📚 IGCSE Maths Practice Book 100: Graphs & Kinematics | IGCSE 数学练习册 100：图像与运动学
Welcome to Practice Book 100 in the IGCSE Maths revision series. This unit focuses on graphs and kinematics, two topics that appear regularly in both Core and Extended papers. You will learn how to read distance-time graphs and speed-time graphs, how to calculate gradients and areas, and how to link these skills to real motion.
欢迎使用 IGCSE 数学复习系列练习册 100。本单元聚焦图像与运动学，这两个主题在核心和扩展试卷中经常出现。你将学会如何读取距离-时间图像和速度-时间图像，如何计算梯度和面积，以及如何将这些技能与实际运动联系起来。
1. Distance-Time Graphs: Core Ideas | 距离-时间图：核心概念
A distance-time graph shows how the distance of an object from a starting point changes over time. The horizontal axis is time, and the vertical axis is distance.
距离-时间图显示物体离起点的距离如何随时间变化。横轴是时间，纵轴是距离。
The gradient of a distance-time graph gives the speed. A steeper line means a higher speed, while a horizontal line means the object is stationary.
距离-时间图像的梯度表示速度。线越陡，速度越大；水平线表示物体静止。
Speed = gradient = (d₂ − d₁) ÷ (t₂ − t₁)
A straight downward slope means the object is moving back towards the starting point. The gradient is negative, so the velocity is negative, but speed is usually given as a positive value.
向下的直线表示物体正在返回起点。梯度为负，因此速度为负，但速率通常以正值表示。
2. Reading Distance-Time Graphs | 读取距离-时间图
Always check the labels on both axes before answering a graph question. A common mistake is to assume a graph is speed-time when it is actually distance-time.
在回答图像题之前，一定要先检查两个坐标轴的标签。常见的错误是把距离-时间图误认为速度-时间图。
If the line is curved on a distance-time graph, the speed is changing. A curve that gets steeper shows acceleration, and a curve that becomes flatter shows deceleration.
如果距离-时间图上的线是弯曲的，说明速度在变化。越来越陡的曲线表示加速，而越来越平的曲线表示减速。
- Horizontal line = stationary.
- Straight sloping line = constant speed.
- Curved line = changing speed.
- 水平线 = 静止。
- 笔直斜线 = 匀速。
- 曲线 = 速度在变化。
3. Speed-Time Graphs: Core Ideas | 速度-时间图：核心概念
A speed-time graph shows how the speed of an object changes over time. The horizontal axis is time, and the vertical axis is speed or velocity.
速度-时间图显示物体的速度如何随时间变化。横轴是时间，纵轴是速度或速率。
The gradient of a speed-time graph gives the acceleration. A positive gradient means acceleration, a negative gradient means deceleration, and a horizontal line means constant speed.
速度-时间图像的梯度表示加速度。正梯度表示加速，负梯度表示减速，水平线表示匀速。
Acceleration = gradient = (v₂ − v₁) ÷ (t₂ − t₁)
Units are important. If speed is in m/s and time is in s, acceleration is in m/s².
单位很重要。如果速度以 m/s 为单位，时间以 s 为单位，那么加速度的单位就是 m/s²。
4. 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 tested ideas in IGCSE graph and kinematics questions.
速度-时间图像下方的面积表示物体运动的总距离。这是 IGCSE 图像与运动学题中最常考的概念之一。
For composite shapes, split the area into rectangles and triangles and add them together.
对于简单的矩形，面积等于速度 × 时间。对于三角形，面积等于 ½ × 底 × 高。对于组合图形，可将面积拆分为矩形和三角形再相加。
Distance = area under speed-time graph = ½ × base × height + width × height
Be careful: if the graph shows velocity rather than speed, an area below the time axis may represent displacement in the opposite direction. In IGCSE speed-time graphs, the line is usually above the axis.
注意：如果图像显示的是速度而不是速率，时间轴下方的面积可能表示反方向的位移。在 IGCSE 的速度-时间图中，线通常位于时间轴上方。
5. Gradients and Rates of Change | 梯度与变化率
A gradient is not only about speed and acceleration. Any straight-line graph has a gradient that tells you how quickly the y-variable changes with respect to the x-variable.
梯度不仅仅涉及速度和加速度。任何直线图像都有梯度，它告诉你 y 变量相对于 x 变量的变化快慢。
In kinematics, the gradient of a distance-time graph is a rate of change of distance with time. The gradient of a speed-time graph is a rate of change of speed with time.
在运动学中，距离-时间图像的梯度是距离随时间的变化率。速度-时间图像的梯度是速度随时间的变化率。
Use the same formula every time: change in the vertical direction divided by change in the horizontal direction. Do not swap the axes.
每次使用相同的公式：竖直方向的变化量除以水平方向的变化量。不要交换坐标轴。
Gradient = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁)
6. Kinematics Formulae in Context | 运动学公式的应用
IGCSE Maths sometimes links graph work to the standard kinematics equations. You do not always need them, but they help you check answers found from a graph.
