Year 12 SQA Physical Education: Formula and Theorem Quick Reference Handbook | SQA 体育公式定理速查手册

📚 Year 12 SQA Physical Education: Formula and Theorem Quick Reference Handbook | SQA 体育公式定理速查手册

This quick reference guide compiles essential formulas and theorems for Year 12 SQA Physical Education, covering exercise physiology, biomechanics, and performance analysis. Use it to reinforce your understanding and prepare for assessments.

本速查手册汇总了 Year 12 SQA 体育学科的核心公式与定理，涵盖运动生理学、生物力学和表现分析，帮助你巩固知识并备考。

1. Maximum Heart Rate (HRmax) Estimation | 最大心率（HRmax）估算

The simplest and most widely used prediction for maximum heart rate is:

最简单和最常用的最大心率预测公式是：

HRmax = 220 − age

This equation provides an estimate, but individual variations exist. It tends to overestimate HRmax in older adults and underestimate in younger individuals.

该公式给出了一个估计值，但存在个体差异。它往往会高估老年人的最大心率，低估年轻人的最大心率。

For older or highly trained individuals, the Tanaka formula may be more accurate:

对于年长者或训练有素的人群，Tanaka 公式可能更准确：

HRmax = 208 − (0.7 × age)

Remember that directly measured HRmax from a graded exercise test remains the gold standard.

请记住，通过递增负荷运动测试直接测得的 HRmax 才是金标准。

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2. Heart Rate Reserve (HRR) and Karvonen Formula | 心率储备与卡氏公式

Heart Rate Reserve (HRR) is the difference between your maximum heart rate and resting heart rate (HRrest):

心率储备（HRR）是最大心率与安静心率（HRrest）之间的差值：

HRR = HRmax − HRrest

The Karvonen formula uses HRR to calculate a target heart rate (THR) for a desired training intensity:

卡氏公式利用 HRR 来计算某一期望训练强度下的目标心率（THR）：

THR = (HRmax − HRrest) × Intensity (%) + HRrest

For example, a 17-year-old with a resting HR of 60 bpm, training at 70% intensity, would have THR = (203 − 60) × 0.70 + 60 = 160 bpm.

例如，一名 17 岁、安静心率为 60 bpm 的学生以 70% 强度训练，其目标心率 = (203 − 60) × 0.70 + 60 = 160 bpm。

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3. Target Heart Rate Zones for Training | 训练目标心率区间

Heart rate zones help you prescribe training for different physiological adaptations. Typical zones as a percentage of HRmax are:

心率区间有助于针对不同生理适应安排训练。典型的心率区间（占 HRmax 的百分比）如下：

Zone	% HRmax	Adaptation
Recovery	< 60%	Active recovery
Aerobic (Endurance)	60–70%	Improves aerobic base
Tempo / Threshold	70–80%	Lactate threshold improvement
High Intensity	80–90%	VO₂max development
Maximum Effort	> 90%	Speed & anaerobic power

The above zones can be applied to both HRmax and HRR methods, although exact boundaries may shift slightly depending on the training model used.

上述区间可应用于 HRmax 和 HRR 方法，但具体边界可能会根据所使用的训练模型略有变化。

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4. Body Mass Index (BMI) | 身体质量指数（BMI）

Body Mass Index is a simple index of weight-for-height used to classify underweight, normal weight, overweight, and obesity:

身体质量指数是一个简单的体重-身高指标，用于分类过轻、正常、超重和肥胖：

BMI = weight (kg) ÷ height² (m²)

e; (1.75)² = 22.9 kg/m²;, which falls within the healthy range (18.5–24.9).

例如，一位体重 70 kg、身高 1.75 m 的运动员 BMI = 70 ÷ (1.75)² = 22.9 kg/m²;，属于健康范围（18.5–24.9）。

BMI does not distinguish between muscle and fat mass, so it should be interpreted with caution in athletic populations.

BMI 无法区分肌肉量和脂肪量，因此在对运动员群体进行解释时应谨慎。

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5. Energy Expenditure and METs | 能量消耗与代谢当量

A Metabolic Equivalent (MET) represents resting energy expenditure. One MET equals an oxygen consumption of 3.5 ml/kg/min.

代谢当量（MET）表示安静时的能量消耗。1 MET 相当于耗氧量 3.5 ml/kg/min。

1 MET = 3.5 ml O₂ · kg&supmin;¹ · min&supmin;¹

To estimate the total energy cost of an activity:

估算一项活动的总能量消耗：

Total energy (kcal) = METs × weight (kg) × time (hours)

plays football (8 METs) for 1.5 hours, energy spent = 8 × 65 × 1.5 = 780 kcal.

如果一名 65 kg 的学生踢足球（8 METs）1.5 小时，能量消耗 = 8 × 65 × 1.5 = 780 kcal。

MET values are useful for comparing the intensity of different activities and for designing training programmes.

MET 值可用于比较不同活动的强度以及设计训练计划。

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6. Predicted VO₂max Equations | 最大摄氧量预测方程

Maximal oxygen uptake (VO₂max) is the gold standard measure of aerobic capacity. Submaximal and field tests estimate VO₂max using regression equations.

最大摄氧量（VO₂max）是衡量有氧能力的金标准。次极量和场地测试借助回归方程估算 VO₂max。

For the Cooper 12-minute run test, one common prediction formula is:

对于 Cooper 12 分钟跑测试，常用的预测公式之一是：

VO₂max (ml/kg/min) = (distance in metres − 504.9) ÷ 44.73

For the Multi-stage Fitness Test (bleep test), a widely used equation is:

对于多阶段体能测试（bleep test），广泛使用的方程为：

VO₂max = 3.46 × (level + shuttle) + 12.2

These predictions are estimates and can be influenced by motivation, technique, and environmental conditions.

