Speed: Definition and Calculation in A-Level Physics | A-Level 物理：速率的定义与计算

📚 Speed: Definition and Calculation in A-Level Physics | A-Level 物理：速率的定义与计算

In A-Level Physics, speed is one of the most fundamental quantities you will meet. It tells us how fast an object is moving, but it does not tell us the direction of motion. This article will define speed precisely, distinguish it from velocity, and show you how to calculate average speed, instantaneous speed and related quantities using graphs and equations.

在 A-Level 物理中，速率是最基本的概念之一。它描述物体运动的快慢，但不涉及运动方向。本文将精确定义速率、区分速率与速度，并讲解如何利用图线和运动学方程计算平均速率、瞬时速率及相关量。

1. Scalars and Vectors | 标量与向量

In physics, quantities are divided into scalars and vectors. A scalar has magnitude only, while a vector has both magnitude and direction. Distance and speed are scalars; displacement and velocity are vectors.

在物理学中，物理量分为标量与向量。标量只有大小，向量既有大小又有方向。距离和速率是标量，位移和速度是向量。

Distance is the total length of the path travelled, while displacement is the straight-line distance from the starting point to the finishing point in a particular direction. Speed is the rate of change of distance, and velocity is the rate of change of displacement.

距离是物体实际运动轨迹的总长度，而位移是从起点到终点的直线距离且有确定方向。速率是距离随时间的变化率，速度是位移随时间的变化率。

The standard unit of speed is the metre per second, written as m s⁻¹ or m/s. Other common units include kilometres per hour (km/h) and miles per hour (mph).

速率的国际单位是米每秒，记作 m s⁻¹ 或 m/s。其他常用单位包括千米每小时（km/h）和英里每小时（mph）。

2. Average Speed | 平均速率

Average speed is defined as the total distance travelled divided by the total time taken. It gives a single value that represents the overall rate of motion over a journey.

平均速率定义为物体运动的总距离除以所用的总时间。它用一个数值表示整个运动过程的总体快慢。

平均速率 = 总距离 ÷ 总时间 = Δs / Δt

Here, Δs is the total distance travelled and Δt is the total time taken. This equation is valid for any motion, whether the speed is constant or changing.

其中 Δs 是总距离，Δt 是总时间。无论物体速率恒定还是变化，该公式都适用。

For example, if a runner completes a 200 m lap in 40 s, the average speed is 200 ÷ 40 = 5 m s⁻¹. Notice that if the runner returns to the starting point, the displacement is zero, so the average velocity would be zero, but the average speed is not zero.

例如，如果一位跑步者用 40 s 跑完 200 m 的跑道一圈，则平均速率为 200 ÷ 40 = 5 m s⁻¹。注意，如果跑步者回到起点，位移为零，平均速度为零，但平均速率不为零。

3. Instantaneous Speed | 瞬时速率

Instantaneous speed is the speed of an object at a particular instant in time. It is found by taking an extremely small time interval Δt and an extremely small distance Δs, then calculating the ratio Δs / Δt as Δt approaches zero.

瞬时速率是物体在某一瞬间的速率。它通过取极短的时间间隔 Δt 和极小的距离 Δs，并在 Δt 趋近于零时计算比值 Δs / Δt 来获得。

On a distance-time graph, the instantaneous speed at any point is the gradient of the tangent to the curve at that point. The steeper the tangent, the greater the speed.

在距离-时间图像中，任意时刻的瞬时速率等于该点切线的斜率。切线越陡，速率越大。

If the object is moving at constant speed, the instantaneous speed is equal to the average speed. If the speed is changing, the two are generally different.

如果物体做匀速运动，瞬时速率等于平均速率；如果速率在变化，两者通常不同。

4. Distance-Time Graphs | 距离-时间图像

A distance-time graph shows how the distance travelled by an object changes with time. The gradient of a distance-time graph gives the speed of the object.

