📚 Distance-Time and Speed-Time Graphs | 距离-时间图与速度-时间图
This Topic T-K-1015 revision note covers distance-time graphs and speed-time graphs, which are essential for describing motion in IGCSE Maths. You need to read axes correctly, interpret gradients and areas, change units, and solve multi-stage journey problems. A clear understanding of these graphs will help you answer graph-based kinematics questions accurately.
本篇 T-K-1015 复习笔记讲解 IGCSE 数学中的距离-时间图与速度-时间图，这是描述运动的核心内容。你需要正确读取坐标轴、理解梯度与面积的含义、会换算单位，并能解决多阶段行程问题。清楚掌握这些图形可以帮助你准确解答基于图形的运动学问题。
1. The Big Idea: Reading Travel Graphs | 核心概念：读懂行程图
A travel graph shows how one quantity changes with time. In IGCSE Maths, the two most common travel graphs are the distance-time graph and the speed-time graph. They look similar, but they answer different questions: a distance-time graph tells you where an object is, while a speed-time graph tells you how fast it is moving.
行程图表示某一物理量随时间的变化。在 IGCSE 数学中，最常见的两种行程图是距离-时间图和速度-时间图。它们看起来相似，但回答的问题不同：距离-时间图告诉你物体在哪里，而速度-时间图告诉你物体运动得有多快。
The key to success is not to memorise the graphs blindly, but to link each feature to a real motion story. Always ask: What does the gradient mean? What does the horizontal section mean? What does the area represent? Once these ideas are clear, the questions become much easier.
成功的关键不是盲目记忆图形，而是把图形特征和真实的运动过程联系起来。始终要问：梯度表示什么？水平段表示什么？面积代表什么？一旦这些概念清楚，题目就会简单很多。
2. Distance-Time Graphs: Axes and Meaning | 距离-时间图：坐标轴与含义
On a distance-time graph, the x-axis represents time and the y-axis represents distance from a starting point. The graph is a visual story of how far an object is from the start at every moment.
在距离-时间图中，x 轴表示时间，y 轴表示到起点的距离。这张图直观地展示了每个时刻物体距离起点有多远。
| Feature on a distance-time graph | Meaning |
|---|---|
| Horizontal straight line | The object is stationary |
| Sloping straight line | The object moves at constant speed |
| Steeper gradient | Higher speed |
| Downward sloping line | The object returns towards the starting point |
上方表格总结了距离-时间图的主要特征：水平直线表示物体静止；倾斜直线表示物体匀速运动；梯度越陡，速度越大；向下倾斜表示物体正在返回起点。
When the graph is made of straight-line segments, the motion happens in stages. For example, a car might drive away from home, stop at traffic lights, then return home. Each segment has its own gradient and therefore its own speed.
当图形由直线段组成时，运动是分阶段进行的。例如，一辆汽车可能先驶离家，在红绿灯处停下，然后返回家中。每一段都有自己的梯度，因此也有自己的速度。
3. Gradient = Speed | 梯度 = 速度
On a distance-time graph, the gradient between two points gives the speed. This is because speed measures how much distance is covered per unit time. The steeper the line, the greater the speed.
在距离-时间图中，两点之间的梯度表示速度。这是因为速度衡量单位时间内经过的距离。直线越陡，速度越大。
Speed = distance moved ÷ time taken
速度 = 移动的距离 ÷ 所用的时间。
0 ÷ 50 = 8 m/s. You must always check whether the units are consistent before calculating.
例如，一名骑行者 3 小时行驶 150 km，速度为 150 ÷ 3 = 50 km/h。如果一名跑步者 50 秒跑 400 m，速度为 400 ÷ 50 = 8 m/s。计算前必须检查单位是否一致。
For a straight-line segment, the gradient is found by choosing two clear points on that segment. Use the change in distance divided by the change in time. If the line slopes downward, the gradient is negative, which means the object is moving back toward the start.
