📚 Kinematics: Reading and Drawing Distance-Time and Speed-Time Graphs | 运动学：读取和绘制距离-时间与速度-时间图像
Kinematics is the branch of mathematics that describes motion without considering its causes. In IGCSE Mathematics, you are often asked to interpret and draw two important types of graph: distance-time graphs and speed-time graphs. Mastering these graphs will help you answer questions on speed, acceleration, and distance covered directly from a diagram.
运动学是描述运动而不考虑其起因的数学分支。在 IGCSE 数学中，你经常需要解读和绘制两种重要的图像：距离-时间图和速度-时间图。掌握这些图像可以让你直接从图中解答关于速度、加速度和行驶距离的问题。
1. Distance-Time Graphs: The Basics | 距离-时间图：基础知识
A distance-time graph shows how far a moving object is from its starting point over time. Time is plotted on the horizontal axis (x-axis) and distance is plotted on the vertical axis (y-axis). The graph itself is always drawn with time increasing to the right.
距离-时间图表示一个运动物体随时间变化与起点之间相距多远。时间标在横轴（x 轴）上，距离标在纵轴（y 轴）上。图像总是从左到右随时间增加而绘制。
The key idea is that the gradient (steepness) of the line tells us the speed of the object. A steep line means fast speed; a shallow line means slow speed; a horizontal line means the object is at rest.
关键思想是：线的斜率（倾斜程度）告诉我们物体的速度。线越陡表示速度越快；线越平缓表示速度越慢；水平线表示物体静止。
For example, if a car travels 100 metres in 20 seconds, the gradient is 100 ÷ 20 = 5 m/s. The line will rise steadily from the origin at 5 metres per second.
例如，如果一辆汽车在 20 秒内行驶 100 米，斜率为 100 ÷ 20 = 5 m/s。这条线会从原点以每秒 5 米的速度均匀上升。
2. Interpreting Different Parts of a Distance-Time Graph | 解读距离-时间图的不同部分
You should be able to read a distance-time graph segment by segment. Each straight-line segment represents a constant speed. When the line is horizontal, the distance is not changing, so the speed is zero (the object is stationary).
你应该能逐段解读距离-时间图。每个直线段代表一个恒定的速度。当线水平时，距离不变，所以速度为零（物体静止）。
When the line slopes downwards, the object is returning towards the starting point. The gradient is negative, but speed is the absolute value of that gradient. So a downhill line still means motion, just in the opposite direction.
当线向下倾斜时，物体正在返回起点。斜率是负数，但速度是斜率的绝对值。因此，向下的线仍然表示运动，只是方向相反。
Let us look at a typical journey: a bus leaves a stop, travels at constant speed, stops to pick up passengers, then continues. The graph will show a rising line, then a horizontal line, then another rising line. The steeper the line, the faster the bus.
让我们看一个典型行程：一辆公共汽车离开车站，以恒定速度行驶，停下来接客，然后继续。图将显示一条上升线，然后一条水平线，再然后另一条上升线。线越陡，汽车越快。
3. Calculating Speed from a Distance-Time Graph | 从距离-时间图计算速度
The speed of an object on a distance-time graph is given by the gradient of the straight-line segment. To find the gradient, choose two points on the segment, then use the formula:
在距离-时间图上，物体的速度由直线段的斜率给出。要求斜率，选线段上的两个点，然后使用公式：
speed = (change in distance) ÷ (change in time) = Δd ÷ Δt
For example, a cyclist moves from 200 m to 800 m in 60 seconds. The change in distance is 800 − 200 = 600 m, and the change in time is 60 seconds. So speed = 600 ÷ 60 = 10 m/s.
例如，一名骑车人从 200 m 移动到 800 m，用了 60 秒。距离变化为 800 − 200 = 600 m，时间变化为 60 秒。所以速度 = 600 ÷ 60 = 10 m/s。
Always include the units. If distance is in metres and time in seconds, speed is in m/s. If distance is in kilometres and time in hours, speed is in km/h.
