📚 AS OCR PE Formula & Theorem Quick Reference | AS OCR 体育：公式定理速查手册
This quick-reference guide covers the essential formulas and principles you need to know for the AS OCR Physical Education specification. Each entry is explained with a clear formula, practical use, and example applications so you can apply them confidently in exam questions and practical contexts.
本速查手册涵盖了 AS OCR 体育课程必备的核心公式与原理。每一个条目都配有清晰的公式、实际用途和示例应用，帮助你在考试和实际情境中自信运用。
1. Maximum Heart Rate (HRmax) | 最大心率（HRmax）
The simplest method to estimate an individual’s maximum heart rate is the age-predicted formula. This value is used to set training intensities and to monitor cardiovascular effort during exercise.
估算个人最大心率最简单的方法是年龄预测公式。该数值用于设定训练强度和监测运动中的心血管负荷。
HRmax = 220 − age
For a 17-year-old athlete, HRmax = 220 − 17 = 203 bpm. Note that this estimate has a standard deviation of about ±10–12 bpm and individual variation can be significant.
对于一名 17 岁的运动员，HRmax = 220 − 17 = 203 次/分。请注意，该估算值的标准差约为 ±10–12 次/分，个体差异可能较大。
2. Karvonen Formula (Target Heart Rate) | 卡沃宁公式（目标心率）
The Karvonen method calculates target heart rate based on heart rate reserve (HRR), incorporating resting heart rate to personalise training zones.
卡沃宁法根据心率储备（HRR）计算目标心率，将安静心率纳入其中，从而定制个性化训练区间。
Target HR = ((HRmax − HRrest) × %intensity) + HRrest
Example: HRmax = 200 bpm, HRrest = 60 bpm, intensity = 70%. Target HR = ((200−60) × 0.7) + 60 = 158 bpm. This method is widely used to prescribe aerobic training zones.
示例：HRmax = 200 次/分，HRrest = 60 次/分，强度 = 70%。目标心率 = ((200−60) × 0.7) + 60 = 158 次/分。该方法广泛用于制定有氧训练区间。
3. Body Mass Index (BMI) | 身体质量指数（BMI）
BMI is a simple index of weight-for-height that is commonly used to classify underweight, normal weight, overweight and obesity in populations.
BMI 是一种简单的体重–身高指数，通常用于在人群中划分偏瘦、正常体重、超重和肥胖。
BMI = mass (kg) ÷ height² (m²)
For a performer weighing 70 kg and 1.75 m tall, BMI = 70 ÷ (1.75×1.75) = 22.9 kg/m², which falls within the normal range. BMI does not distinguish between muscle and fat mass.
对于一名体重 70 公斤、身高 1.75 米的运动员，BMI = 70 ÷ (1.75×1.75) = 22.9 公斤/平方米，属于正常范围。BMI 无法区分肌肉与脂肪质量。
4. Respiratory Exchange Ratio (RER) | 呼吸交换率（RER）
RER is the ratio of carbon dioxide produced to oxygen consumed. It indicates which fuel is predominantly being used during exercise and is essential for understanding energy metabolism.
RER 是生成的二氧化碳与消耗的氧气之比。它反映运动期间主要使用的燃料类型，是理解能量代谢的关键。
RER = VCO₂ ÷ VO₂
An RER of 0.70 indicates primarily fat oxidation, 0.85 for a mix, and 1.00 or above indicates predominantly carbohydrate oxidation. Laboratory gas analysis provides the most accurate RER measurements.
RER 为 0.70 表示主要进行脂肪氧化，0.85 为混合供能，1.00 及以上则表示主要依赖碳水化合物氧化。实验室气体分析可提供最准确的 RER 测量值。
5. Speed, Distance and Time | 速度、距离与时间
This fundamental relationship links the three key variables used to describe linear motion in sporting contexts, from sprint times to endurance pace.
这一基本关系将描述运动中线运动的三个关键变量联系在一起，从短跑用时到耐力配速都适用。
Speed = distance ÷ time
If a runner covers 100 m in 11.2 s, average speed = 100 ÷ 11.2 = 8.93 m/s. Rearranging gives distance = speed × time, and time = distance ÷ speed, which are equally useful for race planning.
如果一名跑者用 11.2 秒跑完 100 米，平均速度 = 100 ÷ 11.2 = 8.93 米/秒。变形后得到距离 = 速度 × 时间，以及时间 = 距离 ÷ 速度，同样适用于比赛规划。
6. Newton’s Second Law of Motion | 牛顿第二运动定律
Newton’s second law describes how the velocity of an object changes when it is subjected to an external force. It underpins all linear acceleration and deceleration in sporting actions.
牛顿第二定律描述了物体受到外力作用时速度如何变化。它是所有体育动作中线性加速与减速的基础。
F = m × a
where F = net force (N), m = mass (kg), a = acceleration (m/s²). To accelerate a 75 kg sprinter at 3 m/s², a net horizontal force of 225 N is required. The same equation explains deceleration when braking forces are applied.
