📚 IGCSE Maths Graphs and Kinematics | IGCSE 数学图像与运动学
Graphs are one of the most common ways to describe motion in IGCSE Mathematics. This topic brings together algebra, geometry and practical problem-solving. You need to read information from distance-time and speed-time graphs, calculate gradients and areas, and link these results to speed, acceleration and distance travelled.
图像是 IGCSE 数学中描述运动最常见的方式之一。这个主题结合了代数、几何和实际解题。你需要从距离-时间图和速度-时间图中读取信息，计算斜率和面积，并将这些结果与速度、加速度和行驶距离联系起来。
1. Distance-Time Graphs: The Basics | 距离-时间图基础
A distance-time graph shows how the distance of an object from a fixed point changes over time. Time is always plotted on the horizontal axis and distance on the vertical axis.
距离-时间图显示物体离固定点的距离随时间的变化。时间总是标在横轴，距离标在纵轴。
The gradient of a distance-time graph gives the speed of the object. A steeper line means a greater speed, while a flat horizontal line means the object is stationary.
距离-时间图的斜率表示物体的速度。线越陡，速度越大；水平线表示物体静止。
If the graph is a straight line, the speed is constant. If the graph is curved, the speed is changing and the gradient at a point must be found by drawing a tangent to the curve.
如果图像是直线，速度恒定；如果图像是曲线，速度在变化，此时需要画切线来求某一点的斜率。
Always check the units given on both axes. For example, distance may be in metres or kilometres, and time may be in seconds, minutes or hours.
一定要检查两个坐标轴上的单位。例如，距离可能是米或千米，时间可能是秒、分钟或小时。
2. Interpreting Distance-Time Graphs | 解读距离-时间图
-time graph, divide the change in distance by the change in time: average speed = Δs / Δt.
要计算距离-时间图上两点间的平均速度，用距离变化量除以时间变化量：平均速度 = Δs / Δt。
otal distance is 60 m and the total time is 20 s, so the average speed is 60 ÷ 20 = 3 m/s.
例题：一辆汽车在前 5 秒行驶 40 米，停车 10 秒，之后 5 秒返回 20 米。总距离为 60 米，总时间为 20 秒，因此平均速率为 60 ÷ 20 = 3 米/秒。
When describing a journey from a distance-time graph, break the graph into sections. For each section, state whether the object is moving away, returning, moving at constant speed, speeding up, slowing down or stationary.
在根据距离-时间图描述一段行程时，要把图像分成若干段。对每一段，说明物体是在远离、返回、匀速运动、加速、减速还是静止。
A return journey is shown by a downward-sloping line on a distance-time graph. The object is moving back towards the starting point, but the speed is still found from the gradient, which is positive when we only consider the steepness.
在距离-时间图上，返回行程用向下倾斜的线表示。物体正在向起点返回，但速率仍然通过斜率的绝对值来求得。
3. Speed-Time Graphs: The Basics | 速度-时间图基础
A speed-time graph plots speed on the vertical axis and time on the horizontal axis. It is also called a velocity-time graph in some IGCSE questions, but for motion in one direction the shape is the same.
速度-时间图把速度标在纵轴，时间标在横轴。在一些 IGCSE 题目中也称为速度-时间图，但对于单向运动，图形是相同的。
The gradient of a speed-time graph gives the acceleration. A positive gradient means speeding up, a negative gradient means slowing down, and a horizontal line means constant speed.
速度-时间图的斜率表示加速度。斜率为正表示加速，斜率为负表示减速，水平线表示匀速。
The area under a speed-time graph between two times gives the distance travelled during that interval.
速度-时间图下两个时间之间的面积表示这段时间内行驶的距离。
4. Interpreting Speed-Time Graphs | 解读速度-时间图
ple, if speed increases from 0 m/s to 20 m/s in 4 s, acceleration = (20 - 0) / 4 = 5 m/s².
从速度-时间图求加速度，使用 a = Δv / Δt。例如，速度在 4 秒内从 0 米/秒增大到 20 米/秒，加速度 = (20 - 0) / 4 = 5 米/秒²。
If the graph is a triangle with base 4 s and height 20 m/s, distance = ½ × 4 × 20 = 40 m.
对于直线型速度-时间图，图下面积可以用三角形或梯形面积公式计算。如果图像是底为 4 秒、高为 20 米/秒的三角形，距离 = ½ × 4 × 20 = 40 米。
When the graph is made of several straight sections, calculate the area of each section separately and then add them. The total area equals the total distance travelled.
当图像由几段直线组成时，分别计算每一段的面积，然后相加。总面积等于行驶的总距离。
Always include the correct units in your answer. Acceleration is measured in m/s², distance in m, speed in m/s and time in s.
在答案中一定要写上正确的单位。加速度的单位是米/秒²，距离是米，速度是米/秒，时间是秒。
5. Gradient = Rate of Change | 斜率就是变化率
In both distance-time and speed-time graphs, the gradient always represents a rate of change: for a distance-time graph it is speed; for a speed-time graph it is acceleration.
在距离-时间图和速度-时间图中，斜率都表示变化率：距离-时间图的斜率是速度；速度-时间图的斜率是加速度。
If the graph is curved, draw a tangent at the required point and find its gradient to estimate the instantaneous speed or acceleration.