始终要带上单位。如果距离以米、时间以秒为单位，速度单位是 m/s。如果距离以千米、时间以小时为单位，速度单位是 km/h。
4. Speed-Time Graphs: The Basics | 速度-时间图：基础知识
In a speed-time graph, speed is plotted on the vertical axis and time on the horizontal axis. This type of graph has a different meaning for its gradient: the gradient represents acceleration, not speed.
在速度-时间图中，速度标在纵轴上，时间标在横轴上。这种图像对斜率有不同含义：斜率代表加速度，而不是速度。
If the line is horizontal, the speed is constant, so acceleration is zero. If the line slopes upward, the object is speeding up (positive acceleration). If the line slopes downward, the object is slowing down (negative acceleration, also called deceleration).
如果线是水平的，速度恒定，因此加速度为零。如果线向上倾斜，物体在加速（正加速度）。如果线向下倾斜，物体在减速（负加速度，也称为减速度）。
The other important fact is that the area between the graph and the time axis represents the total distance travelled.
另一个重要事实是：图像与时间轴之间的面积代表总行驶距离。
5. Finding Acceleration from a Speed-Time Graph | 从速度-时间图求加速度
Acceleration is the rate of change of speed. It is the gradient of the speed-time graph. The formula is:
加速度是速度的变化率。它是速度-时间图的斜率。公式为：
acceleration = (change in speed) ÷ (change in time) = Δv ÷ Δt
For instance, a car goes from 0 m/s to 30 m/s in 10 seconds. Acceleration = (30 − 0) ÷ 10 = 3 m/s². The unit for acceleration is metres per second squared, written as m/s².
例如，一辆车从 0 m/s 加速到 30 m/s，用了 10 秒。加速度 = (30 − 0) ÷ 10 = 3 m/s²。加速度的单位是米每二次方秒，写作 m/s²。
If the speed decreases from 20 m/s to 8 m/s over 4 seconds, the change in speed is 8 − 20 = −12 m/s. So acceleration = (−12) ÷ 4 = −3 m/s². The negative sign means the object is decelerating.
如果速度在 4 秒内从 20 m/s 减小到 8 m/s，速度变化为 8 − 20 = −12 m/s。所以加速度 = (−12) ÷ 4 = −3 m/s²。负号表示物体在减速。
6. Finding Distance from a Speed-Time Graph | 从速度-时间图求距离
The distance covered by an object is represented by the area under the speed-time graph. For a constant-speed segment, the area is a rectangle. For an accelerating segment, it is usually a triangle or a trapezium.
物体通过的距离由速度-时间图下方的面积表示。对于匀速段，面积是一个矩形。对于加速段，通常是三角形或梯形。
Consider a speed-time graph with three parts: constant acceleration to 10 m/s in 5 s, constant speed of 10 m/s for 6 s, then constant deceleration to 0 m/s in 3 s. The total distance is the sum of the areas:
考虑一个速度-时间图包含三段：用 5 s 匀加速到 10 m/s，以 10 m/s 匀速 6 s，然后用 3 s 匀减速到 0 m/s。总距离是各面积之和：
- First triangle: (1/2) × base × height = 0.5 × 5 × 10 = 25 m
- Rectangle: 6 × 10 = 60 m
- Second triangle: 0.5 × 3 × 10 = 15 m
Total distance = 25 + 60 + 15 = 100 m.
第一个三角形：(1/2) × 底 × 高 = 0.5 × 5 × 10 = 25 m；矩形：6 × 10 = 60 m；第二个三角形：0.5 × 3 × 10 = 15 m。总距离 = 25 + 60 + 15 = 100 m。
Remember to check the units: the area under a graph where y-axis is m/s and x-axis is s gives units of (m/s) × s = m, which is distance.