其中 F = 合外力（牛顿），m = 质量（千克），a = 加速度（米/秒²）。要让一名 75 公斤的短跑运动员以 3 米/秒² 的加速度起跑，需要 225 牛顿的水平净力。同理，该方程也解释了施加制动力时的减速过程。
7. Impulse and Momentum | 冲量与动量
Impulse is the product of force and the time over which it acts, and it equals the change in momentum of an object. This relationship explains how technique can maximise force application in throwing, jumping and striking.
冲量是力与其作用时间的乘积，等于物体动量的变化。这一关系解释了如何通过技术最大限度地在投掷、跳跃和击打中施加力量。
Impulse = F × t = Δ(mv)
A long force application time (increased by follow-through) results in a greater change in momentum. In a tennis serve, a 50 N force applied over 0.2 s produces an impulse of 10 N·s, increasing the ball’s momentum.
延长力作用时间（通过随挥动作实现）能够产生更大的动量变化。在网球发球中，50 牛顿的力施加 0.2 秒可产生 10 牛顿·秒的冲量，从而增加球的动量。
8. Power Output | 功率输出
Power is the rate at which work is done. In sport, a high power output is often the key to explosive actions such as jumping, sprinting and throwing.
功率是做功的速率。在运动中，高功率输出通常是跳跃、冲刺和投掷等爆发性动作的关键。
Power = work ÷ time = force × velocity
A vertical jump test measuring force and velocity on a force plate can directly calculate power. If a 700 N ground reaction force is applied at a velocity of 2 m/s, power output = 700 × 2 = 1400 W.
在测力台上测量垂直跳的力量和速度可以直接计算功率。若地面反作用力为 700 牛顿、作用速度为 2 米/秒，则功率输出 = 700 × 2 = 1400 瓦特。
9. Torque (Moment of Force) | 力矩（力的转动效应）
Torque is the rotational equivalent of linear force. It determines the effectiveness of a force in causing rotation around a joint or an axis, crucial for understanding levers in the human body.
力矩是线性力的转动对应量。它决定了力使关节或轴产生转动的效果，对理解人体杠杆至关重要。
Torque = force × perpendicular distance from axis
A bicep curl applies a muscular force at a short perpendicular distance from the elbow, while the dumbbell acts at a longer distance. The athlete must generate sufficient muscle torque to overcome the resistance torque.
二头肌弯举时，肌肉力作用于距肘关节较短垂直距离处，而哑铃则作用在较长的距离上。运动员必须产生足够的肌肉力矩来克服阻力矩。
10. Mechanical Advantage | 机械优势
Mechanical advantage compares the effort arm with the resistance arm in a lever system, determining whether a lever favours force generation or range and speed of movement.
机械优势比较杠杆系统中力臂与阻力臂的长度，决定该杠杆是倾向于产生较大的力还是较大的运动幅度和速度。
Mechanical Advantage = effort arm ÷ resistance arm
In a third-class lever such as the elbow during elbow flexion, the effort arm (bicep insertion to elbow) is shorter than the resistance arm (hand to elbow), giving a mechanical advantage less than 1. This favours speed and range over strength.
在肘关节屈曲这类第三类杠杆中，力臂（肱二头肌附着点至肘）短于阻力臂（手至肘），机械优势小于 1。这类杠杆以牺牲力量为代价换取速度和幅度。
11. Angle of Release for Projectile Motion | 抛射运动的最佳释放角度
The distance travelled by a projectile depends on release velocity, release height and release angle. When release height and landing height are equal, the optimal angle is 45°.
抛射体的飞行距离取决于释放速度、释放高度和释放角度。当释放高度与着地高度相同时，最佳角度为 45°。
Optimal angle = 45° (release height = landing height)
If the release point is above the landing surface (e.g. shot put), the optimal angle is slightly less than 45°. A higher release velocity will always increase range for a given angle. Air resistance and spin also influence the actual flight path.
若释放点高于着地表面（如铅球），最佳角度略小于 45°。在给定角度下，更高的释放速度总是会增加射程。空气阻力和旋转也会影响实际飞行轨迹。
12. Hick’s Law (Reaction Time) | 希克定律（反应时间）
Hick’s law describes the relationship between the number of choices available and the time taken to make a decision. It is a cornerstone of information processing in sport.
希克定律描述了可供选择的数量与做出决定所需时间之间的关系。它是体育信息处理领域的基石。
Reaction Time = a + b log₂(n)
where n = number of stimulus–response alternatives. For a goalkeeper facing a penalty, the large number of possible shot directions increases reaction time. Coaches can reduce response delay by narrowing options through tactical preparation.
其中 n = 刺激–反应选项的数量。对于面对点球的守门员来说，大量可能的射门方向会延长反应时间。教练可以通过战术准备缩小选项，从而缩短反应延迟。
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