A-Level Physics: Speed vs Velocity | A-Level 物理：速率与速度的区别

📚 A-Level Physics: Speed vs Velocity | A-Level 物理：速率与速度的区别

In A-Level Physics, one of the most fundamental distinctions students must master is the difference between speed and velocity. While both describe how fast an object moves, they differ in a crucial way: velocity includes direction, while speed does not. This article will break down the definitions, equations, and graphical interpretations, with worked examples tailored to the CIE syllabus.

在 A-Level 物理中，学生必须掌握的最基本概念之一就是速率与速度的区别。虽然两者都描述物体运动的快慢，但它们在一个关键点上不同：速度包含方向，而速率不包含。本文将详细拆解定义、方程和图像解读，并提供贴合 CIE 考纲的实例。

1. Scalar vs Vector Quantities | 标量与矢量

To understand the difference, you must first understand two categories of physical quantities. A scalar quantity has magnitude (size) only. A vector quantity has both magnitude and direction. Speed is a scalar; velocity is a vector.

要理解两者的区别，首先必须理解物理量的两大类别。标量只有大小；矢量既有大小又有方向。速率是标量；速度是矢量。

Common scalar quantities include distance, speed, mass, time, energy, and temperature. Common vector quantities include displacement, velocity, acceleration, force, and momentum. This classification is not just academic — it determines how the quantities are added and manipulated in calculations.

常见的标量包括距离、速率、质量、时间、能量和温度。常见的矢量包括位移、速度、加速度、力和动量。这种分类并非只是学术性的——它决定了这些量在计算中如何相加和运算。

Scalar = magnitude only | 标量 = 仅有大小
Vector = magnitude + direction | 矢量 = 大小 + 方向

2. Distance vs Displacement | 距离与位移

Before comparing speed and velocity directly, we must distinguish between distance and displacement. Distance is the total length of the path travelled, regardless of direction. It is a scalar measured in metres (m). Displacement is the straight-line distance from the starting point to the finishing point, plus the direction of that line. It is a vector, also measured in metres.

在直接比较速率与速度之前，我们必须先区分距离和位移。距离是所走路径的总长度，不考虑方向，是标量，单位为米（m）。位移是从起点到终点的直线距离，加上该直线的方向，是矢量，单位也为米。

Consider a runner completing one lap of a 400 m track. The distance travelled is 400 m, but the displacement is 0 m because the runner returns to the start. This example alone shows why distance and displacement cannot be used interchangeably.

设想一位跑者在 400 m 跑道上跑完一圈。所走的距离是 400 m，但位移是 0 m，因为跑者回到了起点。仅这个例子就足以说明为什么距离和位移不能混用。

Quantity	Type	Definition
Distance 距离	Scalar 标量	Total path length 路径总长度
Displacement 位移	Vector 矢量	Straight-line change in position 位置直线变化

3. Defining Speed | 速率的定义

Speed is defined as the rate of change of distance. In equation form:

速率定义为距离的变化率。用方程表示为：

speed = distance ÷ time

The SI unit of speed is metres per second (m s⁻¹), though kilometres per hour (km h⁻¹) is also common in everyday contexts. Because distance and time are both scalars, speed is always positive (or zero) and carries no directional information.

速率的 SI 单位是米每秒（m s⁻¹），虽然日常生活中也常用千米每小时（km h⁻¹）。由于距离和时间都是标量，速率始终为正值（或零），不携带任何方向信息。

There are two types of speed you need to distinguish: average speed and instantaneous speed. Average speed is the total distance divided by the total time. Instantaneous speed is the speed at a particular instant, found from the gradient of a distance–time graph at that point.

你需要区分两种速率：平均速率和瞬时速率。平均速率是总距离除以总时间。瞬时速率是某一瞬间的速率，可通过距离–时间图像在该点的斜率求得。

4. Defining Velocity | 速度的定义

Velocity is defined as the rate of change of displacement. In equation form:

速度定义为位移的变化率。用方程表示为：

velocity = displacement ÷ time

Since displacement is a vector, velocity is also a vector. The magnitude of velocity is often called speed, but velocity always carries a direction — for example, “25 m s⁻¹ due north” is a velocity, while “25 m s⁻¹” alone is a speed.

由于位移是矢量，速度也是矢量。速度的大小常被称为速率，但速度总是带有方向——例如，”25 m s⁻¹ 向北”是速度，而仅仅”25 m s⁻¹”是速率。

This directional component becomes crucial when objects change direction. A car driving around a roundabout at constant speed is not moving at constant velocity, because its direction is constantly changing.

当物体改变方向时，方向分量变得至关重要。一辆以恒定速率绕环岛行驶的汽车，其速度并非恒定，因为它的方向在不断改变。

5. The Key Difference: Direction | 关键区别：方向

The single most important difference between speed and velocity is direction. Speed tells you how fast an object is moving. Velocity tells you how fast and in which direction it is moving. When direction is irrelevant, the two terms can be used interchangeably; when direction matters, only velocity can be used.