记得检查单位：y 轴单位为 m/s，x 轴单位为 s 的图像下方面积，其单位为 (m/s) × s = m，也就是距离。
7. Drawing a Distance-Time Graph from Data | 根据数据绘制距离-时间图
To draw a distance-time graph, first set up axes with time on the horizontal axis and distance on the vertical axis. Choose a suitable scale so that the maximum values fit. Then plot the points from a table and join them with straight line segments.
绘制距离-时间图，首先设置坐标轴：横轴为时间，纵轴为距离。选择合适比例，使最大值适合。然后根据表格数据描点，并用直线段连接。
For example, a person walks as follows: 0 s at 0 m, 10 s at 20 m, 20 s at 20 m (rest), 30 s at 40 m. Plot the points (0,0), (10,20), (20,20), (30,40). Then join them in order with straight lines.
例如，一个人行走如下：0 s 在 0 m，10 s 在 20 m，20 s 在 20 m（休息），30 s 在 40 m。描出点 (0,0)、(10,20)、(20,20)、(30,40)，然后按顺序用直线连接。
Check that the graph never goes backwards in time. The x-coordinate (time) must always increase. Sometimes you may need to calculate the slope of each segment for extra accuracy.
检查图像不会在时间上倒退。x 坐标（时间）必须始终增加。有时你可能需要计算每段的斜率以提高精确度。
8. Drawing a Speed-Time Graph from Data | 根据数据绘制速度-时间图
When drawing a speed-time graph, time is on the horizontal axis and speed on the vertical axis. Plot the given speed values at the corresponding times. Then connect the points with straight lines to show the acceleration during each time interval.
绘制速度-时间图时，横轴为时间，纵轴为速度。将给定的速度值在对应时间处描点，然后用直线连接这些点，以显示每个时间区间内的加速度。
A common question gives a table of speed values at equal time intervals. For instance: at t = 0 s, v = 0 m/s; at t = 2 s, v = 6 m/s; at t = 4 s, v = 10 m/s. Plot these points and join them. The line segments show that the acceleration is not necessarily constant between intervals.
常见问题给出等时间间隔的速度值表。例如：在 t = 0 s，v = 0 m/s；在 t = 2 s，v = 6 m/s；在 t = 4 s，v = 10 m/s。描出这些点并连接。线段显示各时间区间内的加速度不一定恒定。
Make sure your graph has a title and labelled axes with units. If the speed does not change for a while, draw a horizontal line at that speed.
确保图像有标题，坐标轴有标签和单位。如果速度在一段时间内不变，就在该速度处画水平线。
9. Comparing Distance-Time and Speed-Time Graphs | 比较距离-时间图与速度-时间图
Students often confuse the two types of graph. A horizontal line means “stationary” in a distance-time graph, but it means “constant speed” in a speed-time graph. The gradient means “speed” in the first graph and “acceleration” in the second graph.
学生经常混淆这两种图。水平线在距离-时间图中表示“静止”，但在速度-时间图中表示“匀速”。斜率在第一种图中表示“速度”，在第二种图中表示“加速度”。
| Feature | Distance-Time | Speed-Time |
|---|---|---|
| Gradient | Speed | Acceleration |
| Area under graph | No simple meaning | Distance |
| Horizontal line | Stationary | Constant speed |
| Downward slope | Returning to start | Deceleration |
Feature comparison table: gradient, area, horizontal line, downward slope.
功能比较表：斜率、面积、水平线、向下倾斜。
10. Real-World Example: A Bus Journey | 现实例子：公共汽车旅程
Let us apply everything to a single example. A bus leaves the station at 9:00 and accelerates uniformly from rest to 20 m/s in 10 s. It then travels at 20 m/s for 30 s. Finally, it brakes uniformly to stop in 5 s.
让我们把所有的内容应用到一个例子中。一辆公共汽车在 9:00 离开车站，从静止开始匀加速，在 10 s 内达到 20 m/s。然后以 20 m/s 行驶 30 s。最后，它匀减速，在 5 s 内停下。
First, draw the speed-time graph. There are three line segments: from (0,0) to (10,20); from (10,20) to (40,20) because 10 + 30 = 40 s; and from (40,20) to (45,0) because 40 + 5 = 45 s.
