Graphs and Kinematics: IGCSE Mathematics Teacher's Guide | 图形与运动学：IGCSE数学教师用书

📚 Graphs and Kinematics: IGCSE Mathematics Teacher’s Guide | 图形与运动学：IGCSE数学教师用书

In IGCSE Mathematics, kinematics is the study of motion. Graphs provide a visual way to describe how distance and speed change over time. A distance-time graph shows how far an object has travelled from a fixed starting point at each instant.

在IGCSE数学中，运动学是研究物体运动的学科。图线提供了一种直观的方式来描述距离和速度随时间的变化。路程-时间图显示物体从固定起点出发，在每个时刻已经走过的距离。

1. Introduction to Distance-Time Graphs | 路程-时间图入门

On a distance-time graph, time is plotted on the horizontal axis and distance on the vertical axis. Every point on the graph gives the distance travelled from the start at a particular moment. A straight horizontal line means the object is stationary; a straight sloped line means the object is moving at constant speed.

在路程-时间图中，横轴表示时间，纵轴表示路程。图上的每一个点都表示在某一特定时刻从起点出发走过的距离。水平直线表示物体静止；倾斜直线表示物体做匀速运动。

Gradient = speed | 梯度（斜率）= 速度
Steeper line → faster motion | 图线越陡 → 运动越快
Flat line → object at rest | 水平线 → 物体静止

2. Speed-Time Graphs | 速度-时间图

A speed-time graph shows how speed varies with time. The gradient of a speed-time graph represents acceleration. A horizontal line means constant speed; a line with positive gradient means acceleration; a line with negative gradient means deceleration. The area between the graph and the time axis gives the distance travelled.

速度-时间图显示速度随时间的变化。速度-时间图上的梯度表示加速度。水平线表示匀速；具有正梯度的直线表示加速；具有负梯度的直线表示减速。图线与时间轴所围成的面积表示行进的距离。

Positive gradient → acceleration | 正梯度 → 加速
Negative gradient → deceleration | 负梯度 → 减速
Zero gradient → constant speed | 零梯度 → 匀速

3. Gradient and Area Under a Graph | 梯度和图线下的面积

For a distance-time graph, the gradient at any point equals the speed at that instant. For a speed-time graph, the area under the graph between two times t₁ and t₂ equals the distance travelled in that interval. This idea links algebra, geometry, and real-world motion.

对于路程-时间图，任意一点的梯度等于该时刻的速度。对于速度-时间图，图线下在t₁和t₂之间的面积等于该时间段内走过的距离。这一概念将代数、几何与真实运动联系在一起。

Speed = Gradient of distance-time graph × (scale factor) | 速度 = 路程-时间图的梯度 ×（比例因子）

Distance = Area under speed-time graph | 路程 = 速度-时间图下的面积

4. Using Tangents for Non-Linear Speed | 用切线求非匀速运动的速度

When the distance-time graph is curved, the speed changes continuously. To find the speed at a specific time, draw a tangent to the curve at that time and calculate its gradient. The area under a curved speed-time graph can be estimated by counting grid squares or by dividing the region into trapezia.

当路程-时间图为曲线时，速度是连续变化的。要计算某一时刻的速度，可在该时刻画出曲线的切线，并计算切线的梯度。曲线速度-时间图下的面积可以通过数方格或将区域分割成梯形来估算。

Draw tangent → gradient = instantaneous speed | 画切线 → 梯度 = 瞬时速度
Count squares → estimate area | 数方格 → 估算面积

5. Constant Acceleration Formulae (SUVAT) | 匀加速运动公式（SUVAT）

When acceleration is constant, the SUVAT equations provide powerful tools for solving kinematic problems. Here s represents displacement, u is initial velocity, v is final velocity, a is acceleration, and t is time. The equations are valid only when acceleration is uniform.

当加速度恒定时，SUVAT方程组为求解运动学问题提供了有力工具。其中s表示位移，u是初速度，v是末速度，a是加速度，t是时间。这些公式仅在匀加速条件下成立。

v = u + at
s = ((u + v)/2) × t
s = ut + ½at²
v² = u² + 2as
s = vt − ½at²

6. Interpreting Real-World Graphs | 解读现实世界中的图线

Real journeys often involve multiple stages: starting, speeding up, travelling at constant speed, slowing down, and stopping. Teach students to break the graph into distinct intervals and analyse each part separately. For a full journey, the total distance is the sum of the distances for each stage.

实际旅程通常包含多个阶段：启动、加速、匀速行驶、减速和停止。教导学生将图线划分为不同的区间，并分别分析每个部分。对于完整旅程，总路程是各阶段路程之和。

Identify each stage | 识别每个阶段
Calculate gradients or areas per stage | 分别计算每个阶段的梯度或面积
Combine results for the whole journey | 汇总得到全程结果

7. Displacement vs Distance | 位移与路程的区别

Distance is a scalar quantity that measures the total length of the path travelled, without regard to direction. Displacement is a vector quantity that describes the overall change in position from start to finish. In one-dimensional motion, a negative displacement means the object has moved in the opposite direction.

