Graphs and Kinematics | 图形与运动学

📚 Graphs and Kinematics | 图形与运动学

Graphs are one of the most powerful tools in mathematics, especially when they are used to describe the motion of objects. In the IGCSE syllabus, you will meet two important types of graph: distance-time graphs and speed-time graphs. Understanding these graphs, how to read them, and how to calculate values from them is essential for success in the exams.

图形是数学中最有力的工具之一，尤其当它们用于描述物体运动的时候。在 IGCSE 课程中，你将会遇到两类重要的图形：距离-时间图和速度-时间图。理解这些图形、如何读取它们以及如何从中计算数值，对于在考试中取得成功至关重要。

1. Coordinate Geometry Basics | 坐标几何基础

Before studying motion graphs, you must be confident with the Cartesian coordinate system. Any point on a plane can be described by an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

在学习运动图形之前，你必须熟练掌握笛卡尔坐标系。平面上的任意一点都可以用有序数对 (x, y) 来描述，其中 x 是水平坐标，y 是垂直坐标。

The distance between two points (x₁, y₁) and (x₂, y₂) can be found using the distance formula. This formula is derived from Pythagoras’ theorem.

两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离可以用距离公式求出。该公式是由勾股定理推导而来的。

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The midpoint of a segment joining two points is found by averaging the x-coordinates and averaging the y-coordinates. This gives the point exactly halfway between them.

连接两点的线段的中点，可以通过对 x 坐标取平均、对 y 坐标取平均来求得。这样得到的就是恰好位于两者之间的点。

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Practice plotting points carefully and labelling axes clearly.
Always write coordinates in the correct order: (x, y).

认真练习描点，并清楚标注坐标轴。
始终按正确顺序书写坐标：(x, y)。

2. Gradients of Straight Lines | 直线斜率

The gradient of a straight line measures how steep the line is. It is defined as the change in y divided by the change in x between two points on the line.

直线的斜率衡量直线的陡峭程度。它定义为直线上两点之间的 y 变化量除以 x 变化量。

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line slopes upwards from left to right. A negative gradient means the line slopes downwards. A horizontal line has a gradient of zero, and a vertical line has an undefined gradient.

正斜率表示直线从左向右上升。负斜率表示直线从左向右下降。水平线的斜率为零，垂直线的斜率没有定义。

If two lines are parallel, their gradients are equal. If two lines are perpendicular, the product of their gradients is −1.

如果两条直线平行，则它们的斜率相等。如果两条直线垂直，则它们斜率的乘积为 −1。

m₁ × m₂ = −1

For parallel lines: m₁ = m₂.
For perpendicular lines: m₁ × m₂ = −1.

对于平行线：m₁ = m₂。
对于垂直线：m₁ × m₂ = −1。

3. Equations of Straight Lines | 直线方程

The most common form of the equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept, the point where the line crosses the y-axis.

直线方程最常见的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距，即直线与 y 轴相交的位置。

y = mx + c

To find the equation of a line, you need the gradient and one point on the line. Substitute the coordinates of the point into y = mx + c to find the value of c, then write the full equation.

要求一条直线的方程，你需要知道斜率以及直线上一个点的坐标。将该点坐标代入 y = mx + c 求出 c 的值，然后写出完整方程。

Sometimes the equation is given in the form ax + by = c. You can rearrange it into the form y = mx + c to read off the gradient and intercept more easily.

有时方程以 ax + by = c 的形式给出。你可以将其整理成 y = mx + c 的形式，以便更方便地读出斜率和截距。

If a line passes through (2, 5) with gradient 3, then c = 5 − 3×2 = −1, so y = 3x − 1.

Always check your equation by substituting the given point.

如果一条直线经过点 (2, 5) 且斜率为 3，那么 c = 5 − 3×2 = −1，因此 y = 3x − 1。
始终通过代入已知点来检验你的方程是否正确。

However, I must correct the previous bullet: the formula should be c = 5 − 3(2) = −1, which gives y = 3x − 1. Checking: 3(2) − 1 = 5, correct.

不过，我需要更正上一条要点的内容：公式应为 c = 5 − 3(2) = −1，得到 y = 3x − 1。检验：3(2) − 1 = 5，正确。

4. Distance-Time Graphs | 距离-时间图

A distance-time graph shows how the distance travelled by an object changes over time. The horizontal axis represents time (t), usually measured in seconds, and the vertical axis represents distance (d), usually measured in metres.

距离-时间图展示物体所经过的路程如何随时间变化。横轴表示时间 (t)，通常以秒为单位；纵轴表示距离 (d)，通常以米为单位。

The key fact is that the gradient of a distance-time graph represents the speed of the object. If the gradient is constant, the object is moving at a constant speed.