对于直线段，选择该线段上两个清晰的点来求梯度。用距离的变化量除以时间的变化量。如果直线向下倾斜，梯度为负，表示物体正在返回起点。
4. Horizontal Lines Mean Stationary | 水平线段表示静止
A horizontal line on a distance-time graph means the distance from the start does not change. The object is not moving; it is stationary. The gradient of a horizontal line is zero, so the speed is zero.
距离-时间图中的水平线段表示到起点的距离没有变化。物体没有运动，处于静止状态。水平线的梯度为零，因此速度为零。
This appears in many exam questions. A car might travel for 1 hour, then stop for 30 minutes, then continue. The stop is shown as a flat section. The length of the flat section on the time axis tells you how long the object was stationary.
这一点在许多考试题目中出现。一辆汽车可能行驶 1 小时，然后停下来休息 30 分钟，再继续行驶。停车过程用一段水平线表示。水平段在时间轴上的长度告诉你物体静止了多长时间。
A straight sloped line means constant speed, but a steeper slope means a higher constant speed. If the slope changes, the speed changes. A distance-time graph can therefore show a journey with several speeds.
倾斜直线表示匀速运动，但斜率越陡表示速度越大。如果斜率发生变化，说明速度发生变化。因此，距离-时间图可以表示包含多个速度的行程。
5. Speed-Time Graphs: Axes and Meaning | 速度-时间图：坐标轴与含义
On a speed-time graph, the x-axis represents time and the y-axis represents speed. The graph shows how speed changes over time. It is sometimes called a velocity-time graph when direction is also considered, but IGCSE usually uses speed-time graphs with non-negative speed values.
在速度-时间图中，x 轴表示时间，y 轴表示速度。图形展示速度随时间的变化。当同时考虑方向时，它有时被称为速度-时间图，但 IGCSE 通常使用速度值不为负的速度-时间图。
A horizontal line on a speed-time graph means constant speed. A rising straight line means constant acceleration. A falling straight line means constant deceleration, also called retardation. The line can be made of several straight segments to represent different stages of motion.
速度-时间图中的水平直线表示匀速运动。上升直线表示匀加速运动。下降直线表示匀减速运动，也叫减速。图形可以由多个直线段组成，表示不同的运动阶段。
You should be careful: a distance-time graph and a speed-time graph can look identical, but their meanings are completely different. Always check the label on the y-axis before answering.
你需要小心：距离-时间图和速度-时间图可能看起来相同，但含义完全不同。答题前一定要先看 y 轴的标签。
6. Gradient = Acceleration | 梯度 = 加速度
On a speed-time graph, the gradient gives the acceleration. Acceleration is the rate of change of speed. A positive gradient means the object is speeding up, and a negative gradient means the object is slowing down.
在速度-时间图中，梯度表示加速度。加速度是速度的变化率。正梯度表示物体正在加速，负梯度表示物体正在减速。
Acceleration = change in speed ÷ time taken
加速度 = 速度变化量 ÷ 时间。
m/s in 5 s. The change in speed is 25 - 10 = 15 m/s. The acceleration is 15 ÷ 5 = 3 m/s².
例如，一辆公交车在 5 秒内速度从 10 m/s 增加到 25 m/s。速度变化量为 25 - 10 = 15 m/s。加速度为 15 ÷ 5 = 3 m/s²。
40 = -30 m/s, so the acceleration is -30 ÷ 6 = -5 m/s². This is a deceleration of 5 m/s².
如果速度减小，加速度为负。例如，一列火车在 6 秒内从 40 m/s 减速到 10 m/s。速度变化量为 10 - 40 = -30 m/s，因此加速度为 -30 ÷ 6 = -5 m/s²，也就是减速度为 5 m/s²。
7. Area Under a Speed-Time Graph = Distance | 速度-时间图下方面积 = 距离
The area between the speed-time line and the time axis represents the distance travelled. This is one of the most important ideas for IGCSE graph questions. You find the total distance by calculating the area of the shapes under the graph.
