📚 IGCSE Kinematics: Distance-Time and Speed-Time Graphs (T-K-1050) | IGCSE 运动学：距离-时间图与速度-时间图
Kinematics graphs are a core part of the IGCSE Mathematics syllabus. They test your ability to interpret motion from visual information, calculate gradients and areas, and switch between distance-time and speed-time graphs. This article explains the key ideas, common pitfalls, and exam techniques you need for topic T-K-1050.
运动学图像是 IGCSE 数学大纲中的核心内容。它们考查你从图像中读取运动信息、计算斜率和面积，以及在距离-时间图与速度-时间图之间进行转换的能力。本文将讲解你掌握 T-K-1050 专题所需的关键概念、常见误区和考试技巧。
1. Understanding Distance-Time Graphs | 理解距离-时间图
A distance-time graph shows how far an object has travelled from a starting point over a period of time. Time is always plotted on the horizontal axis, while distance is plotted on the vertical axis. The shape of the graph tells you whether the object is stationary, moving at constant speed, or changing speed.
距离-时间图显示物体从起点开始在一段时间内移动了多远。时间总是画在横轴上，距离画在纵轴上。图像的形状可以告诉你物体是静止、匀速运动还是速度发生变化。
On a distance-time graph, a horizontal line means the object is not moving because the distance does not change. A straight sloping line means the object is moving at constant speed. A curved line means the speed is changing.
在距离-时间图中，水平线表示物体没有移动，因为距离没有变化。一条笔直的斜线表示物体在做匀速运动。曲线则表示速度在发生变化。
2. Gradient of a Distance-Time Graph | 距离-时间图的斜率
The gradient of a distance-time graph is equal to the speed of the object. To find the gradient, choose two clear points on the line and divide the change in distance by the change in time.
距离-时间图的斜率等于物体的速度。要求斜率，在直线上选取两个清晰的点，用距离的变化量除以时间的变化量。
Speed = Change in distance ÷ Change in time
速度 = 距离变化量 ÷ 时间变化量
Always remember to include the correct unit, such as m/s or km/h, when you state the gradient.
例如，如果一个物体在 6 秒内移动了 30 米，那么速度为 30 ÷ 6 = 5 m/s。在写出斜率时，一定要记得带上正确单位，例如 m/s 或 km/h。
3. Understanding Speed-Time Graphs | 理解速度-时间图
A speed-time graph shows how the speed of an object changes with time. Time is on the horizontal axis and speed is on the vertical axis. This graph is especially useful for finding acceleration and distance travelled.
速度-时间图显示物体速度随时间变化的关系。时间在横轴上，速度在纵轴上。该图像对于求解加速度和运动距离尤其有用。
A horizontal line on a speed-time graph represents constant speed. An upward sloping straight line represents uniform acceleration, and a downward sloping straight line represents uniform deceleration or negative acceleration.
速度-时间图上的水平线表示匀速运动。向上倾斜的直线表示匀加速运动，向下倾斜的直线表示匀减速运动或负加速度。
4. Gradient of a Speed-Time Graph | 速度-时间图的斜率
On a speed-time graph, the gradient gives the acceleration of the object. Acceleration measures how quickly the speed changes. A positive gradient means the object is speeding up, while a negative gradient means it is slowing down.
在速度-时间图中，斜率表示物体的加速度。加速度衡量速度变化的快慢。正斜率表示物体在加速，负斜率表示物体在减速。
Acceleration = Change in speed ÷ Change in time
加速度 = 速度变化量 ÷ 时间变化量
seconds, the acceleration is (0 - 12) ÷ 4 = -3 m/s². The negative sign shows deceleration.
如果一个物体的速度在 4 秒内从 12 m/s 减速到 0 m/s，那么加速度为 (0 - 12) ÷ 4 = -3 m/s²。负号表示减速。
5. Area Under a Speed-Time Graph | 速度-时间图下的面积
The area under a speed-time graph represents the total distance travelled. This is one of the most important rules in kinematics graphs. You often need to split the area into rectangles, triangles, or trapeziums to calculate it.
速度-时间图下方的面积表示物体的总运动距离。这是运动学图像中最重要的规则之一。你通常需要把面积拆分成矩形、三角形或梯形来进行计算。
Rectangle area = Length × Width
矩形面积 = 长 × 宽
Triangle area = ½ × Base × Height
三角形面积 = ½ × 底 × 高
Trapezium area = ½ × (a + b) × Height
梯形面积 = ½ × (上底 + 下底) × 高
Always check the units on both axes. The area unit is the product of the speed unit and the time unit, such as metres when speed is in m/s and time is in seconds.
始终检查两个轴上的单位。面积单位是速度单位与时间单位的乘积，例如当速度为 m/s、时间为秒时，面积单位就是米。
6. Interpreting Curved Graphs | 曲线图的解读
When a distance-time graph is curved, the speed is not constant. A curve that becomes steeper means the object is speeding up. A curve that becomes flatter means the object is slowing down.