如果图像是曲线，在要求的点画切线并求其斜率，以估算瞬时速度或瞬时加速度。
ange / horizontal change, so always check the scales on both axes before calculating. Do not simply count grid squares if the scales are not equal.
记住斜率 = 纵轴变化量 / 横轴变化量，所以在计算前一定要检查两轴的刻度。如果刻度不相同，不要简单地数格子的数量。
In distance-time graphs, a decreasing positive gradient means the object is slowing down. In speed-time graphs, a decreasing positive gradient means the acceleration is decreasing.
在距离-时间图中，正的斜率减小表示物体正在减速。在速度-时间图中，正的斜率减小表示加速度正在减小。
6. Area Under a Speed-Time Graph | 速度-时间图下的面积
The area under a speed-time graph can be split into simple shapes such as rectangles, triangles and trapeziums. This makes the distance calculation easier.
速度-时间图下的面积可以分割成矩形、三角形和梯形等简单图形，这样计算距离更容易。
becomes distance = ½ × (u + v) × t, where u is the initial speed and v is the final speed.
对于梯形，面积 = ½ × (两底之和) × 高。在运动学中，这可以写成距离 = ½ × (u + v) × t，其中 u 是初速度，v 是末速度。
h × width. For example, a constant speed of 12 m/s for 5 s gives distance = 12 × 5 = 60 m.
如果图像的一部分是矩形，使用面积 = 长 × 宽。例如，以 12 米/秒的匀速行驶 5 秒，距离 = 12 × 5 = 60 米。
For IGCSE Mathematics, speed-time graphs normally stay on or above the time axis, so all areas are positive. If a velocity-time graph goes below the axis, the area below represents motion in the opposite direction.
在 IGCSE 数学中，速度-时间图通常位于时间轴及其上方，因此所有面积为正。如果速度-时间图下穿到时间轴以下，轴下面积表示反方向运动。
7. Kinematics Equations and Graphs | 运动学公式与图像
For constant acceleration, the main equations linking initial speed u, final speed v, acceleration a, time t and distance s are:
对于匀加速运动，联系初速度 u、末速度 v、加速度 a、时间 t 和距离 s 的主要公式有：
v = u + at
s = ½(u + v)t
s = ut + ½at²
v² = u² + 2as
These equations are consistent with the fact that the area under a speed-time graph with constant acceleration is a trapezium. The first equation comes directly from the gradient, and the second from the trapezium area.
这些公式与匀加速运动速度-时间图下面积是梯形这一事实一致。第一个公式直接来自斜率，第二个公式来自梯形面积。
You only need to use the formula that contains the three quantities you know and the one quantity you need to find. This is often tested in IGCSE word problems.
你只需要使用包含已知三个量和待求一个量的公式。这在 IGCSE 应用题中经常考查。
8. Converting Between Graphs | 图像之间的转换
If you are given a distance-time graph, you can sketch the corresponding speed-time graph by looking at the gradient in each section.
如果给出距离-时间图，你可以通过观察每一段的斜率来画出相应的速度-时间图。
A constant positive gradient in a distance-time graph becomes a horizontal line at positive speed in a speed-time graph. A horizontal distance-time graph becomes a zero-speed line.
距离-时间图中恒为正的斜率，在速度-时间图中变为位于正速度处的水平线；距离-时间图的水平段变为速度为零的线。
If the distance-time graph has a decreasing gradient, the speed-time graph shows a decreasing speed. If the gradient increases, the speed-time graph rises.
如果距离-时间图的斜率逐渐减小，速度-时间图显示速度减小；如果斜率逐渐增大，速度-时间图上升。
When converting from a speed-time graph to a distance-time graph, remember that the distance is the accumulated area. A constant positive speed produces a straight line with a constant positive gradient on the distance-time graph.
当从速度-时间图转换为距离-时间图时，记住距离是累积的面积。恒定的正速度会在距离-时间图上产生一条斜率恒为正的直线。
9. Common Mistakes and Misconceptions | 常见错误与误区
A common mistake is to confuse distance-time and speed-time graphs: the gradient of a distance-time graph is speed, not acceleration, and the gradient of a speed-time graph is acceleration, not distance.
常见错误是混淆距离-时间图和速度-时间图：距离-时间图的斜率是速度而非加速度，速度-时间图的斜率是加速度而非距离。
Students sometimes calculate average speed by adding speeds and dividing by the number of values. This is only correct if each speed lasts for the same time interval. Otherwise use total distance / total time.
学生有时通过把速度相加再除以数量来计算平均速度。这仅在各速度持续时间相等时才正确。否则应使用总距离 / 总时间。
When finding the area under a graph, make sure you use the correct base and height from the axes. Do not forget to include units in the final answer, and check that the units are consistent.
计算图下面积时，确保使用坐标轴上正确的底和高。不要忘记在最终答案中写上单位，并检查单位是否一致。
Another error is reading the graph at the wrong point. Always draw dashed lines from the point on the curve to both axes before reading off values.
另一个错误是在错误的点读取图像。在读值之前，一定要从曲线上该点向两个坐标轴画出虚线。
10. Exam Technique and Practice | 应试技巧与练习
In the exam, always annotate the graph: label each section as 'constant speed
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