📚 G-K Workbook 100: Graphs and Kinematics | G-K练习册第100题：图像与运动学
Welcome to the TutorHao revision series for IGCSE Mathematics. This article focuses on Question 100 from the G-K workbook, a challenging problem that combines graph interpretation with kinematic principles. We will break down the key concepts, work through the solution step by step, and provide exam-ready strategies.
欢迎阅读 TutorHao 国际教育 IGCSE 数学复习系列。本文聚焦 G-K 练习册第 100 题，这是一道结合图形解读与运动学原理的综合性难题。我们将逐步拆解核心概念、完整推导解题过程，并提供考场实用策略。
1. Understanding Distance-Time Graphs | 理解距离-时间图
A distance-time graph shows how far an object travels over time. The horizontal axis represents time (t), and the vertical axis represents distance (s). The gradient of this graph equals the speed of the object. A straight line means constant speed, while a curve indicates changing speed.
距离-时间图表示物体在一段时间内行进的距离。横轴代表时间 t，纵轴代表距离 s。该图形的斜率等于物体的速度。直线表示匀速运动，曲线表示变速运动。
For the G-K Workbook 100 problem, you must first identify the type of motion described. If the graph is a straight horizontal line, the object is stationary. If the gradient is positive, the object moves away from the starting point; if negative, it returns.
在 G-K 练习册第 100 题中，你首先要判断所描述的运动类型。若图形为水平直线，物体静止；若斜率为正，物体远离起点；若斜率为负，物体返回起点。
Speed = Gradient of distance-time graph = Δs ÷ Δt
2. Understanding Velocity-Time Graphs | 理解速度-时间图
A velocity-time graph plots velocity (v) against time (t). The gradient of this graph represents acceleration. The area under the graph represents displacement (or distance if velocity is always positive).
速度-时间图以时间 t 为横轴、速度 v 为纵轴。该图形的斜率代表加速度。图形下方的面积代表位移（若速度始终为正则为路程）。
In Question 100, you may need to switch between distance-time and velocity-time graphs. Remember that the area under a velocity-time graph can be found by splitting it into rectangles and triangles.
第 100 题中，你可能需要在距离-时间图和速度-时间图之间转换。记住：速度-时间图的面积可通过分割为矩形和三角形来求解。
Acceleration = Gradient of velocity-time graph = Δv ÷ Δt
3. Calculating Gradient and Area | 计算斜率与面积
The gradient between two points on a graph is calculated as the vertical difference divided by the horizontal difference. For a curved graph, you may need to draw a tangent to find the instantaneous gradient.
图形上两点间的斜率等于纵坐标之差除以横坐标之差。对于曲线，你需要画切线来求瞬时斜率。
Area calculation on velocity-time graphs is essential. For a triangle, area = ½ × base × height; for a rectangle, area = length × width. Sum these areas to find the total displacement.
速度-时间图上的面积计算至关重要。三角形面积 = ½ × 底 × 高；矩形面积 = 长 × 宽。将这些面积相加即可得到总位移。
- Positive gradient: positive acceleration | 正斜率：正加速度
- Negative gradient: negative acceleration (deceleration) | 负斜率：负加速度（减速）
- Area above horizontal axis: positive displacement | 水平轴上方面积：正位移
- Area below horizontal axis: negative displacement | 水平轴下方面积：负位移
4. Kinematic Equations | 运动学方程
Kinematics is the branch of mechanics that describes motion without considering forces. In IGCSE mathematics, you often use four equations for uniform acceleration:
运动学是描述运动而不考虑力作用的分支学科。在 IGCSE 数学中，你经常需要用到以下四个匀加速运动方程：
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Here u is initial velocity, v is final velocity, a is acceleration, t is time, and s is displacement. These equations require constant acceleration, so be careful when acceleration changes.
其中 u 为初速度，v 为末速度，a 为加速度，t 为时间，s 为位移。这些方程要求加速度恒定，因此当加速度变化时需格外小心。
In the G-K workbook question 100, you might be given a velocity-time graph and asked to find values such as acceleration, displacement, or final velocity. The equations above convert graph information into algebraic answers.
在 G-K 练习册第 100 题中，你可能会获得一个速度-时间图，并需要求解加速度、位移或末速度等量。上述方程将图像信息转化为代数答案。
5. Interpreting Graphs in Practice | 实际图形解读
Real exam questions often combine multiple stages of motion. A car might accelerate, maintain a constant speed, then decelerate to stop. The velocity-time graph would contain a slope, a horizontal line, and a negative slope.
