Understanding Distance-Time and Speed-Time Graphs | 理解距离-时间图与速度-时间图

📚 Understanding Distance-Time and Speed-Time Graphs | 理解距离-时间图与速度-时间图

In your IGCSE Mathematics studies, the topics of distance-time graphs and speed-time graphs are essential for describing motion. These graphs help you turn real-world movement into clear mathematical pictures, and they appear frequently in exam questions.

在 IGCSE 数学学习中，距离-时间图和速度-时间图是描述运动的重要内容。这些图帮助你以数学图像的方式清晰地呈现真实世界的运动，而且它们在考试题目中频繁出现。

1. Why Do We Use Motion Graphs? | 为什么使用运动图像？

Motion graphs allow us to summarise how an object moves in one dimension. Instead of writing long sentences, we can draw a graph and immediately see whether the object is stationary, moving at a constant speed, speeding up, or slowing down.

运动图像让我们能够概括一个物体在一维空间中的运动方式。比起书写冗长的描述，我们可以画一张图，立刻看出物体是静止、匀速运动、加速还是减速。

A distance-time graph shows how distance changes with time.
A speed-time graph shows how speed changes with time.

距离-时间图显示距离随时间如何变化。
速度-时间图显示速度随时间如何变化。

distance = speed × time

距离 = 速度 × 时间

2. Axes and Units | 坐标轴与单位

Before reading any graph, check the labels and scales on both axes. For a distance-time graph, the horizontal axis is always time (t) and the vertical axis is distance (d). For a speed-time graph, the horizontal axis is time (t) and the vertical axis is speed (v).

在阅读任何图像之前，先检查两个坐标轴的标注和刻度。对于距离-时间图，横轴始终是时间 t，纵轴是距离 d。对于速度-时间图，横轴是时间 t，纵轴是速度 v。

Graph Type	Horizontal axis	Vertical axis
Distance-time	Time (s, min, h)	Distance (m, km)
Speed-time	Time (s, min, h)	Speed (m/s, km/h)

Always include units in your answers, and be careful to convert between seconds and hours when needed.

在答案中始终要包含单位，并且在需要时小心地在秒和小时之间进行换算。

3. Distance-Time Graphs: Horizontal Lines | 距离-时间图：水平线

On a distance-time graph, a horizontal line means that the distance is not changing. This tells us that the object is stationary, or at rest.

在距离-时间图中，水平线表示距离没有变化。这告诉我们物体是静止的，即处于静止状态。

Gradient = 0 → stationary

斜率 = 0 → 静止

For example, a car stopped at a red traffic light produces a horizontal segment on its distance-time graph.

例如，一辆汽车在红绿灯前停下，其距离-时间图上就会产生一段水平线段。

4. Distance-Time Graphs: Gradient = Speed | 距离-时间图：斜率 = 速度

The most important idea for a distance-time graph is that the gradient (slope) represents the speed of the object. A steeper line means a faster speed, and a positive gradient means moving away from the start.

距离-时间图最重要的概念是：斜率代表物体的速度。线越陡，速度越快；斜率为正表示远离起点。

Speed = gradient = (change in distance) / (change in time) = Δd / Δt

速度 = 斜率 = (距离变化量) / (时间变化量) = Δd / Δt

If the line slopes downward, the object is returning towards the starting point; the gradient is negative, but speed is reported as a positive value.

如果直线向下倾斜，说明物体正在返回起点；此时斜率为负，但速度通常取正值。

Example: A cyclist travels 100 m in 20 s. The gradient is 100 ÷ 20 = 5 m/s.

示例：一名骑自行车的人在 20 秒内行驶了 100 米。斜率为 100 ÷ 20 = 5 米/秒。

5. Curved Distance-Time Graphs | 弯曲的距离-时间图

If a distance-time graph is curved, the speed is changing. A curve that becomes steeper means acceleration; a curve that becomes flatter means deceleration.

如果距离-时间图是弯曲的，说明速度在变化。曲线变得越来越陡表示加速；曲线变得越来越平缓表示减速。

To find the speed at a particular instant, draw a tangent to the curve at that point and calculate its gradient.

要找出某一瞬间的速度，需要在该点画一条切线，然后计算切线的斜率。

Instantaneous speed = gradient of tangent

瞬时速度 = 切线的斜率

6. Speed-Time Graphs: Horizontal Lines | 速度-时间图：水平线

On a speed-time graph, a horizontal line means that the speed is constant. However, a horizontal line here does not mean the object is stationary; it means the object moves at a steady speed.

