📚 14.1 Distance-Time Graphs | 14.1 距离-时间图
Distance-time graphs are a key way of representing motion in Edexcel IGCSE Mathematics. They show how the distance of an object from its starting point changes during a journey. With a simple glance, you can tell whether the object is moving, stationary, going away or returning, and even how fast it is travelling. Mastering these graphs will sharpen your problem-solving skills for both papers and practical reasoning.
距离-时间图是爱德思IGCSE数学中表示运动的关键方式之一。它们展示了物体距离起点的远近如何随时间变化。只需简单一瞥，你就能判断物体是在移动、静止、远离还是返回，甚至能看出它运动得有多快。掌握这些图表能磨砺你的解题技巧，无论是应对考试还是进行实际推理都大有益处。
1. What Is a Distance-Time Graph? | 什么是距离-时间图？
A distance-time graph plots the distance of an object from a fixed starting point on the vertical y-axis against the elapsed time on the horizontal x-axis. It tells the story of a journey in visual form. Each point on the line corresponds to the object's distance at a specific moment.
距离-时间图将物体离固定起点的距离绘制在垂直 y 轴上，将经过的时间绘制在水平 x 轴上。它以可视化的方式讲述了一段行程的故事。图线上的每一个点都对应着物体在特定时刻的距离。
Unlike speed-time graphs, distance-time graphs can slope downwards if the object moves back towards the starting point, reducing its distance from the start. This is perfectly normal for return journeys and is widely tested in Edexcel IGCSE exams.
与速度-时间图不同，如果物体朝起点返回使得离起点的距离减小，距离-时间图可以向下倾斜。这在往返行程中十分常见，也是爱德思IGCSE考试中经常考查的内容。
2. Axes and Labels | 坐标轴与标签
Always check what the axes represent. Time is the independent variable, so it goes on the x-axis. Distance is the dependent variable, shown on the y-axis. Common units include seconds (s) or minutes (min) for time, and metres (m) or kilometres (km) for distance.
一定要先查看坐标轴表示什么。时间是自变量，因此放在 x 轴上；距离是因变量，展示在 y 轴上。常见的时间单位有秒 (s) 或分钟 (min)，距离单位有米 (m) 或千米 (km)。
The axes must be clearly labelled with the quantity and unit, for example 'Time (s)' and 'Distance (m)'. The scale should be linear and consistent. If the axes are not labelled correctly, any interpretation or calculation of speed becomes unreliable.
坐标轴必须清楚标明物理量与单位，例如“Time (s)”和“Distance (m)”。刻度应为线性且匀称。如果坐标轴标注不正确，任何对速度的解读或计算都不可靠。
3. Interpreting Straight Lines | 解读直线段
A straight line on a distance-time graph means the object is moving at a constant speed. If the line slopes upwards from left to right, the object is moving away from the starting point. If it slopes downwards, the object is returning towards the start.
距离-时间图上的直线段表示物体正以恒定速度运动。若线段从左向右上倾，物体正远离起点；若线段下倾，物体正在返回起点。
The steepness (gradient) of the line tells us the speed. A steeper line indicates a higher speed, while a gentler slope indicates a slower speed. For example, a line going from (0, 0) to (4, 100) is steeper than one going from (0, 0) to (10, 100), so the first journey was faster.
线段的陡峭程度（梯度）反映了速度的大小。线越陡，速度越高；线越平缓，速度越慢。例如，从 (0, 0) 到 (4, 100) 的线段比从 (0, 0) 到 (10, 100) 的更陡，因此前者行程更快。
4. Horizontal Lines and Stationary Objects | 水平线与静止物体
A horizontal line (flat segment) shows that time is increasing but distance stays the same. This means the object is stationary – it is at rest. On a journey graph, this often represents a stop, such as waiting at traffic lights or taking a break.
水平线段（平坦段）表明时间在增加但距离保持不变。这意味着物体处于静止状态。在行程图中，这通常代表一次停顿，比如等红绿灯或休息。
The longer the horizontal line, the longer the object remains stopped. When interpreting a multi-stage journey, always check the length of each stationary period on the time axis to describe how long the stop lasted.
水平线段延续得越长，物体停止的时间就越久。解释多段行程时，务必从时间轴上查看每一段静止期的长度，以描述停顿持续了多久。
5. Calculating Speed Using the Gradient | 利用梯度计算速度
The speed of an object during a straight-line section is found by calculating the gradient of that segment. The formula is:
物体在直线段运动时的速度可以通过计算该段的梯度得出。公式为：
Speed = Distance ÷ Time
More precisely, choose two points on the segment, (t₁, d₁) and (t₂, d₂). Then:
更准确地说，选取该段上的两点 (t₁, d₁) 和 (t₂, d₂)，然后：
Speed = (d₂ – d₁) ÷ (t₂ – t₁)
For example, if a section runs from (0 s, 0 m) to (8 s, 200 m), the speed is (200 – 0) ÷ (8 – he units of speed will be the distance unit divided by the time unit, such as m/s or km/h.