速率与速度之间最重要的区别就是方向。速率告诉你物体移动多快。速度告诉你物体以多快速度朝哪个方向移动。当方向无关紧要时，两者可以互换使用；当方向重要时，只能用速度。

A simple example: two cars travel at 60 km h⁻¹ — one heading east, one heading west. Their speeds are identical (60 km h⁻¹), but their velocities are different (+60 km h⁻¹ and −60 km h⁻¹ if east is taken as positive). This sign convention is a standard way to represent direction in one-dimensional motion.

一个简单的例子：两辆车都以 60 km h⁻¹ 行驶——一辆向东，一辆向西。它们的速率相同（60 km h⁻¹），但速度不同（若以向东为正，则分别为 +60 km h⁻¹ 和 −60 km h⁻¹）。这种正负号约定是在一维运动中表示方向的标准方法。

Speed: how fast | 速率：有多快
Velocity: how fast + which way | 速度：有多快 + 往哪去

6. Average vs Instantaneous Values | 平均值与瞬时值

Both speed and velocity can be expressed as average or instantaneous values, and CIE exams frequently test the distinction. Average velocity is total displacement divided by total time; average speed is total distance divided by total time. Instantaneous velocity is the limit of average velocity as the time interval approaches zero — in practice, the gradient of a displacement–time graph at a specific point.

速率和速度都可以用平均值或瞬时值表示，CIE 考试经常考查这一区别。平均速度是总位移除以总时间；平均速率是总距离除以总时间。瞬时速度是时间间隔趋近于零时平均速度的极限——实际上就是位移–时间图像在某一点的斜率。

Consider a classic exam scenario: an object travels 100 m north in 20 s, then 100 m south in another 20 s. The total distance is 200 m, so the average speed is 200 ÷ 40 = 5 m s⁻¹. The total displacement is 0 m, so the average velocity is 0 m s⁻¹. This dramatic contrast is a favourite exam question.

考虑一个经典考试场景：一个物体向北走 100 m 用了 20 s，然后向南走 100 m 又用了 20 s。总距离为 200 m，所以平均速率为 200 ÷ 40 = 5 m s⁻¹。总位移为 0 m，所以平均速度为 0 m s⁻¹。这个鲜明的对比是考试中非常喜欢考查的题目。

7. Graphical Interpretation | 图像解读

Graphs are essential tools in A-Level Physics. A distance–time graph has gradient equal to speed. A displacement–time graph has gradient equal to velocity. If the displacement–time graph is a straight line with a positive slope, the velocity is constant and positive. If the slope becomes steeper, the velocity is increasing; if the line curves downwards, the object is slowing down.

图像是 A-Level 物理中必不可少的工具。距离–时间图像的斜率等于速率。位移–时间图像的斜率等于速度。如果位移–时间图像是一条斜率为正的直线，则速度恒定且为正。如果斜率变得更陡，则速度在增大；如果曲线向下弯曲，则物体在减速。

One subtle point: a distance–time graph can never have a negative gradient, because distance cannot decrease. However, a displacement–time graph can have a negative gradient, indicating motion in the negative direction. This is because displacement is a vector and can take negative values.

一个微妙的要点：距离–时间图像的斜率永远不可能为负，因为距离不能减少。然而，位移–时间图像的斜率可以为负，表示向负方向运动。这是因为位移是矢量，可以取负值。

8. Worked Example 1: Straight-Line Motion | 实例 1：直线运动

A cyclist travels 240 m east in 30 s, then 120 m west in 20 s. Calculate (a) the total distance travelled, (b) the total displacement, (c) the average speed, and (d) the average velocity.

一名骑行者向东骑行 240 m 用了 30 s，然后向西骑行 120 m 用了 20 s。计算 (a) 总路程，(b) 总位移，(c) 平均速率，(d) 平均速度。

Solution 解答：
(a) Total distance = 240 + 120 = 360 m
(b) Taking east as positive, total displacement = 240 + (−120) = 120 m east
(c) Average speed = 360 ÷ (30 + 20) = 360 ÷ 50 = 7.2 m s⁻¹
(d) Average velocity = 120 ÷ 50 = 2.4 m s⁻¹ east

解答：
(a) 总路程 = 240 + 120 = 360 m
(b) 以向东为正，总位移 = 240 + (−120) = 120 m 向东
(c) 平均速率 = 360 ÷ (30 + 20) = 360 ÷ 50 = 7.2 m s⁻¹
(d) 平均速度 = 120 ÷ 50 = 2.4 m s⁻¹ 向东

Notice the average speed (7.2 m s⁻¹) is much larger than the average velocity (2.4 m s⁻¹). This is normal: average speed ≥ magnitude of average velocity, with equality only when motion is in a single straight line without reversing direction.

注意平均速率（7.2 m s⁻¹）远大于平均速度的大小（2.4 m s⁻¹）。这是正常的：平均速率 ≥ 平均速度的大小，只有沿单一方向直线运动且不折返时两者才相等。

9. Worked Example 2: Circular Motion | 实例 2：圆周运动

A particle moves at a constant speed of 5 m s⁻¹ around a circular track of circumference 200 m. After completing exactly one quarter of the circle (50 m of arc), what are the magnitude of its velocity and its average velocity?