这些预测值是估计值，可能会受到动机、技术和环境条件的影响。

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7. Speed, Velocity and Acceleration | 速率、速度与加速度

In sports biomechanics, speed and velocity describe how fast a performer or object moves, while acceleration quantifies the rate of change of velocity.

在运动生物力学中，速率和速度描述运动员或物体的移动快慢，而加速度则量化速度变化的快慢。

Speed = distance ÷ time   (v = d / t)

Velocity is a vector quantity and considers displacement:

速度是矢量，需考虑位移：

Velocity = displacement ÷ time   (v = s / t)

Acceleration is given by:

加速度公式为：

ion = (final velocity − initial velocity) / time   (a = (v − u) / t)

Negative acceleration (deceleration) occurs when a sprinter slows down after crossing the finish line.

当短跑运动员冲过终点后减速时，即出现负加速度（减速度）。

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8. Newton's Laws of Motion in Sports | 运动中的牛顿定律

Newton's three laws of motion govern the forces acting on athletes and sports equipment.

牛顿三大运动定律支配着作用于运动员和运动器材上的力。

First Law (Inertia): An object remains at rest or in uniform motion in a straight line unless acted upon by an external force. In football, a stationary ball stays still until kicked.

第一定律（惯性）： 物体将保持静止或匀速直线运动状态，除非受到外力作用。在足球中，静止的球在被踢之前会保持不动。

Second Law (Acceleration):

第二定律（加速度）：

Force = mass × acceleration   (F = m a)

A greater force applied to a shot put results in a greater acceleration, producing a longer throw.

施加在铅球上的力越大，加速度越大，从而掷得更远。

Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.

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。

When a swimmer pushes water backwards, the water pushes the swimmer forwards with an equal force.

当游泳者向后推水时，水以相等的力将游泳者向前推进。

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9. Momentum, Impulse and Collisions | 动量、冲量与碰撞

Momentum is the product of an object's mass and velocity. It is a vector quantity conserved in collisions.

动量是物体质量与速度的乘积。它是一个矢量，在碰撞过程中守恒。

Momentum (p) = mass (m) × velocity (v)

Impulse equals the change in momentum and is the product of force and the time over which it is applied:

冲量等于动量的变化量，等于力与该力作用时间的乘积：

Impulse = Force × time = Δp = m v − m u

In a tackle, extending the impact time reduces the peak force, decreasing injury risk. In striking, a shorter follow-through time increases the force delivered.

在铲球中，延长撞击时间可降低峰值力，从而减少受伤风险。在击打中，缩短随挥时间可增加施加的力。

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10. Work, Power and Kinetic Energy | 功、功率与动能

When a force moves an object, mechanical work is done:

力使物体移动时，就做了机械功：

Work (W) = Force (F) × distance (d) × cos θ

Power is the rate of doing work:

功率是做功的速率：

Power (P) = Work / time = F v

Kinetic energy (KE) is the energy of motion:

动能（KE）是物体由于运动而具有的能量：

KE = ½ m v²;

Gravitational potential energy (PE) is stored energy due to height:

重力势能（PE）是由于高度而储存的能量：

PE = m g h

These principles explain why a pole vaulter's run-up converts kinetic energy into the elastic potential of the pole and then into gravitational potential energy to clear the bar.

这些原理解释了撑杆跳高运动员的助跑如何将动能转化为撑杆的弹性势能，再转化为越过横杆所需的重力势能。

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11. Projectile Motion Parameters | 抛射体运动参数

Projectile motion describes the flight of an object or athlete through the air under gravity, assuming negligible air resistance. The key equations for a projectile launched from ground level at velocity u and angle θ are:

抛射体运动描述的是物体或运动员在重力作用下在空气中的飞行轨迹，假设空气阻力可忽略。对于从地面以速度 u 和角度 θ 发射的抛射体，关键方程有：

Time of flight:

飞行时间：

T = (2 u sin θ) / g

Maximum height reached:

达到的最大高度：

H = (u² sin²θ) / (2 g)

Horizontal range:

水平射程：

R = (u² sin 2θ) / g

To maximise range in a long jump or shot put, athletes aim for a launch angle close to 45° and maximise release velocity.

为了在跳远或铅球中获得最大射程，运动员需要使出手角度接近 45° 并最大化出手速度。

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12. Levers and Torque | 杠杆与力矩

Levers in the human body magnify the effect of muscle forces. A lever consists of a fulcrum, an effort force, and a resistance load.

人体中的杠杆能够放大肌力的作用效果。杠杆由支点、动力和阻力组成。

Torque (moment of force) is the turning effect produced by a force:

力矩是力产生的转动效应：

Torque = Force × perpendicular distance from fulcrum

There are three classes of levers:

杠杆分为三类：

• First-class: fulcrum between effort and load (e.g., neck extension).
• 第一类： 支点在动力和阻力之间（如颈部后伸）。
• Second-class: load between fulcrum and effort (e.g., standing calf raise – ball of foot is fulcrum).
• 第二类： 阻力在支点和动力之间（如站立提踵——跖球为支点）。
• Third-class: effort between fulcrum and load (e.g., bicep curl). Most skeletal levers are third-class, favouring range of motion and speed over force.
• 第三类： 动力在支点和阻力之间（例如肱二头肌弯举）。大多数骨骼杠杆属于第三类，以活动度和速度换取力。

Understanding levers helps coaches improve technique and reduce injury risk by optimising force application.

了解杠杆原理有助于教练通过优化力的应用来改进技术并降低受伤风险。

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