📚 Gradients and Kinematics: Interpreting Motion Graphs | 斜率与运动学：解读运动图像
In IGCSE Mathematics, motion graphs link algebra, gradients and area to real-world movement. Students usually meet distance-time graphs and speed-time graphs in the same topic because they describe journeys using straight lines, slopes and areas. This article explains how to read these graphs, calculate speed and acceleration, find distance from a speed-time graph, and avoid the most common exam errors.
在 IGCSE 数学中，运动图像把代数、斜率和面积与现实世界的运动联系起来。距离-时间图和速度-时间图通常出现在同一主题中，因为它们用直线、斜率和面积来描述行程。本文将解释如何读懂这些图像、计算速度和加速度、从速度-时间图中求距离，并避开最常见的考试错误。
1. Distance-Time Graphs: The Big Picture | 距离-时间图全貌
A distance-time graph shows how far an object has travelled from a starting point over time. Time is plotted on the horizontal x-axis, and distance is plotted on the vertical y-axis. A straight sloping line indicates constant speed; a horizontal line indicates that the object is stationary; a steeper line indicates a higher speed. If the line slopes downwards, the object is moving back towards the start.
距离-时间图显示一个物体从起点出发随时间移动的距离。时间标在水平 x 轴上，距离标在垂直 y 轴上。一条倾斜直线表示匀速运动；水平线表示物体静止；越陡的线表示速度越快。如果线向下倾斜，则表示物体正在返回起点。
Always begin by checking the labels on both axes. Many mistakes happen because students assume the y-axis is speed when it is actually distance. The shape of the graph only makes sense when you know which quantity is being plotted.
解题时首先要检查两个坐标轴上的标签。很多错误都是因为学生把 y 轴误认为速度，而它实际上是距离。只有知道图上表示的是哪个量，图像的形状才有意义。
2. Gradient of a Distance-Time Graph = Speed | 距离-时间图的斜率 = 速度
The gradient of a distance-time graph gives speed. To calculate the speed on a straight section, choose two points on that section, find the change in distance and the change in time, then divide. For a straight section from (t₁, d₁) to (t₂, d₂), the speed is:
距离-时间图的斜率表示速度。要计算直线段上的速度，请在该线段上选取两个点，求出距离变化量和时间变化量，然后相除。对于从 (t₁, d₁) 到 (t₂, d₂) 的直线段，速度为：
Speed = (d₂ - d₁) ÷ (t₂ - t₁)
The units depend on the axes, such as m s⁻¹ or km h⁻¹. If the graph is curved, the gradient is not constant, so the gradient of a tangent at a point gives the instantaneous speed at that point.
单位取决于坐标轴，例如 m s⁻¹ 或 km h⁻¹。如果图像是曲线，斜率不是恒定的，因此某点切线的斜率表示该点的瞬时速度。
3. Speed-Time Graphs: Reading the Axes | 速度-时间图：读懂坐标轴
A speed-time graph has time on the x-axis and speed on the y-axis. A horizontal line means constant speed. A straight upward sloping line means constant acceleration, and a straight downward sloping line means constant deceleration, which is also called negative acceleration. The y-intercept gives the initial speed, and the point where the line crosses the x-axis can show when the object stops.
速度-时间图的 x 轴是时间，y 轴是速度。水平线表示匀速运动。向上的倾斜直线表示匀加速，向下的倾斜直线表示匀减速，也称为负加速度。y 轴截距表示初速度，图像与 x 轴的交点可以表示物体停止运动的时刻。
On this graph, the slope does not represent speed; it represents how quickly the speed is changing. This is a key difference from the distance-time graph, and exam questions often test whether students can separate the two ideas.
在这张图上，斜率不表示速度，而表示速度变化的快慢。这是与距离-时间图的关键区别，考试题经常考查学生是否能区分这两个概念。
4. Gradient of a Speed-Time Graph = Acceleration | 速度-时间图的斜率 = 加速度
The gradient of a speed-time graph equals acceleration. For a straight section, acceleration is calculated by dividing the change in speed by the change in time. A positive gradient means the object is speeding up, a negative gradient means it is slowing down, and a zero gradient means constant speed. Common units are m s⁻² or km h⁻¹ per second.