一个质点以恒定速率 5 m s⁻¹ 沿周长为 200 m 的圆形轨道运动。在恰好完成四分之一圈（50 m 弧长）后，其速度大小和平均速度大小各是多少？

Solution 解答：

The magnitude of instantaneous velocity equals the speed, since speed is the magnitude of velocity. So the instantaneous velocity magnitude is 5 m s⁻¹. For average velocity, we need displacement after a quarter circle. The displacement is the chord of the quarter circle. If the radius is R, then 2πR = 200, so R = 200 ÷ 2π ≈ 31.8 m. The quarter-circle chord length is R√2 ≈ 45.0 m. The time taken is 50 ÷ 5 = 10 s. Therefore average velocity = 45.0 ÷ 10 = 4.5 m s⁻¹ (at 45° to the initial direction).

瞬时速度的大小等于速率，因为速率是速度的大小。所以瞬时速度大小为 5 m s⁻¹。对于平均速度，我们需要四分之一圈后的位移。位移是四分之一圆的弦长。若半径为 R，则 2πR = 200，所以 R = 200 ÷ 2π ≈ 31.8 m。四分之一圆的弦长为 R√2 ≈ 45.0 m。所用时间为 50 ÷ 5 = 10 s。因此平均速度 = 45.0 ÷ 10 = 4.5 m s⁻¹（方向与初始方向成 45°）。

Speed remains constant: 5 m s⁻¹ | 速率保持恒定：5 m s⁻¹
Average velocity magnitude: ≈ 4.5 m s⁻¹ | 平均速度大小：≈ 4.5 m s⁻¹

This example demonstrates that even with constant speed, the velocity changes continuously because direction changes. A velocity–time graph for circular motion would not be a horizontal line even though speed is constant.

这个例子说明，即使速率恒定，速度也在不断变化，因为方向在改变。匀速圆周运动的速度–时间图像不会是一条水平线，即使速率恒定。

10. Common Misconceptions | 常见误解

Students often make the following mistakes in exams. First, assuming velocity is always positive — velocity can be negative depending on the chosen reference direction. Second, believing that constant speed implies constant velocity — false, as circular motion proves. Third, confusing “magnitude of average velocity” with “average speed” — these are different quantities computed from displacement and distance respectively.

学生在考试中经常犯以下错误。第一，以为速度总为正——速度可以为负，取决于所选参考方向。第二，认为恒定速率意味着恒定速度——错误，圆周运动就是反例。第三，混淆”平均速度的大小”与”平均速率”——这是两个不同的量，分别由位移和距离计算。

Another frequent error: using distance in a velocity equation or using displacement in a speed equation. Always check: if the word “velocity” appears, use displacement; if “speed” appears, use distance. This simple check can save many marks.

另一个常见错误：在速度方程中使用距离，或在速率方程中使用位移。始终检查：如果题目中出现”速度”，就用位移；如果出现”速率”，就用距离。这个简单的检查可以帮你避免丢分。

11. Exam Tips for CIE | CIE 考试提示

CIE examiners report that candidates lose marks for: omitting units, failing to state direction for vector answers, and writing vectors as scalars in final answers. When a question asks for velocity, your final answer must include direction — for example, “3.0 m s⁻¹ to the right” or “3.0 m s⁻¹ in the positive x-direction.” Magnitude alone will not earn full marks.

CIE 考官报告指出，考生因以下原因丢分：漏写单位、矢量答案未注明方向、把矢量写成标量。当题目要求速度时，最终答案必须包含方向——例如，”3.0 m s⁻¹ 向右”或”3.0 m s⁻¹ 沿 x 轴正方向”。仅有大小无法得到满分。

Also remember: if displacement is zero, average velocity is zero regardless of the distance travelled. Sign convention should be stated clearly at the start of your solution. If you define a positive direction, stick to it consistently throughout the calculation.

还要记住：如果位移为零，无论走了多少路程，平均速度都为零。在解题开始时明确说明正方向约定。如果你定义了一个正方向，在整个计算过程中必须始终如一地使用它。

12. Summary Table | 总结表

Feature 特征	Speed 速率	Velocity 速度
Type 类型	Scalar 标量	Vector 矢量
Definition 定义	Rate of change of distance 距离变化率	Rate of change of displacement 位移变化率
Formula 公式	distance ÷ time 距离 ÷ 时间	displacement ÷ time 位移 ÷ 时间
Direction 方向	No 无	Yes 有
Can be negative? 可为负？	Never 不可	Yes 可以
Graph gradient 图像斜率	Distance–time graph 距离–时间图	Displacement–time graph 位移–时间图

The distinction between speed and velocity is a cornerstone of kinematics. Master this concept, and you will find related topics — acceleration, momentum, and circular motion — far more approachable. Remember the golden rule: speed is scalar, velocity is vector; velocity = speed + direction.

速率与速度的区别是运动学的基石。掌握这个概念，你会发现后续的相关主题——加速度、动量和圆周运动——都容易得多。记住黄金法则：速率是标量，速度是矢量；速度 = 速率 + 方向。

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A-Level Physics: Speed vs Velocity | A-Level 物理：速率与速度的区别