例如，如果图上一段从 (0 s, 0 m) 到 (8 s, 200 m)，速度就是 (200 – 0) ÷ (8 – 0) = 25 m/s。速度的单位是距离单位除以时间单位，如 m/s 或 km/h。
When the graph slopes downwards, the change in distance is negative, giving a negative gradient. In distance-time graphs, the speed is taken as the magnitude of the gradient (positive value) to describe how fast the object returns.
当图线向下倾斜时，距离变化量为负，梯度也是负值。在距离-时间图中，速度取梯度的绝对值（正值）来描述物体返回得多快。
6. Curved Lines and Acceleration | 曲线与加速度
Not all journeys happen at constant speed. A curved line on a distance-time graph tells us that the speed is changing – the object is accelerating or decelerating. If the curve gets steeper, the object is speeding up; if it becomes less steep, it is slowing down.
并非所有行程都匀速进行。距离-时间图上的曲线表示速度在变化——物体在加速或减速。若曲线变陡，物体在加速；若曲线变缓，物体在减速。
To find the speed at a particular instant on a curve, you draw a tangent to the curve at that point and calculate its gradient. The steeper the tangent, the higher the instantaneous speed. This method is often tested with carefully drawn graphs on Edexcel IGCSE papers.
要找出曲线上某一瞬间的速度，你可以在该点处画出曲线的切线，并计算切线的梯度。切线越陡，瞬时速度越高。爱德思IGCSE试卷中常通过精确绘制的图形考查这一方法。
A curve that becomes flatter over time shows deceleration before stopping. Conversely, a curve starting gently and becoming steeper shows acceleration from rest.
随时间推移逐渐变平的曲线表示减速直至停下。相反，起初平缓而后变陡的曲线表示从静止开始加速。
7. Finding Average Speed | 计算平均速度
The average speed over a whole journey is not simply the average of different segment speeds. Instead, it is calculated using the total distance travelled and the total time taken:
整个行程的平均速度并不是各段速度的简单平均。而是用总行驶距离和总耗时来计算：
Average Speed = Total Distance ÷ Total Time
On a distance-time graph, the total distance is the sum of all distance changes (ignoring direction), or simply the final distance from start if the object returns to the start, the total distance includes the outward and return journeys. The total time is the duration from the start to the end of the motion on the time axis.
在距离-时间图上，总距离是所有距离变化量之和（忽略方向），或者若物体返回起点，总距离包含去程与回程。总时间则是在时间轴上从运动开始至结束的持续时间。
is 600 ÷ 120 = 5 m/s, even though the starting and ending distance readings are both zero.
例如，若物体离开 300 m 又返回 300 m，总距离为 600 m。若行程耗时 120 s，平均速度就是 600 ÷ 120 = 5 m/s，尽管起止距离读数均为零。
8. Multi-Stage Journeys | 多段行程
Many distance-time graphs consist of several straight-line segments connected together, each representing a different stage of motion. You must be able to describe what happens in each stage clearly: moving away, stopping, moving away faster, returning, stopping again and so on.
许多距离-时间图由若干直线段连接而成，每一段代表运动的不同阶段。你必须能够清楚地描述每一阶段发生了什么：远离、停顿、加速远离、返回、再次停顿等。
When answering exam questions, break the graph into sections using the points where the line changes direction. For each section, state the type of motion (constant speed, at rest, constant return speed) and calculate the speed if required, using the gradient formula on the coordinates at the ends of that segment.
回答考题时，根据图线改变方向的点把图形划分为若干段。对每一段，陈述运动类型（匀速、静止、匀速返回），如有需要，利用该段端点坐标和梯度公式计算速度。
It is also common to be asked for the total time spent stationary. Simply add up the lengths of all horizontal line segments on the graph, reading from the time axis.
考题还经常要求找出静止的总时长。只要把图上所有水平线段时间轴上的长度加总即可。
9. Drawing a Distance-Time Graph | 绘制距离-时间图
To draw a distance-time graph from a description of a journey, first set up a pair of axes with time on the horizontal and distance on the vertical. Choose an appropriate scale so the graph fits well in the space and the points can be plotted easily.
要根据行程描述绘制距离-时间图，首先搭建坐标轴，水平轴为时间，垂直轴为距离。选择合适的刻度，使图形能够充分展开且易于描点。
Plot each key event as a coordinate. For constant speed, join consecutive points with a straight line. If the object is stationary between two times, draw a horizontal line segment. Make sure the line is continuous and any changes in slope are clearly shown at the correct times.