真实考试题通常包含多阶段运动。汽车可能加速、保持匀速、然后减速停止。其速度-时间图将包含一段上升直线、一段水平线和一段下降直线。
For the G-K Workbook 100, you must break the graph into segments and analyse each separately. Use the appropriate equations for each phase, then combine the results to answer the full question.
对于 G-K 练习册第 100 题，你必须将图形分成若干区段，并分别分析每一段。为每个阶段选用合适的方程，然后将结果合并以回答完整问题。
6. Common Pitfalls | 常见错误
One common mistake is confusing distance and displacement. Distance is a scalar quantity (only magnitude), while displacement is a vector quantity (magnitude and direction). On a velocity-time graph, the total area gives displacement, not necessarily total distance.
一个常见错误是混淆路程与位移。路程是标量（仅大小），位移是矢量（大小和方向）。在速度-时间图上，总面积给出的是位移，不一定是总路程。
Another mistake is reading the gradient of a distance-time graph as acceleration. The gradient here is speed, not acceleration. Acceleration only comes from the gradient of a velocity-time graph.
另一个错误是将距离-时间图的斜率读作加速度。该斜率是速度，不是加速度。加速度只能由速度-时间图的斜率得到。
Also, be careful with units. If time is in minutes and distance in kilometres, speed must be in km/min unless you convert to seconds or hours. Always convert to consistent units before calculating.
同时注意单位。如果时间以分钟、距离以千米为单位，那么速度单位应为千米/分钟，除非你换算成秒或小时。计算前务必统一单位。
7. Worked Example: The G-K Workbook Question 100 | 练习册第 100 题示例
Let us solve a similar version of the G-K Workbook 100. A cyclist starts from rest, accelerates uniformly at 0.5 m/s² for 8 seconds, maintains a constant speed for 20 seconds, then decelerates uniformly to rest in 4 seconds.
让我们求解一道与 G-K 练习册第 100 题类似的题目。一名骑行者从静止开始，以 0.5 m/s² 的加速度匀加速 8 秒，然后以恒定速度骑行 20 秒，最后匀减速在 4 秒内停止。
First, find the maximum speed. Using v = u + at, with u=0, a=0.5, t=8: v = 0 + 0.5 × 8 = 4 m/s.
首先求最大速度。利用 v = u + at，其中 u=0，a=0.5，t=8：v = 0 + 0.5 × 8 = 4 m/s。
Next, calculate the total displacement using the area under the velocity-time graph. The graph consists of a triangle (first 8 s), a rectangle (next 20 s), and a triangle (last 4 s).
接下来，利用速度-时间图下方面积计算总位移。该图形由一个三角形（前 8 秒）、一个矩形（中间 20 秒）和一个三角形（后 4 秒）组成。
Triangle 1 area = ½ × 8 × 4 = 16 m. Rectangle area = 20 × 4 = 80 m. Triangle 2 area = ½ × 4 × 4 = 8 m. Total displacement = 16 + 80 + 8 = 104 m.
三角形 1 面积 = ½ × 8 × 4 = 16 m。矩形面积 = 20 × 4 = 80 m。三角形 2 面积 = ½ × 4 × 4 = 8 m。总位移 = 16 + 80 + 8 = 104 m。
The average speed during the whole journey is total distance ÷ total time = 104 ÷ (8+20+4) = 104 ÷ 32 = 3.25 m/s.
整个旅程的平均速度 = 总路程 ÷ 总时间 = 104 ÷ (8+20+4) = 104 ÷ 32 = 3.25 m/s。
8. Practice Tips and Exam Strategy | 练习技巧与考试策略
When tackling graph and kinematics questions, always begin by labelling axes and identifying known values. Write down the given data and the symbol you need to find. Choose the correct equation or area method.
在解答图像与运动学题目时，首先标注坐标轴并识别已知量。写下给定数据以及需要求得的符号。选择正确的方程或面积方法。
Draw a sketch of the graph if it is not supplied. This helps you visualise the motion and break it into segments. Mark each segment with its time interval and velocity or distance at the boundaries.