速度-时间图的斜率等于加速度。对于直线段，加速度等于速度变化量除以时间变化量。正斜率表示物体在加速，负斜率表示物体在减速，零斜率表示匀速。常见单位是 m s⁻² 或 km h⁻¹ 每秒。
Acceleration = Δv ÷ Δt
write the units clearly, because a numerical value without units is incomplete in IGCSE answers.
例如，如果速度在 3 秒内从 5 m s⁻¹ 增加到 20 m s⁻¹，则加速度为 (20 - 5) ÷ 3 = 5 m s⁻²。请始终写清单位，因为在 IGCSE 答案中，没有单位的数值是不完整的。
5. Area Under a Speed-Time Graph = Distance | 速度-时间图下的面积 = 距离
The area between the speed-time graph and the time axis represents the distance travelled. For a horizontal line, the area is a rectangle. For a sloping line, the area may be a triangle or a trapezium. In IGCSE problems, you usually split the graph into rectangles, triangles and trapezia, calculate each area, and then add them.
速度-时间图与时间轴之间的面积表示运动距离。水平线下方是矩形；斜线下方可能是三角形或梯形。在 IGCSE 题目中，通常需要把图像拆分成矩形、三角形和梯形，分别计算面积，再相加。
Area of a trapezium = ½ × (a + b) × h
If the graph shows speed in m s⁻¹ and time in seconds, the area is in metres. If the speed is always positive, the total area is the total distance. If a graph goes below the time axis, you would need to be careful about direction, but IGCSE speed-time graphs usually keep speed non-negative.
如果图像中速度的单位是 m s⁻¹，时间单位是秒，那么面积的单位就是米。如果速度始终为正值，总面积就是总距离。如果图像出现在时间轴下方，则需要注意方向，但 IGCSE 的速度-时间图通常保持速度非负。
6. Curved Lines and Instantaneous Values | 曲线与瞬时值
If a distance-time graph is curved, the motion is not uniform. The average speed over an interval is the gradient of the chord connecting the two end points, while the instantaneous speed at a point is the gradient of the tangent at that point. You may be asked to draw a tangent to a curve and estimate its gradient in an exam.
如果距离-时间图是曲线，运动就是非匀速的。一段时间内的平均速度是连接两个端点的弦的斜率，而某一点的瞬时速度是该点切线的斜率。考试中可能会要求你在曲线上画一条切线并估算其斜率。
On a speed-time graph, a curved line means the acceleration is changing. The gradient at a point still gives the acceleration, but you must use a tangent. The area under a curved speed-time graph can be estimated by counting squares or by splitting the region into approximate shapes.
在速度-时间图中，曲线表示加速度正在变化。某一点的斜率仍然表示加速度，但必须使用切线。曲线速度-时间图下的面积可以通过数方格或把区域拆分成近似图形来估算。
7. Comparing Distance-Time and Speed-Time Graphs | 比较距离-时间图和速度-时间图
It is easy to confuse the two graph types. On a distance-time graph, a horizontal line means 'stopped', whereas on a speed-time graph, a horizontal line means 'constant speed'. A straight diagonal line on a distance-time graph means constant speed, while a straight diagonal line on a speed-time graph means constant acceleration. Always check the y-axis label before interpreting the graph.
这两种图像很容易混淆。在距离-时间图中，水平线表示“静止”，而在速度-时间图中，水平线表示“匀速”。距离-时间图中的倾斜直线表示匀速，而速度-时间图中的倾斜直线表示匀加速。解读图像前一定要检查 y 轴标签。
| Feature | 特征 | Distance-time graph | 距离-时间图 | Speed-time graph | 速度-时间图 |
|---|---|---|
| Horizontal line | 水平线 | Stationary | 静止 | Constant speed | 匀速 |
| Straight diagonal line | 倾斜直线 | Constant speed | 匀速 | Constant acceleration | 匀加速 |
| Gradient | 斜率 | Speed | 速度 | Acceleration | 加速度 |
| Area under graph | 图下面积 | No direct meaning | 无直接意义 | Distance travelled | 运动距离 |
8. Non-Uniform Motion and Turning Points | 非匀速运动与折返点
A real journey can include several legs: acceleration, constant speed, deceleration, rest, and return. On a distance-time graph, a downward slope indicates returning to the start. A turning point is where the graph changes from upward to downward, meaning the object has changed direction. The gradient at a turning point is zero because the object is momentarily stopped.