首先绘制速度-时间图。有三条线段：(0,0) 到 (10,20)；(10,20) 到 (40,20)，因为 10 + 30 = 40 s；以及 (40,20) 到 (45,0)，因为 40 + 5 = 45 s。
Acceleration in the first stage = (20 − 0) ÷ 10 = 2 m/s². The second stage has acceleration 0. The braking stage has acceleration (0 − 20) ÷ 5 = −4 m/s².
第一阶段加速度 = (20 − 0) ÷ 10 = 2 m/s²。第二阶段加速度为 0。刹车阶段加速度 = (0 − 20) ÷ 5 = −4 m/s²。
The total distance is the area under the graph: triangle (0.5 × 10 × 20 = 100 m), rectangle (30 × 20 = 600 m), triangle (0.5 × 5 × 20 = 50 m). Total = 100 + 600 + 50 = 750 m.
总距离为图像下的面积：三角形 (0.5 × 10 × 20 = 100 m)，矩形 (30 × 20 = 600 m)，三角形 (0.5 × 5 × 20 = 50 m)。总计 = 100 + 600 + 50 = 750 m。
11. Common Mistakes in Kinematics Graphs | 运动学图像中的常见错误
One common mistake is treating the distance-time graph the same as a speed-time graph. For example, a student may think that a steeper distance-time graph means “accelerating”. Actually, a steeper graph just means a higher constant speed; acceleration appears as a curve in a distance-time graph.
一个常见错误是认为距离-时间图与速度-时间图相同。例如，学生可能认为距离-时间图越陡表示“加速”。实际上，更陡的图只表示更大的恒定速度；加速在距离-时间图中会表现为曲线。
Another mistake is forgetting to convert units. If time is given in minutes and speed in km/h, you must convert to the same units before calculating distance from the area.
另一个错误是忘记换算单位。如果时间以分钟、速度以 km/h 给出，你必须先换成相同单位，才能从面积计算距离。
A third mistake is ignoring the sign of acceleration. A negative value does not mean the object is moving backwards; it simply means the speed is decreasing. In a speed-time graph, speed itself is always non-negative.
第三个错误是忽略加速度的符号。负值并不表示物体向后移动；仅表示速度在减小。在速度-时间图中，速度本身始终是非负的。
12. Exam Tips and Summary | 考试技巧与总结
When you see a graph question, always label the axes and note the units. Underline what the question asks: speed, acceleration, distance, or a description of the motion. Then choose the correct formula: gradient for speed or acceleration, area for distance.
当看到图像题目时，先标注坐标轴并注意单位。在问题要求下划线：速度、加速度、距离或运动描述。然后选择正确的公式：斜率求速度或加速度，面积求距离。
Practice reading graphs by drawing them from tables and by finding the gradient and area of simple shapes. Make sure you can also handle graphs that do not start at the origin.
通过从表格绘制图像，以及求简单图形的斜率和面积，练习读取图像。确保你也能处理不从原点开始的图像。
Finally, always check your answer for realism. A car cannot cover 1000 km in 2 seconds in a normal IGCSE question, so if you get a huge number, re-check the scale and your calculations.
最后，始终检查答案的合理性。在正常的 IGCSE 题中，汽车不可能在 2 秒内行驶 1000 km，所以如果你得到很大的数字，重新检查比例和你的计算。
In summary, keep these three golden rules in mind: distance-time graph → gradient = speed; speed-time graph → gradient = acceleration and area = distance. With these, you can solve most kinematics graph questions in the IGCSE Extended syllabus.
总结一下，记住三条黄金法则：距离-时间图 → 斜率 = 速度；速度-时间图 → 斜率 = 加速度，面积 = 距离。有了这些，你就能解决 IGCSE 扩展大纲中的大多数运动学图像问题。
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