Distance-Time and Speed-Time Graphs | 距离-时间图像与速度-时间图像

📚 Distance-Time and Speed-Time Graphs | 距离-时间图像与速度-时间图像

In IGCSE Mathematics, kinematics graphs are essential tools for describing motion in everyday contexts. Two types of graphs are especially important: distance-time graphs and speed-time graphs. This article explains how to interpret, calculate, and sketch them accurately.

在IGCSE数学中，运动学图像是描述日常运动的重要工具。其中两类图像尤其关键：距离-时间图像和速度-时间图像。本文将解释如何准确地解读、计算和绘制这些图像。

1. What are Kinematics Graphs? | 什么是运动学图像？

Kinematics graphs show how a moving object’s position or velocity changes over time. The horizontal axis always represents time, usually measured in seconds (s) or minutes (min). The vertical axis represents either distance (m) or speed (m/s).

运动学图像展示移动物体的位置或速度随时间的变化。横轴始终表示时间，通常以秒（s）或分钟（min）为单位。纵轴表示距离（m）或速度（m/s）。

There are two main types: distance-time graphs and speed-time graphs. They look similar but convey different information. Always check the axes before interpreting any graph.

主要有两类：距离-时间图像和速度-时间图像。它们外观相似但传达的信息不同。在解读任何图像前，一定要先看清坐标轴。

2. Distance-Time Graphs: The Basics | 距离-时间图像基础

A distance-time graph plots the total distance travelled against time. The gradient (slope) of the graph at any point gives the speed at that moment.

距离-时间图像绘制的是总行驶距离随时间的变化。图像上任意一点的梯度（斜率）表示该时刻的速度。

A horizontal line means the object is stationary, so speed is zero.

水平线表示物体静止，速度为零。
A straight line with a constant slope means constant speed.

具有恒定斜率的直线表示匀速运动。
A curved line means the speed is changing (accelerating or decelerating).

曲线表示速度正在变化（加速或减速）。

3. Speed from Distance-Time Graphs | 从距离-时间图像求速度

For a straight-line segment, speed is calculated as change in distance ÷ change in time. In mathematical terms, speed = gradient = Δd / Δt.

对于直线段，速度可通过距离变化量 ÷ 时间变化量计算。在数学术语中，速度 = 梯度 = Δd / Δt。

speed = Δd ÷ Δt

速度 = Δd ÷ Δt

For example, if an object travels 50 m in 10 s, the speed is 50 ÷ 10 = 5 m/s. If the graph is curved, you must draw a tangent at the point of interest and find its gradient.

例如，如果物体在10秒内行驶了50米，速度为50 ÷ 10 = 5米/秒。如果图像是曲线，你必须在目标点处画切线并求其梯度。

4. Speed-Time Graphs: The Basics | 速度-时间图像基础

A speed-time graph plots speed against time. Here, the gradient gives the acceleration, and the area under the graph gives the total distance travelled.

速度-时间图像绘制速度随时间的变化。这里，梯度表示加速度，图像下方的面积表示行驶的总距离。

A horizontal line indicates constant speed (zero acceleration).

水平线表示匀速运动（加速度为零）。
A straight line with positive slope means constant acceleration.

具有正斜率的直线表示匀加速运动。
A straight line with negative slope means constant deceleration.

具有负斜率的直线表示匀减速运动。

5. Distance from Speed-Time Graphs | 从速度-时间图像求距离

On a speed-time graph, the distance travelled is represented by the area under the graph between two times. This may be a rectangle, triangle, trapezium, or a combination of shapes.

在速度-时间图像上，行驶的距离由图像在两个时刻之间下方的面积表示。该区域可能是矩形、三角形、梯形，或这些形状的组合。

distance = area under speed-time graph

距离 = 速度-时间图像下方的面积

For a triangle, area = ½ × base × height. For a trapezium, area = ½ × (a + b) × height. Remember to use the units correctly.

对于三角形，面积 = ½ × 底 × 高。对于梯形，面积 = ½ × (上底 + 下底) × 高。注意正确使用单位。

6. Acceleration from Speed-Time Graphs | 从速度-时间图像求加速度

Acceleration is the rate of change of speed. On a speed-time graph, the gradient at any point equals the acceleration. A positive gradient means acceleration, a negative gradient means deceleration.

加速度是速度的变化率。在速度-时间图像上，任意一点的梯度等于加速度。正梯度表示加速，负梯度表示减速。

acceleration = Δv ÷ Δt

加速度 = Δv ÷ Δt

For example, if speed changes from 0 to 20 m/s in 4 s, acceleration = (20 – 0) ÷ 4 = 5 m/s². The unit is metres per second squared (m/s²).