真实的行程可以包括多个阶段：加速、匀速、减速、静止和返回。在距离-时间图中，向下倾斜表示返回起点。转折点是图像由向上变为向下的位置，表示物体的运动方向发生了改变。转折点处的斜率为零，因为物体瞬间停止。
On a speed-time graph, zero speed is shown by a point on the time axis. The graph cannot usually go below the time axis unless direction is being modelled with signed velocity. For IGCSE, keep 'speed' as a non-negative quantity and use the distance-time graph to show return journeys.
在速度-时间图中，速度为零表示为时间轴上的一点。除非用带符号的速度表示方向，否则图像通常不会出现在时间轴下方。在 IGCSE 中，通常把“速度”作为非负量，并用距离-时间图来表示返回行程。
9. Worked Example: Two-Stage Journey | 例题：两段行程
t line down to (15, 0). The acceleration in the second stage is (0 - 12) ÷ 5 = -2.4 m s⁻².
一辆汽车以 12 m s⁻¹ 的恒速行驶了 10 秒，然后在接下来的 5 秒内匀减速到静止。画出速度-时间图：一条从 (0, 12) 到 (10, 12) 的水平线，然后一条向下到 (15, 0) 的直线。第二阶段的加速度为 (0 - 12) ÷ 5 = -2.4 m s⁻²。
The second part is a triangle: ½ × 5 × 12 = 30 m. Therefore, the total distance is 150 m.
总距离是图下的面积。第一部分是矩形：12 × 10 = 120 m。第二部分是三角形：½ × 5 × 12 = 30 m。因此总距离为 150 m。
Total distance = 120 m + 30 m = 150 m
This type of two-stage question is very common in IGCSE papers. It tests gradient, area, unit conversion and graph interpretation together, so practise drawing the graph before doing any calculations.
这类两阶段问题在 IGCSE 试卷中非常常见。它同时考查斜率、面积、单位换算和图像解读，所以在计算之前先练习画出图像。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common errors include reading the wrong axis, using distance units for speed, forgetting to convert hours to seconds, and calculating the gradient as time ÷ distance. Always state the units with your final answer. When estimating a tangent, use a long tangent line and choose points far apart to reduce measurement error.
常见错误包括读错坐标轴、把距离单位当作速度单位、忘记把小时换算成秒，以及把斜率算成时间除以距离。最终答案一定要写明单位。估算切线时，要画较长的切线并选择相距较远的点，以减少测量误差。
If the graph uses km and hours, speeds will be in km h⁻¹. If the graph uses m and seconds, speeds will be in m s⁻¹. Do not mix unit systems. In area calculations, write down each shape separately and show the sum, because method marks are often awarded even if a small arithmetic error occurs.
如果图像使用千米和小时，速度单位就是 km h⁻¹。如果图像使用米和秒，速度单位就是 m s⁻¹。不要混用单位系统。在面积计算中，要把每个形状单独写出并展示求和过程，因为即使出现小的算术错误，通常也会得到方法分。
11. Summary Table | 总结表
Use the following summary to check your understanding before an assessment. The table connects the graph feature to the physical quantity it represents.
在测评前，请使用下面的总结检查自己的理解。该表格将图像特征与其所表示的物理量联系起来。
| Graph type | 图像类型 | Gradient gives | 斜率表示 | Area gives | 面积表示 |
|---|---|---|
| Distance-time | 距离-时间 | Speed | 速度 | No direct meaning | 无直接意义 |
| Speed-time | 速度-时间 | Acceleration | 加速度 | Distance | 距离 |
Memorising this table helps you choose the correct calculation quickly, especially under exam pressure when graph questions are mixed with algebra and ratio problems.
记住这张表可以帮助你快速选择正确的计算方法，尤其是在考试压力下，图像题与代数和比例题混合出现时。
12. Final Practice Check | 最后练习检查
Before the exam, ensure you can do all of the following: find speed from a distance-time graph; find acceleration from a speed-time graph; find distance from the area under a speed-time graph; draw a tangent on a curve and estimate its gradient; and interpret horizontal
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