若题目未给出图形，请自行绘制草图。这有助于你直观理解运动并分段处理。在每段标记时间区间以及边界处的速度或距离。
In exam conditions, check your answer for reasonableness. A displacement cannot be negative unless the direction changes. A velocity graph that goes below the time axis indicates the object is moving backward.
在考试条件下，检查答案是否合理。除非方向改变，位移不可能为负。速度图若低于时间轴，表示物体在反向运动。
Practice with past papers and the G-K workbook exercises repeatedly. The more graphs you analyse, the faster you will recognise patterns and choose efficient methods.
反复练习历年真题和 G-K 练习册习题。你分析的图形越多，就越能快速识别规律并选择高效的方法。
9. Advanced: Non-Uniform Acceleration | 进阶：非匀加速运动
Sometimes the velocity-time graph is a curve, meaning acceleration is not constant. Then the gradient at a point gives instantaneous acceleration. You can estimate the gradient by drawing a tangent and calculating its slope.
有时速度-时间图是曲线，表示加速度不恒定。此时，某一点的斜率给出瞬时加速度。你可以通过画切线并计算其斜率来估算。
The area under a curved velocity-time graph can be estimated using the trapezium rule. This is a standard IGCSE technique for numerical integration.
曲线速度-时间图下方的面积可以使用梯形法则估算。这是 IGCSE 中数值积分的一种标准技巧。
The G-K Workbook 100 may include a non-uniform section to test your understanding of limits and tangents. Read the question carefully to see whether you need an exact value or an approximation.
G-K 练习册第 100 题可能包含非匀速部分，以测试你对切线和极限的理解。仔细读题，判断题目需要精确值还是近似值。
10. Linking Graphs to Real-World Problems | 将图像与实际应用联系
Kinematic graphs are not just abstract exercises. They model real-world phenomena such as free-falling objects, vehicle motion, and even population growth when modified. Understanding the graph shapes helps you reason about physical events.
运动学图形不仅仅是抽象练习。它们模拟真实世界现象，例如自由落体、车辆运动，甚至改进后的人口增长模型。理解图形形状有助于你推理物理事件。
For example, a constant negative acceleration appears as a straight line with negative slope on a velocity-time graph. On a distance-time graph, the same motion appears as a downward-opening parabola.
例如，恒定的负加速度在速度-时间图上表现为一条斜率为负的直线。在距离-时间图上，同样的运动表现为一条开口向下的抛物线。
In the G-K workbook 100 context, you might be asked to sketch a graph from a written description. Always start from rest or a given initial condition, then apply each phase in order.
在 G-K 练习册第 100 题的背景下，你可能会被要求根据文字描述绘制图形。始终从静止或给定初始状态开始，然后按顺序应用每个阶段。
11. Revision Checklist | 复习清单
Before the exam, verify that you can complete each of the following tasks:
考试前，请确认你能够完成以下每一项任务：
- Calculate gradient of a straight line | 计算直线斜率
- Draw a tangent to a curve and estimate its gradient | 画曲线切线并估算斜率
- Convert between distance-time and velocity-time graphs | 在距离-时间图和速度-时间图之间转换
- Find displacement from a velocity-time graph using area | 利用面积从速度-时间图求位移
- Apply the four kinematic equations correctly | 正确应用四个运动学方程
- Identify acceleration/deceleration from graph slope | 根据图形斜率判断加速/减速
Use this checklist before moving on to mixed practice questions. The G-K workbook 100 itself is a good diagnostic test of all these skills.
在进入混合练习前使用此清单。G-K 练习册第 100 题本身就是检验这些技能的良好诊断题。
12. Conclusion | 结论
Graphs and kinematics are a core topic in IGCSE Mathematics. From distance-time graphs to velocity-time graphs and kinematic equations, each element connects to form a complete picture of motion. The G-K workbook 100 serves as an excellent example of how these ideas merge in a single challenging problem.
图像与运动学是 IGCSE 数学的核心主题。从距离-时间图到速度-时间图，再到运动学方程，每个要素相互联系，构成完整的运动图景。G-K 练习册第 100 题很好地示范了这些概念如何融合在一道有挑战性的题目中。
We hope this breakdown helps you master the topic with confidence. For more revision resources and exam tips, stay with TutorHao.
希望这次解析能帮助你自信地掌握这一主题。如需更多复习资源和考试技巧，请持续关注 TutorHao。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导