关键事实是：距离-时间图的斜率代表物体的速度。如果斜率恒定，则物体以恒定速度运动。

If the line is horizontal, the gradient is zero, so the speed is zero: the object is stationary. If the line is curved, the speed is changing, meaning the object is accelerating or decelerating.

如果图线是水平的，斜率为零，因此速度为零：物体静止。如果图线是弯曲的，则速度在变化，说明物体在加速或减速。

Speed = gradient of distance-time graph = change in distance / change in time

A steep line means a fast speed.
A horizontal line means the object is not moving.
A curved line means acceleration or deceleration.

陡峭的图线意味着速度快。
水平图线意味着物体没有移动。
弯曲的图线意味着加速或减速。

5. Speed-Time Graphs | 速度-时间图

A speed-time graph shows how the speed of an object changes over time. Here the horizontal axis is time (t) and the vertical axis is speed (v), usually measured in metres per second (m/s).

速度-时间图展示物体的速度如何随时间变化。这里横轴是时间 (t)，纵轴是速度 (v)，通常以米每秒 (m/s) 为单位。

On a speed-time graph, the gradient represents acceleration. A positive gradient means the object is speeding up; a negative gradient means it is slowing down; a zero gradient means the speed is constant.

在速度-时间图上，斜率代表加速度。正斜率表示物体在加速；负斜率表示物体在减速；零斜率表示速度恒定。

The area under a speed-time graph represents the distance travelled. This is one of the most important concepts in kinematics and is frequently tested.

速度-时间图下的面积代表物体所经过的路程。这是运动学中最重要的概念之一，也是考试中经常考查的内容。

Distance travelled = area under speed-time graph

The gradient gives acceleration: m = (v₂ − v₁) / (t₂ − t₁).
The area under the graph gives distance.

斜率给出加速度：m = (v₂ − v₁) / (t₂ − t₁)。
图线下方面积给出距离。

6. Finding Speed from Distance-Time Graphs | 从距离-时间图求速度

To find the speed of an object from a distance-time graph, you must calculate the gradient of the straight line segment. Choose two points on the line, read their coordinates, and apply the gradient formula.

要从距离-时间图求物体的速度，你必须计算直线线段的斜率。在直线上选择两个点，读取它们的坐标，然后应用斜率公式。

For example, suppose an object is at 20 m at t = 2 s and at 60 m at t = 7 s. The speed is (60 − 20) / (7 − 2) = 40 / 5 = 8 m/s.

例如，假设物体在 t = 2 s 时位于 20 m 处，在 t = 7 s 时位于 60 m 处。速度为 (60 − 20) / (7 − 2) = 40 / 5 = 8 m/s。

If the graph is curved, the speed at a particular instant is found by drawing a tangent to the curve at that point and finding the gradient of the tangent.

如果图线是弯曲的，则某一瞬间的速度需要通过在该点画切线，并求出切线的斜率来得到。

Instantaneous speed = gradient of tangent on distance-time graph

Use a ruler to draw tangents accurately.
Choose large, easily readable coordinate points to reduce errors.

使用直尺准确绘制切线。
选择大而易于读数的坐标点以减少误差。

7. Finding Distance and Acceleration from Speed-Time Graphs | 从速度-时间图求距离与加速度

To find acceleration from a speed-time graph, calculate the gradient. For example, if the speed increases from 10 m/s to 30 m/s over 5 seconds, the acceleration is (30 − 10) / 5 = 4 m/s².

要从速度-时间图求加速度，直接计算斜率。例如，如果速度在 5 秒内从 10 m/s 增加到 30 m/s，则加速度为 (30 − 10) / 5 = 4 m/s²。

To find distance, split the area under the graph into simple shapes: rectangles, triangles, and trapeziums. Calculate each area and add them together.

要求距离，请将图线下方的区域分割成简单形状：矩形、三角形和梯形。计算每个面积然后相加。

Consider a speed-time graph with a triangle at the start, a rectangle in the middle, and a triangle at the end. The total distance is the sum of the three areas.

考虑一个速度-时间图，开始是三角形，中间是矩形，末尾是三角形。总距离就是这三个面积之和。

Area of trapezium = (a + b) / 2 × h

For example, a car accelerates from rest at 2 m/s² for 5 s, travels at 10 m/s for 10 s, then brakes to rest in 5 s. The total distance is (0.5 × 5 × 10) + (10 × 10) + (0.5 × 5 × 10) = 25 + 100 + 25 = 150 m.

例如，一辆汽车从静止开始以 2 m/s² 加速 5 秒，然后以 10 m/s 匀速行驶 10 秒，最后在 5 秒内刹车至静止。总距离为 (0.5 × 5 × 10) + (10 × 10) + (0.5 × 5 × 10) = 25 + 100 + 25 = 150 m。

Identify shapes carefully before calculating areas.
Use consistent units: metres and seconds for distance and time.

