Cd。
The Magnus effect explains sideways forces on spinning balls. When a ball spins, it drags air on one side, creating a pressure difference that produces a lift force perpendicular to the flow. A topspin tennis ball dips faster; backspin creates lift.
马格努斯效应解释了旋转球体所受的侧向力。球体旋转时带动一侧流体，产生压力差，形成垂直于流动方向的升力。上旋网球加速下坠，下旋则产生升力。
8. Cardiovascular Formulas | 心血管系统公式
Understanding heart rate (HR) responses is crucial for training design. Maximum heart rate can be estimated using the simple age‑based formula:
理解心率 (HR) 反应对训练设计至关重要。最大心率可用基于年龄的简易公式估算：
HRmax = 220 − age
More precise protocols exist, but this equation is widely used in SQA Physical Education.
尽管存在更精确的方案，此公式仍广泛用于 SQA 体育课程。
The Karvonen formula determines target heart rate using resting heart rate (HRrest) and heart rate reserve (HRR):
卡沃宁公式借助安静心率 (HRrest) 和心率储备 (HRR) 计算目标心率：
HRR = HRmax − HRrest
Target HR = HRrest + (Desired % × HRR)
For instance, a 17‑year‑old with HRrest of 60 bpm aiming for 80% 143 bpm, Target HR = 60 + 0.80 × 143 = 174.4 bpm. This method personalises training zones.
例如，一名 17 岁青年安静心率 60 bpm，目标强度 80%：HRmax = 203 bpm，HRR = 143 bpm，目标心率 = 60 + 0.80 × 143 = 174.4 bpm。该方法能个性化训练强度区间。
9. VO₂max Estimation | 最大摄氧量估算
VO₂max, the maximal volume of oxygen the body can utilise per minute per kilogram of body mass, is the gold standard of aerobic fitness. While direct gas analysis is most accurate, several field tests provide useful estimates.
最大摄氧量 VO₂max 是每分钟每公斤体重所能利用的最大氧气量，是有氧能力黄金标准。尽管直接气体分析最精准，多种场地测试可提供有效估算。
The Cooper 12‑minute run estimates VO₂max (ml/kg/min) from distance covered in metres (D):
库珀 12 分钟跑根据跑动距离（米，D）估算 VO₂max (ml/kg/min)：
VO₂max = (D − 504.9) / 44.73
The multi‑stage fitness test (bleep test) also correlates with VO₂max. A frequently referenced prediction equation is:
多级有氧耐力测试（蜂鸣测试）也常与 VO₂max 关联，常用预测公式为：
VO₂max = 3.46 × (level + shuttle / total shuttles in level) + 12.2
These estimates allow coaches and performers to monitor aerobic development over time.
这些估算使教练和运动员能持续监测有氧能力的发展。
10. Body Mass Index (BMI) | 身体质量指数
BMI is a simple ratio of mass to height² used to classify individuals into weight categories. While it does not distinguish between fat and lean mass, it remains a convenient screening tool in population studies and some fitness assessments.
BMI 是体重与身高平方的简单比值，用于划分体重类别。虽然不能区分脂肪与瘦体重，但依然是群体研究和部分体能评估中便捷的筛查工具。
BMI = mass (kg) / (height in m)²
should interpret BMI with caution, particularly in athletic populations with high muscle mass.
例如，一名体重 72 kg、身高 1.80 m 的运动员，其 BMI = 72 / (1.80)² ≈ 22.2 kg/m²。SQA 考生需谨慎解读 BMI，尤其对于肌肉量高的运动员群体。
11. Angular Kinematics | 角运动学
Rotational movements such as somersaults, discus spins, and golf swings are described by angular kinematics.
旋转运动如空翻、铁饼投掷和高尔夫挥杆等通过角运动学进行描述。
Angular velocity (ω) is the rate of change of angular displacement (θ): ω = Δθ / Δt, measured in rad/s. One full rotation equals 2π rad.
角速度 (ω) 是角位移 (θ) 的变化率：ω = Δθ / Δt，单位 rad/s；一整圈相当于 2π 弧度。
Angular acceleration (α) describes how quickly angular velocity changes: α = (ω₂ − ω₁) / t.
角加速度 (α) 描述角速度变化的快慢：α = (ω₂ − ω₁) / t。
Linear and angular quantities are directly linked. The linear velocity of a point on a rotating object is: v = r ω, where r is the radius of rotation. Centripetal acceleration required to keep an object moving in a circle is: ac = v² / r = r ω².
线量与角量直接关联。旋转物体上一点的线速度：v = r ω，r 为旋转半径。维持圆周运动所需的向心加速度：ac = v² / r = r ω²。
The centripetal force producing this acceleration is Fc = m v² / r. A hammer thrower generates this force to keep the hammer moving in a circle before release.
产生该加速度的向心力为 Fc = m v² / r。链球运动员在出手前产生向心力使链球保持圆周运动。
Published by TutorHao | Physical Education Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图