当距离-时间图是曲线时，速度不是恒定的。曲线越来越陡表示物体在加速；曲线越来越平缓表示物体在减速。
For curved speed-time graphs, the acceleration is changing. The gradient at a specific point gives the instantaneous acceleration. You can estimate this by drawing a tangent to the curve and finding its gradient.
对于曲线形的速度-时间图，加速度是变化的。某一点的斜率表示该时刻的瞬时加速度。你可以通过在该点作曲线的切线并求切线斜率来进行估算。
7. Converting Between Distance-Time and Speed-Time Graphs | 距离-时间图与速度-时间图的转换
You can convert a distance-time graph into a speed-time graph by looking at the gradient of each section. A straight line segment with constant gradient becomes a horizontal line segment on the speed-time graph.
你可以通过观察距离-时间图每一段的斜率，将其转换为速度-时间图。一条斜率恒定的直线段在速度-时间图上会变成一条水平线段。
When the distance-time graph is curved, the speed-time graph will have a sloping line because the speed is changing. The steeper the distance-time curve, the higher the corresponding speed.
当距离-时间图是曲线时，速度-时间图将有一条斜线，因为速度在变化。距离-时间曲线越陡，对应的速度就越大。
To convert back, use the area under each section of the speed-time graph to determine how much distance is added in each time interval.
反向转换时，可以利用速度-时间图每一段下方的面积来确定每个时间间隔内增加了多少距离。
8. Average Speed and Instantaneous Speed | 平均速度与瞬时速度
Average speed is the total distance travelled divided by the total time taken. It is useful when an object changes speed during a journey. Instantaneous speed is the speed at a particular moment, found from the gradient at that point on a distance-time graph.
平均速度等于总运动距离除以总时间。当物体在整个过程中速度发生变化时，平均速度非常有用。瞬时速度是物体在某一瞬间的速度，可以通过距离-时间图上该点的斜率得出。
Average speed = Total distance ÷ Total time
平均速度 = 总距离 ÷ 总时间
Do not confuse average speed with the average of the speeds shown on a speed-time graph. You must work out the total distance first, then divide by the total time.
不要把平均速度与速度-时间图上显示的速度平均值混淆。你必须先求出总距离，然后除以总时间。
9. Common Misconceptions and Exam Tips | 常见误区与考试技巧
One common mistake is trying to find distance from the area under a distance-time graph. That area has no physical meaning in kinematics. Distance is found only from the area under a speed-time graph.
一个常见误区是试图从距离-时间图下方的面积来求距离。这个面积在运动学中没有物理意义。距离只能从速度-时间图下方的面积得出。
Another mistake is forgetting to convert units. If time is given in minutes and speed in m/s, convert the time to seconds before any calculation.
另一个常见错误是忘记换算单位。如果时间以分钟为单位，而速度以 m/s 为单位，就要在计算前把时间换算成秒。
Always label each section of a graph carefully. In exam questions, write the values you know on the graph and show clear working for both gradient and area calculations.
始终仔细标注图像的每一个部分。在考试中，把已知的数值写在图上，并在计算斜率和面积时展示清晰的解题步骤。
10. Worked Example | 例题解析
A car starts from rest and accelerates uniformly for 5 seconds until it reaches 10 m/s. It then travels at a constant speed for 10 seconds. Finally, it decelerates uniformly and stops after 5 seconds. Find the total distance travelled and the average speed.
一辆汽车从静止开始匀加速 5 秒，达到 10 m/s。然后以匀速行驶 10 秒。最后匀减速，5 秒后停下。求总运动距离和平均速度。
First, sketch the speed-time graph. It consists of a triangle for the acceleration stage, a rectangle for the constant speed stage, and a triangle for the deceleration stage.
首先，画出速度-时间图。它由加速阶段的三角形、匀速阶段的矩形和减速阶段的三角形组成。
s 10 × 10 = 100 m. The final triangle has base 5 s and height 10 m/s, so its area is 25 m.
第一个三角形的底为 5 秒，高为 10 m/s，因此面积为 ½ × 5 × 10 = 25 m。矩形的宽为 10 秒，高为 10 m/s，因此面积为 10 × 10 = 100 m。最后一个三角形的底为 5 秒，高为 10 m/s，因此面积为 25 m。
25 + 100 + 25 = 150 m. Total time = 5 + 10 + 5 = 20 s. Average speed = 150 ÷ 20 = 7.5 m/s.
总距离 = 25 + 100 + 25 = 150 m。总时间 = 5 + 10 + 5 = 20 秒。平均速度 = 150 ÷ 20 = 7.5 m/s。
This example shows how splitting the area into simple shapes makes distance calculations straightforward in an exam.
这个例题说明，把面积拆分成简单图形可以使考试中的距离计算变得更加直接。
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