距离-时间图像表示物体运动的距离随时间的变化关系。该图像的斜率表示物体的速率。

If the graph is a straight line through the origin, the object is moving at constant speed. If the graph is horizontal, the object is stationary. If the graph curves upwards, the speed is increasing; if it curves downwards, the speed is decreasing.

如果图像是通过原点的直线，物体做匀速运动；如果图像是水平线，物体静止；如果图像向上弯曲，速率增大；如果向下弯曲，速率减小。

Gradient = rise / run = Δdistance / Δtime = speed
斜率 = 纵坐标变化量 ÷ 横坐标变化量 = Δ距离 ÷ Δ时间 = 速率

When calculating the speed from a curved graph, draw a tangent at the point of interest and measure the gradient of that tangent. Do not use the chord between two distant points unless you want the average speed over that interval.

当从弯曲图像中求速率时，应在感兴趣的点画切线并测量切线斜率。除非需要某段时间内的平均速率，否则不要使用两个远点之间的割线。

5. Speed-Time Graphs | 速率-时间图像

A speed-time graph plots speed on the vertical axis and time on the horizontal axis. The gradient of a speed-time graph gives the acceleration of the object.

速率-时间图像以速率为纵轴、时间为横轴。该图像的斜率表示物体的加速度。

The area under a speed-time graph represents the total distance travelled. This is because distance = speed × time, and the area of each small strip of the graph equals the distance covered in that time interval.

速率-时间图像与时间轴围成的面积表示物体运动的总距离。原因在于距离 = 速率 × 时间，图像中每个小条的面积等于该时间间隔内通过的距离。

Gradient of speed-time graph = acceleration
Area under speed-time graph = distance travelled
速率-时间图像的斜率 = 加速度
速率-时间图像下方的面积 = 通过的总距离

If the graph shows velocity instead of speed, the area gives displacement, not distance. Since speed is always positive, a speed-time graph never goes below the time axis.

如果图像表示的是速度而不是速率，则面积表示位移而非距离。由于速率始终为正，速率-时间图像不会低于时间轴。

6. Kinematic Equations and Speed Calculations | 运动学方程与速率计算

For motion with constant acceleration, we can use the kinematic equations, also called SUVAT equations. In these equations, u is the initial velocity, v is the final velocity, a is the acceleration, t is the time, and s is the displacement.

对于匀加速运动，可以使用运动学方程，也称 SUVAT 方程。在这些方程中，u 是初速度，v 是末速度，a 是加速度，t 是时间，s 是位移。

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

These equations are vector equations, so direction matters. However, if motion is along a straight line and the direction does not change, the magnitude of velocity equals speed. In such cases, you can use these equations to calculate speed.

这些方程是向量方程，因此方向很重要。但如果物体沿直线运动且方向不变，速度的大小就等于速率。在这种情况下，可以用这些方程计算速率。

For example, a car accelerates from rest at 2 m s⁻² for 5 s. The final speed is v = 0 + 2 × 5 = 10 m s⁻¹. The distance covered is s = 0 × 5 + ½ × 2 × 5² = 25 m.

例如，一辆汽车从静止开始以 2 m s⁻² 的加速度行驶 5 s。末速率 v = 0 + 2 × 5 = 10 m s⁻¹。通过的距离为 s = 0 × 5 + ½ × 2 × 5² = 25 m。

7. Unit Conversions | 单位换算

Speed can be expressed in many units. In A-Level physics, you must be able to convert between m s⁻¹ and km/h and vice versa.

速率可以用多种单位表示。在 A-Level 物理中，必须能在 m s⁻¹ 与 km/h 之间进行换算。

1 m s⁻¹ = 3.6 km/h

1 km/h = 1 / 3.6 m s⁻¹ ≈ 0.278 m s⁻¹

To convert from km/h to m s⁻¹, divide by 3.6. To convert from m s⁻¹ to km/h, multiply by 3.6.