将每一个关键事件描成坐标点。对于匀速运动，用直线连接相邻点。若物体在两个时间点之间静止，画一条水平线段。确保图线连续，且在正确的时间点处清晰表现出斜率的变化。
Label the axes with quantity and units, and give the graph a title. If you are asked to estimate a value from a curve, use a neatly drawn smooth curve passing through the plotted points, not a series of sharp, jagged lines.
给坐标轴标上物理量与单位，并为图形加上标题。如果要求根据曲线估算数值，应画一条通过描点的平滑曲线，而非一系列锯齿状折线。
10. Common Pitfalls | 常见错误
One of the most frequent mistakes is confusing a distance-time graph with a speed-time graph. Remember: a horizontal line here means stationary, not constant speed. The gradient gives speed directly, not acceleration. Always read the y-axis label carefully to avoid mixing them up.
最常见的错误之一是将距离-时间图与速度-时间图混淆。请记住：这里的水平线表示静止，而非匀速；梯度直接给出速度，而非加速度。始终仔细查看 y 轴标签以避免混淆。
Another error is using the wrong coordinates or misreading scales when calculating gradient. For example, if the scale is in kilometres and minutes, the calculated speed will be in km/min. If the question expects m/s, you must convert units properly. Always state the units of your answer.
另一个错误是计算梯度时用错坐标或读错刻度。例如，若刻度为千米和分钟，算出的速度单位是 km/min。如果题目要求 m/s，就必须正确换算单位。始终要写明答案的单位。
Students also sometimes forget that a line sloping downwards still represents motion, just in the reverse direction. The speed is the absolute value of the gradient, and the total distance still increases even if the distance from start decreases.
学生有时还会忘记向下倾斜的线段仍然表示运动，只是方向相反。速度取梯度的绝对值，并且即便离起点的距离减小，总路程仍在增加。
11. Exam Tips | 考试技巧
In Edexcel IGCSE papers, distance-time graph questions often appear alongside other graph types. Highlight or underline what the axes represent before starting. Identify stationary sections by looking for horizontal parts and label them 'stop' to avoid misinterpretation.
在爱德思IGCSE试卷中，距离-时间图问题常与其他图类同时出现。动手前先高亮或下划横轴与纵轴的含义。找出水平部分，认定其为静止段，并标上“停”以免误读。
When calculating speed, show clearly which two points you are using – write their coordinates, the formula, and the working. Even if the final answer is wrong, you can earn method marks. Double-check that you have read the time values correctly, especially if the scale uses multiples like 2, 5 or 10 seconds per division.
计算速度时，要清晰展示所用的两点——写出它们的坐标、公式和计算过程。即便最终答案有误，也可能获得方法分。仔细复核读出的时间值，尤其是当刻度以每格 2、5 或 10 秒为单位时。
If a question asks you to 'describe the journey', use concise statements such as 'constant speed away from start', 'stationary for 30 seconds', 'faster constant speed returning to start'. Avoid vague terms like 'going up' without mentioning what that means in the context of distance and time.
若题目要求“描述行程”，请使用简明的陈述，如“匀速远离起点”“静止 30 秒”“以更快的匀速返回起点”。避免使用“上升”这样含糊的词语而没有结合距离和时间的语境说明其含义。
12. Worked Example | 实例讲解
A cyclist leaves home and cycles 400 m in 60 seconds at a constant speed to a shop. She stays at the shop for 30 seconds. She then cycles straight back home at a faster constant speed, taking only 40 seconds for the return trip. Interpret and construct the distance-time graph, then calculate the speed for each moving stage and the average speed for the whole journey.
一位骑车人离开家，用 60 秒匀速骑行了 400 m 到达商店。她在商店停留 30 秒，然后以更快的匀速直接骑回家，仅用 40 秒完成返程。解读并构建距离-时间图，然后计算每一运动阶段的速度以及整个行程的平均速度。
Stage 1 – Outward: From (0, 0) to (60, 400) . Speed = 400 ÷ 60 = 6.67 m/s (or 6 ⅔ m/s) . Stage 2 – Stationary: From (60, 400) to (90, 400) , horizontal line. Stage 3 – Return: From (90, 400) to (130, 0) . The gradient is (0 – 400) ÷ (130 – 90) = –400 ÷ 40 = – o speed = 10 m/s . The graph consists of an upward line, a flat line, and a downward line.
阶段 1 去程：从 (0, 0) 到 (60, 400)。速度 = 400 ÷ 60 = 6.67 m/s。阶段 2 静止：从 (60, 400) 到 (90
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