例如，如果速度在4秒内从0变到20米/秒，加速度 = (20 – 0) ÷ 4 = 5米/秒²。单位是米每二次方秒（m/s²）。

7. Interpreting Curves | 曲线图像的解释

Not all motion is at constant speed or acceleration. A curved distance-time graph means changing speed. A curved speed-time graph means changing acceleration.

并非所有运动都是匀速或匀加速。曲线距离-时间图像意味着速度在变化。曲线速度-时间图像意味着加速度在变化。

On a distance-time graph, a convex curve (steeper over time) means accelerating. A concave curve (flatter over time) means decelerating.

在距离-时间图像上，凸曲线（随时间变陡）表示加速。凹曲线（随时间变平）表示减速。

On a speed-time graph, a curved line indicates non-uniform acceleration. The gradient of a tangent gives the instantaneous acceleration.

在速度-时间图像上，曲线表示非匀加速度。切线的梯度给出瞬时加速度。

8. Sketching Graphs from Given Motion | 根据给定运动画出图像

You may be asked to sketch a distance-time graph or a speed-time graph from a description of motion. Break the motion into stages: stationary, constant speed, accelerating, or decelerating.

你可能会被要求根据运动描述画出距离-时间图像或速度-时间图像。将运动分成几个阶段：静止、匀速、加速或减速。

For distance-time graphs: stationary → horizontal line; constant speed → straight diagonal line; accelerating → curve upwards; decelerating → curve flattening.

对于距离-时间图像：静止→水平线；匀速→斜直线；加速→向上弯曲的曲线；减速→变平的曲线。
For speed-time graphs: stationary or constant speed → horizontal line; constant acceleration → straight diagonal line; changing acceleration → curved line.

对于速度-时间图像：静止或匀速→水平线；匀加速→斜直线；变加速度→曲线。

Always label axes with the appropriate quantities and units. Use a pencil and ruler when drawing straight-line segments.

始终在坐标轴上标注适当的量和单位。画直线段时使用铅笔和直尺。

9. Worked Example: A Car Journey | 例题解析：汽车行程

A car starts from rest, accelerates uniformly for 4 s until it reaches 12 m/s, then maintains this speed for 6 s, then brakes uniformly to rest in 2 s. Sketch the speed-time graph and find the total distance travelled.

一辆汽车从静止开始，匀加速4秒，速度达到12米/秒，然后保持这个速度6秒，最后匀减速2秒停下来。画出速度-时间图像，并求总行驶距离。

Draw a graph with time on the horizontal axis (0 to 12 s) and speed on the vertical axis (0 to 12 m/s). The first segment is a straight line from (0,0) to (4,12). The second segment is horizontal from (4,12) to (10,12). The third segment is a straight line from (10,12) to (12,0).

画一个横轴为时间（0到12秒）、纵轴为速度（0到12米/秒）的图像。第一段是从(0,0)到(4,12)的直线。第二段是从(4,12)到(10,12)的水平线。第三段是从(10,12)到(12,0)的直线。

Total distance = area of trapezium: ½ × (6 + 12) × 12 = 108 m. Equivalently, sum of triangle (24 m), rectangle (72 m), and triangle (12 m).

总距离 = 梯形面积：½ × (6 + 12) × 12 = 108米。也可分为三角形（24米）、矩形（72米）和三角形（12米）之和。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always check whether the graph is distance-time or speed-time. Many students confuse the two and accidentally use area on a distance-time graph or gradient on a speed-time graph incorrectly.

始终检查图像是距离-时间还是速度-时间。许多学生混淆两者，错误地在距离-时间图像上使用面积，或在速度-时间图像上使用梯度。

Use correct units: distance in metres (m), time in seconds (s), speed in m/s, acceleration in m/s².

使用正确的单位：距离为米（m），时间为秒（s），速度为米/秒（m/s），加速度为米/秒²（m/s²）。
For a curved distance-time graph, the gradient of the tangent gives instantaneous speed, not average speed.

对于曲线距离-时间图像，切线的梯度给出瞬时速度，而非平均速度。
On a speed-time graph, the area under the graph is positive even if speed is positive; if speed is always positive, distance equals area without sign issues.

在速度-时间图像上，图像下方的面积为正值；如果速度始终为正，距离等于面积，无需考虑符号。

Finally, when sketching graphs, mark key points clearly and show all working for calculations. Practice past paper questions to improve accuracy and speed.

最后，画图时清晰标出关键点，并展示所有计算过程。通过练习历年真题来提高准确度和速度。

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Distance-Time and Speed-Time Graphs | 距离-时间图像与速度-时间图像