在速度-时间图中，水平线表示速度恒定。不过，这里的水平线并不表示物体静止；它表示物体以稳定的速度运动。

If the line lies on the time axis at v = 0, then the object is at rest.

如果这条线位于时间轴上，即 v = 0，那么物体处于静止状态。

7. Gradient of a Speed-Time Graph = Acceleration | 速度-时间图的斜率 = 加速度

For 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 constant speed.

对于速度-时间图，斜率代表加速度。斜率为正表示物体在加速；斜率为负表示物体在减速；斜率为零表示速度恒定。

Acceleration = gradient = Δv / Δt

加速度 = 斜率 = Δv / Δt

The unit of acceleration is m/s² (metres per second squared).

加速度的单位是米/秒²（m/s²）。

8. Area Under a Speed-Time Graph = Distance | 速度-时间图下的面积 = 距离

The area enclosed between the speed-time graph and the time axis tells us the total distance travelled. This works for any shape: rectangles, triangles, trapeziums, or a combination of shapes.

速度-时间图与时间轴之间所围成的面积告诉我们总行驶距离。这对任何形状都成立：矩形、三角形、梯形，或者多个形状的组合。

Distance = area under speed-time graph

距离 = 速度-时间图下的面积

For a constant speed v over time t, the area is v × t, which matches the standard formula.

对于匀速 v 在时间 t 内的运动，面积是 v × t，这与标准公式一致。

9. Worked Example | 例题详解

A train starts from rest, accelerates uniformly to 20 m/s over 10 s, travels at 20 m/s for 30 s, then brakes to a stop in 5 s. Find the total distance.

一列火车从静止开始，在 10 秒内匀加速到 20 米/秒，然后以 20 米/秒匀速行驶 30 秒，最后在 5 秒内刹车停止。求总距离。

Break the speed-time graph into three regions:

将该速度-时间图分成三个区域：

Region 1: triangle with base 10 s and height 20 m/s, area = ½ × 10 × 20 = 100 m
Region 2: rectangle with length 30 s and height 20 m/s, area = 30 × 20 = 600 m
Region 3: triangle with base 5 s and height 20 m/s, area = ½ × 5 × 20 = 50 m

区域 1：三角形，底 10 秒，高 20 米/秒，面积 = ½ × 10 × 20 = 100 米
区域 2：矩形，长 30 秒，高 20 米/秒，面积 = 30 × 20 = 600 米
区域 3：三角形，底 5 秒，高 20 米/秒，面积 = ½ × 5 × 20 = 50 米

Total distance = 100 + 600 + 50 = 750 m

总距离 = 100 + 600 + 50 = 750 米

10. Common Mistakes | 常见错误

Confusing distance-time gradients with speed-time gradients.
Forgetting that the gradient of a distance-time graph gives speed, not acceleration.
Using the area of a distance-time graph; this area has no standard meaning in IGCSE mathematics.
Reading scales incorrectly, especially when each grid square stands for a non-unit value.

混淆距离-时间图斜率和速度-时间图斜率。
忘记距离-时间图的斜率给出的是速度，而不是加速度。
使用距离-时间图的面积；在 IGCSE 数学中这个面积没有标准含义。
错误读取刻度，特别是当每个小方格代表非单位数值时。

Always state the formula you are using and show your calculations clearly.

始终写出你使用的公式，并清楚地展示计算过程。

11. Exam Tips | 考试技巧

Read the question carefully to identify which type of graph is described.
For distance-time: find gradient to get speed.
For speed-time: find gradient to get acceleration, and find area to get distance.
When a graph is curved, use tangents for instantaneous values.
Convert units if necessary: 1 km = 1000 m, 1 hour = 3600 s.

仔细读题，判断题目描述的是哪种类型的图。
对于距离-时间图：求斜率得到速度。
对于速度-时间图：求斜率得到加速度，求面积得到距离。
当图像为曲线时，用切线求瞬时值。
必要时进行单位换算：1 千米 = 1000 米，1 小时 = 3600 秒。

12. Summary | 内容总结

Distance-time graphs and speed-time graphs are powerful tools for describing motion. The key rules are: gradient of distance-time = speed; gradient of speed-time = acceleration; area under speed-time = distance. Mastering these simple links will help you solve a wide range of IGCSE mathematics questions with confidence.

距离-时间图和速度-时间图是描述运动的强大工具。关键规则是：距离-时间图的斜率 = 速度；速度-时间图的斜率 = 加速度；速度-时间图下的面积 = 距离。掌握这些简单联系，将帮助你自信地解决各种 IGCSE 数学问题。

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Understanding Distance-Time and Speed-Time Graphs | 理解距离-时间图与速度-时间图