将 km/h 换算为 m s⁻¹ 时，除以 3.6；将 m s⁻¹ 换算为 km/h 时，乘以 3.6。

Example: A train travels at 108 km/h. Dividing by 3.6 gives 30 m s⁻¹. This is a common exam skill.

示例：一列火车以 108 km/h 行驶。除以 3.6 得到 30 m s⁻¹。这是常见的考试技巧。

8. Worked Examples | 典型例题

Let us apply these ideas to typical exam-style problems.

下面将上述知识应用于典型考试风格的题目中。

Example 1: A cyclist travels 360 m north in 60 s, then 480 m south in 120 s. Find the average speed and the average velocity for the whole journey.

例 1：一名骑车人先向北骑行 360 m，用时 60 s；再向南骑行 480 m，用时 120 s。求全程的平均速率和平均速度。

Average speed = total distance ÷ total time = (360 + 480) ÷ (60 + 120) = 840 ÷ 180 = 4.67 m s⁻¹.

平均速率 = 总距离 ÷ 总时间 = (360 + 480) ÷ (60 + 120) = 840 ÷ 180 = 4.67 m s⁻¹。

For average velocity, net displacement = 360 m north + 480 m south = 120 m south. Average velocity = 120 ÷ 180 = 0.67 m s⁻¹ south.

对于平均速度，合位移 = 向北 360 m + 向南 480 m = 向南 120 m。平均速度 = 120 ÷ 180 = 0.67 m s⁻¹，方向向南。

Example 2: The distance-time graph of a particle is a straight line passing through (0 s, 0 m) and (4 s, 20 m). Calculate the speed.

例 2：某质点的距离-时间图像是过 (0 s, 0 m) 和 (4 s, 20 m) 两点的直线。求速率。

Speed = gradient = (20 – 0) ÷ (4 – 0) = 5 m s⁻¹.

速率 = 斜率 = (20 – 0) ÷ (4 – 0) = 5 m s⁻¹。

Example 3: A ball is dropped from rest and falls with constant acceleration 9.8 m s⁻². Find its speed after 3.0 s.

例 3：一个小球从静止开始下落，加速度恒为 9.8 m s⁻²。求 3.0 s 后小球的速率。

Using v = u + at, v = 0 + 9.8 × 3.0 = 29.4 m s⁻¹.

由 v = u + at，v = 0 + 9.8 × 3.0 = 29.4 m s⁻¹。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks by confusing speed and velocity, or by misreading graphs. Careful attention to definitions and units will improve your score.

很多学生因为混淆速率与速度，或读错图像而失分。仔细理解定义并注意单位可以提高分数。

Do not say speed is negative; speed is a scalar and is always positive. Velocity can be negative.
When using a distance-time graph, the gradient always gives speed; when using a displacement-time graph, the gradient gives velocity.
Always convert time to seconds unless the question requires another unit.
For speed-time graphs, remember that the area under the graph gives distance, not displacement.
When calculating the average speed, always use total distance, not net displacement.
如果题目要求向量，别忘了注明方向。

在多项选择题中，常会给出一个距离-时间图像并询问哪个时间段速率最大。此时应比较各段线段的斜率绝对值，斜率绝对值最大处速率最大。

In multiple-choice questions, you may be shown a distance-time graph and asked which section has the greatest speed. Compare the absolute gradients of each section; the steepest section has the greatest speed.

在计算题中，若物体先加速后匀速再减速，可将运动分成几段，分别计算每段的距离，再求和得到总距离，最后除以总时间得到平均速率。

In calculation questions, if an object speeds up, moves at constant speed, and then slows down, split the motion into sections, calculate each distance, add them to find total distance, and divide by total time to find the average speed.

对于图像题，务必标出坐标轴的单位，并检查是否已经使用正确的公式。切线斜率与割线斜率含义不同，不可混淆。

For graph questions, always label axis units and verify that you have used the correct formula. Tangent gradient and chord gradient have different meanings and must not be confused.

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Speed: Definition and Calculation in A-Level Physics | A-Level 物理：速率的定义与计算