IGCSE 数学有时会将图像题与标准运动学方程联系起来。你并不总是需要这些公式，但它们可以帮助你检查从图像中得到的答案。
The three common formulae for constant acceleration are:
匀加速运动的三个常用公式如下：
v = u + at
s = ut + ½at²
v² = u² + 2as
Here, u is initial speed, v is final speed, a is acceleration, t is time, and s is distance or displacement.
这里，u 是初速度，v 是末速度，a 是加速度，t 是时间，s 是距离或位移。
7. Worked Example: Distance-Time Graph | 例题：距离-时间图
A cyclist travels from home. For the first 10 seconds she travels 40 m in a straight line. She then stops for 10 seconds. Finally, she returns home in 20 seconds at a steady speed.
一名骑车人从家出发。前 10 秒她沿直线前进了 40 米。然后她停了 10 秒。最后，她用 20 秒匀速回到家。
On a distance-time graph, this would show a straight rising line for 10 s, a horizontal line from 10 s to 20 s, and a straight falling line from 20 s to 40 s.
在距离-时间图上，这会显示：前 10 秒为上升直线，10 秒到 20 秒为水平线，20 秒到 40 秒为下降直线。
= 4 m/s. The speed on the return is 40 m ÷ 20 s = 2 m/s. The flat section has speed 0 m/s.
第一段的速度是 40 m ÷ 10 s = 4 m/s。返回时的速度是 40 m ÷ 20 s = 2 m/s。水平段的速度为 0 m/s。
8. Worked Example: Speed-Time Graph | 例题：速度-时间图
A car accelerates uniformly from rest to 20 m/s in 5 seconds. It then travels at a constant speed for 10 seconds, and finally slows down uniformly to rest in 5 seconds.
一辆汽车从静止开始匀加速，5 秒内达到 20 m/s。然后以匀速行驶 10 秒，最后在 5 秒内匀减速至静止。
s) ÷ 5 s = 4 m/s². The deceleration in the last stage is (0 m/s − 20 m/s) ÷ 5 s = −4 m/s².
第一阶段的加速度是 (20 m/s − 0 m/s) ÷ 5 s = 4 m/s²。最后阶段的减速度是 (0 m/s − 20 m/s) ÷ 5 s = −4 m/s²。
The total distance travelled is the area under the speed-time graph. The shape is a trapezium, which can be split into two triangles and one rectangle:
总行驶距离是速度-时间图下的面积。该图形为梯形，可拆分为两个三角形和一个矩形：
Area = ½ × 5 × 20 + 10 × 20 + ½ × 5 × 20 = 50 + 200 + 50 = 300 m
So the car travels 300 m in total.
因此汽车总共行驶了 300 m。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is confusing the distance-time graph with the speed-time graph. Always write down what the gradient and area represent for the graph you are given.
一个常见错误是将距离-时间图与速度-时间图混淆。一定要先写下给定图像中梯度和面积分别代表什么。
Another mistake is ignoring units when calculating speed or acceleration. Write units on every answer, and convert minutes to seconds when needed.
另一个错误是在计算速度或加速度时忽略了单位。每个答案都要写单位，并在需要时将分钟换算为秒。
- Distance-time graph: gradient = speed.
- Speed-time graph: gradient = acceleration, area = distance.
- Use seconds for time unless the question asks otherwise.
- 距离-时间图：梯度 = 速度。
- 速度-时间图：梯度 = 加速度，面积 = 距离。
- 除非题目另有要求，时间使用秒。
10. Practice Set 100: Quick Check | 练习集 100：快速检测
Test yourself with these quick questions. Try to answer each one before reading the solution below.
用下面的快速问题测试自己。先尝试回答，再看下面的答案。
Question 1: A runner covers 100 m in 25 s. What is the average speed?
问题 1：一名跑步者 25 秒跑了 100 m。平均速度是多少？
Average speed = total distance ÷ total time = 100 m ÷ 25 s = 4 m/s
Question 2: A car accelerates from 10 m/s to 30 m/s in 4 s. Find the acceleration.
问题 2：一辆汽车 4 秒内从 10 m/s 加速到 30 m/s。求加速度。
Acceleration = (30 m/s − 10 m/s) ÷ 4 s = 20 m/s ÷ 4 s = 5 m/s²
Question 3: A speed-time graph shows a horizontal line at 15 m/s for 12 s. How far does the object travel?
问题 3：一张速度-时间图显示一条位于 15 m/s 的水平线，持续 12 s。物体运动的距离是多少？
Distance = speed × time = 15 m/s × 12 s = 180 m
Practice Book 100 gives you the tools to handle graph and kinematics questions confidently. Return to these worked examples before any test.
练习册 100 为你提供了自信应对图像与运动学题目的工具。在任何测试前反复回顾这些例题。
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