路程是标量，只量度路径的总长度，不考虑方向。位移是矢量，描述从起点到终点的总体位置变化。在一维运动里，负位移表示物体沿相反方向运动。

Quantity \| 物理量	Type \| 类型	Example \| 示例
Distance \| 路程	Scalar \| 标量	5 km along a winding road \| 沿弯曲道路走5 km
Displacement \| 位移	Vector \| 矢量	3 km east of start \| 起点以东3 km

8. Average Speed and Average Velocity | 平均速度与平均速率

Average speed is the total distance divided by the total time. Average velocity is the total displacement divided by the total time. The magnitude of average velocity is usually less than or equal to average speed, because displacement cannot exceed distance.

平均速率等于总路程除以总时间；平均速度（矢量）等于总位移除以总时间。平均速度的大小通常小于或等于平均速率，因为位移不可能超过路程。

Average speed = Total distance ÷ Total time | 平均速率 = 总路程 ÷ 总时间

Average velocity = Total displacement ÷ Total time | 平均速度 = 总位移 ÷ 总时间

9. Worked Example 1: Distance-Time Graph | 例题1：路程-时间图

A cyclist travels 15 km in the first hour, rests for 30 minutes, then travels 5 km in the next 15 minutes. Find the average speed for the whole journey.

一名骑车人第一个小时骑行15 km，休息30分钟，随后15分钟又骑行5 km。求全程的平均速率。

Total distance = 15 + 5 = 20 km
总路程 = 15 + 5 = 20 km
Total time = 1 + 0.5 + 0.25 = 1.75 h
总时间 = 1 + 0.5 + 0.25 = 1.75 h
Average speed = 20 ÷ 1.75 ≈ 11.4 km/h
平均速率 = 20 ÷ 1.75 ≈ 11.4 km/h

10. Worked Example 2: Speed-Time Graph and Total Distance | 例题2：速度-时间图与总路程

A car accelerates uniformly from rest to 20 m/s in 10 s, maintains 20 m/s for 20 s, then brakes uniformly to rest in 5 s. Calculate the total distance travelled.

汽车从静止开始匀加速，在10 s内达到20 m/s，保持20 m/s匀速行驶20 s，然后匀减速，在5 s内停下来。计算总路程。

First triangle | 第一个三角形: ½ × 10 × 20 = 100 m
Rectangle | 长方形: 20 × 20 = 400 m
Second triangle | 第二个三角形: ½ × 5 × 20 = 50 m
Total distance | 总路程 = 100 + 400 + 50 = 550 m

Total distance = ½ × 10 × 20 + 20 × 20 + ½ × 5 × 20 = 550 m

11. Common Misconceptions and Teaching Tips | 常见误区与教学建议

Students often confuse distance with displacement, forget to convert units, or take the gradient from the wrong scale. They may also think a curved distance-time graph means changing direction. Use real motion demonstrations, interactive graphs, and small-step questioning to correct these ideas.

学生常常混淆路程与位移，忘记进行单位换算，或者从错误的比例尺上读取梯度。他们还可能误以为弯曲的路程-时间图就表示物体改变方向。请使用真实运动演示、交互式图线以及小步递进的问题来纠正这些错误认识。

Always check units (km/h, m/s, etc.) | 始终检查单位（km/h、m/s 等）
Label axes and scales clearly | 清楚标注坐标轴与比例尺
Use technology to visualise graphs | 利用技术手段直观展示图线
Emphasise ‘area’ as multiplication of speed and time | 强调“面积”是速度与时间的乘积

12. Practice Questions for the Classroom | 课堂练习建议

These questions are suitable for in-class reinforcement or homework. Solutions may be found in the accompanying teacher resources on aleveler.com.

以下题目适合课堂巩固或课后作业。配套答案可在 aleveler.com 的教师资源中获取。

A train travels 120 km in 2 hours, stops for 30 minutes, then travels 60 km in 40 minutes. Draw the distance-time graph and find the average speed.
一辆火车2小时行驶120 km，停车30分钟，再以40分钟行驶60 km。画出路程-时间图并求平均速率。
A sprinter reaches a top speed of 10 m/s after 4 s of uniform acceleration. She maintains this speed for 6 s, then slows down to rest in 2 s. Find the total distance covered.
一名短跑选手经过4 s匀加速后达到最大速度10 m/s，保持该速度6 s，再用2 s匀减速停止。求总路程。
A car accelerates uniformly from 5 m/s to 25 m/s in 10 s. Calculate (a) the acceleration, (b) the distance travelled in this time.
汽车在10 s内从5 m/s匀加速到25 m/s。求 (a) 加速度；(b) 这段时间内行驶的路程。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Quantity \| 物理量	Type \| 类型	Example \| 示例
Distance \| 路程	Scalar \| 标量	5 km along a winding road \| 沿弯曲道路走5 km
Displacement \| 位移	Vector \| 矢量	3 km east of start \| 起点以东3 km

Graphs and Kinematics: IGCSE Mathematics Teacher’s Guide | 图形与运动学：IGCSE数学教师用书