在计算面积之前认真识别图形的形状。
使用一致的单位：距离用米，时间用秒。

8. Average Speed and Velocity | 平均速度与速率

Average speed is calculated by dividing the total distance travelled by the total time taken. This is different from the instantaneous speed, which is the speed at a particular moment.

平均速度通过总路程除以总时间来计算。这与瞬时速度不同，瞬时速度是指某一特定瞬间的速度。

Average speed = total distance / total time

Velocity is a vector quantity: it has both magnitude and direction. Speed is a scalar quantity, meaning it only has magnitude. In IGCSE mathematics, you are usually asked to work with speed, but you should understand the distinction.

速度是一个矢量：它既有大小也有方向。速率是一个标量，意味着它只有大小。在 IGCSE 数学中，通常要求你处理速率，但你应该理解两者之间的区别。

For example, if a person walks 300 m north in 60 s, the average speed is 300 / 60 = 5 m/s. The velocity is also 5 m/s north, but the direction must be stated.

例如，如果一个人向北走 300 m 用时 60 s，平均速率为 300 / 60 = 5 m/s。速度也是 5 m/s，方向向北，但必须说明方向。

Speed = distance / time, always positive.
Velocity = displacement / time, can be positive or negative depending on direction.

速率 = 距离 / 时间，始终为正值。
速度 = 位移 / 时间，可正可负，取决于方向。

9. Solving Kinematics Problems with Graphs | 用图像解决运动学问题

Many exam problems combine both types of graph. A typical question gives you a distance-time graph and asks for speed, or gives a speed-time graph and asks for acceleration and distance. Always start by reading the axes carefully to determine which type of graph you have.

许多考试题目将两类图像结合在一起。典型问题会给你一个距离-时间图并要求求速度，或者给你一个速度-时间图并要求求加速度和距离。开始解题时务必仔细阅读坐标轴，以确定你面对的是哪种类型的图。

A common strategy is to highlight the relevant section of the graph, then decide whether you need to find a gradient or an area. If the question asks for speed or acceleration, find the gradient. If it asks for distance, find the area under the curve.

一个常见策略是突出图中相关的部分，然后决定你需要求斜率还是面积。如果问题问的是速度或加速度，就求斜率。如果问的是距离，就求曲线下方的面积。

When dealing with a speed-time graph, remember that the gradient gives acceleration and the area gives distance. When dealing with a distance-time graph, the gradient gives speed, and there is no meaningful interpretation of area.

处理速度-时间图时，记住斜率给出加速度，面积给出距离。处理距离-时间图时，斜率给出速度，而面积没有有意义的解释。

Quantity	Distance-Time Graph	Speed-Time Graph
Gradient	Speed	Acceleration
Area under graph	No meaning	Distance
Horizontal line	Stationary	Constant speed

物理量	距离-时间图	速度-时间图
斜率	速度	加速度
图线下方区域	无意义	距离
水平线	静止	匀速

10. Common Pitfalls and Exam Tips | 常见错误与应试技巧

One of the most common mistakes is confusing distance-time graphs with speed-time graphs. In a distance-time graph, the gradient is speed; in a speed-time graph, the gradient is acceleration and the area is distance. Remember these facts repeatedly.

最常见的错误之一是将距离-时间图与速度-时间图混淆。在距离-时间图中，斜率是速度；在速度-时间图中，斜率是加速度，面积是距离。请反复记住这些事实。

Another common error is using the wrong units. If time is given in minutes, convert it to seconds before calculating speed in m/s. If distance is in kilometres, convert to metres when necessary.

另一个常见错误是使用错误的单位。如果时间是以分钟给出的，在计算以 m/s 为单位的速度之前，先将其转换为秒。如果距离是千米，必要时转换为米。

When calculating the area under a speed-time graph, check the shape carefully. A trapezium requires the formula (a + b) / 2 × h, while a triangle uses 0.5 × base × height. Misidentifying shapes leads to wrong answers.

在计算速度-时间图下的面积时，请仔细检查形状。梯形需要使用公式 (a + b) / 2 × h，而三角形使用 0.5 × 底 × 高。错误识别形状会导致结果不正确。

Always label your axes and units in any sketch.
Show your working step by step; marks are awarded for method.
Check whether the answer is reasonable: a car cannot travel at 1000 m/s in normal conditions.

在任何草图中都标注坐标轴和单位。
逐步展示你的计算过程；方法正确可以获得步骤分。
检查答案是否合理：正常条件下汽车不可能以 1000 m/s 行驶。

Graphs and kinematics are not just abstract exercises; they help us understand how objects move in the real world. By mastering the relationship between gradients, areas, and motion, you will be well prepared for the IGCSE examinations.

图形与运动学不仅仅是抽象的练习；它们帮助我们理解物体在现实世界中是如何运动的。通过掌握斜率、面积与运动之间的关系，你将能够从容应对 IGCSE 考试。

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Graphs and Kinematics | 图形与运动学