速度-时间图与时间轴之间的面积表示行驶的距离。这是 IGCSE 图形题中最重要的概念之一。通过计算图形下方各形状的面积，可以求出总距离。
Distance = area under the speed-time graph
距离 = 速度-时间图下方的面积。
n 8 s, the graph forms a triangle. The area is ½ × 8 × 20 = 80 m, so the car travels 80 m.
例如，一辆汽车在 8 秒内从 0 均匀加速到 20 m/s，图形形成一个三角形。面积为 ½ × 8 × 20 = 80 m，因此汽车行驶了 80 m。
For multi-stage journeys, split the area into rectangles, triangles and trapeziums. Calculate each area separately and add them together. If the graph is curved, you may need to estimate the number of squares under the curve or use a given method.
对于多阶段行程，将面积分成矩形、三角形和梯形。分别计算每个图形的面积，然后相加。如果图形是曲线，你可能需要估算曲线下方的方格数，或使用题目给出的方法。
8. Average Speed and Total Distance | 平均速度与总距离
Average speed is found by dividing the total distance travelled by the total time taken. You cannot always simply average two speeds unless the time spent at each speed is the same.
平均速度等于总路程除以总时间。不能总是简单地把两个速度平均，除非在两个速度下运动的时间相同。
Average speed = total distance travelled ÷ total time taken
平均速度 = 总行驶距离 ÷ 总用时。
rs. The average speed is 120 ÷ 3 = 40 km/h, not simply the average of 45 km/h and 30 km/h.
例如，一辆卡车 2 小时行驶 90 km，然后 1 小时行驶 30 km。总路程为 120 km，总时间为 3 小时。平均速度为 120 ÷ 3 = 40 km/h，而不是简单地把 45 km/h 和 30 km/h 平均。
When a question asks for average speed, remember to include stationary periods in the total time. A stop for rest increases the total time and therefore reduces the average speed.
当题目要求计算平均速度时，记得把静止的时间也算入总时间。休息停留会增加总时间，从而降低平均速度。
9. Curved Graphs and Instantaneous Rate | 曲线图形与瞬时变化率
Some travel graphs are curved rather than made of straight lines. A curved distance-time graph means the speed is not constant. A curved speed-time graph means the acceleration is not constant.
有些行程图是曲线，而不是由直线组成。曲线距离-时间图意味着速度不是恒定的。曲线速度-时间图意味着加速度不是恒定的。
To find the instantaneous speed on a curved distance-time graph, draw a tangent to the curve at the chosen time. The gradient of the tangent gives the speed at that exact moment.
要在曲线距离-时间图上求瞬时速度，在所选时刻画出曲线的切线。切线的梯度就是该时刻的瞬时速度。
Similarly, on a curved speed-time graph, the gradient of the tangent gives the instantaneous acceleration. If you need the distance under a curved speed-time graph, you may count squares or use a given approximation method.
类似地，在曲线速度-时间图上，切线的梯度表示瞬时加速度。如果需要求曲线速度-时间图下方的距离，可以数方格或使用题目给出的近似方法。
10. Common Exam Mistakes and Tips | 常见考试错误与应对技巧
One common mistake is confusing the distance-time graph with the speed-time graph. Students often use area on a distance-time graph, which does not give a meaningful quantity for IGCSE motion questions. Always read the y-axis label first.
一个常见错误是混淆距离-时间图和速度-时间图。学生常常在距离-时间图上使用面积，这在 IGCSE 运动题中没有意义。一定要先看 y 轴标签。
Another mistake is using gradient on a speed-time graph to find distance. Gradient gives acceleration, not distance. Distance is found by area. Also, check your units: time may be in seconds or hours, and speed may be in m/s or km/h.
另一个错误是在速度-时间图上用梯度求距离。梯度表示加速度，而不是距离。距离要用面积来求。此外，检查单位：时间可能是秒或小时，速度可能是
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