Kinematics for IGCSE AQA Maths: Key Points Explained | IGCSE AQA 数学：运动学 考点精讲

📚 Kinematics for IGCSE AQA Maths: Key Points Explained | IGCSE AQA 数学：运动学 考点精讲

Kinematics is the study of motion without considering the forces that cause it. In IGCSE AQA Mathematics, you are expected to interpret and analyse movement using graphs, equations, and real-life contexts. This article covers every essential concept, from constant speed to variable acceleration, ensuring you are fully prepared for your exam.

运动学是研究物体运动而不考虑引起该运动的力的学科。在 IGCSE AQA 数学中，你需要使用图表、方程和实际情境来解释和分析运动。本文涵盖从匀速运动到变加速度的每一个基本概念，确保你为考试做好充分准备。

1. What is Kinematics? | 什么是运动学？

Kinematics deals with quantities like distance, displacement, speed, velocity, and acceleration. In the AQA IGCSE Maths specification, you will mainly work with graphs of motion and formulas that link these quantities. The key difference between a vector and a scalar is often tested: distance is a scalar (magnitude only), while displacement is a vector (magnitude and direction). Similarly, speed is scalar and velocity is vector.

运动学处理距离、位移、速率、速度和加速度等物理量。在 AQA IGCSE 数学大纲中，你主要处理运动图形以及连接这些量的公式。矢量与标量的关键区别经常被考查：距离是标量（仅有大小），而位移是矢量（大小和方向）。类似地，速率是标量，速度是矢量。

2. The Speed, Distance, Time Relationship | 速度、距离与时间的关系

For constant speed motion, the fundamental formula is speed = distance ÷ time. This is often rearranged depending on the unknown: distance = speed × time, or time = distance ÷ speed. Make sure your units are consistent; for example, if speed is in m/s, time must be in seconds and distance in metres.

对于匀速运动，基本公式是 速度 = 距离 ÷ 时间。通常可以根据未知量重新排列：距离 = 速度 × 时间，或 时间 = 距离 ÷ 速度。确保单位一致；例如，如果速度单位是 m/s，时间单位必须是秒，距离单位必须是米。

v = d ÷ t

This triangle method helps you remember the three variations quickly. Cover the quantity you want to find, and the remaining two show the operation: d = v × t, v = d ÷ t, t = d ÷ v.

这个三角形方法帮助你快速记住三种变形。遮住你想要找的量，剩下的两个就表示运算：d = v × t，v = d ÷ t，t = d ÷ v。

3. Constructing a Distance-Time Graph | 绘制距离-时间图

A distance-time graph shows how an object’s distance changes over time. The horizontal axis represents time, and the vertical axis represents distance from a starting point. A straight, sloping line indicates constant speed. A horizontal line means the object is stationary. A curve suggests acceleration or deceleration.

距离-时间图显示物体距离如何随时间变化。横轴代表时间，纵轴代表距起点的距离。倾斜的直线表示匀速运动。水平线表示物体静止。曲线表示加速或减速。

Key features to look for: the steeper the line, the greater the speed. If the line curves upward, the speed is increasing (accelerating). If it curves downward, the object is decelerating.

要寻找的关键特征：线越陡，速度越大。如果线向上弯曲，速度在增加（加速）。如果向下弯曲，物体在减速。

4. Interpreting Distance-Time Graphs | 解读距离-时间图

The gradient of a distance-time graph gives the speed. To find the speed at a specific time on a curved graph, you must draw a tangent at that point and calculate its gradient. This gradient represents the instantaneous speed. The average speed over a time interval is simply the total distance travelled divided by the total time taken.

距离-时间图的斜率给出速度。要在曲线上找到某一特定时刻的速度，必须在该点作切线，并计算其斜率。该斜率代表瞬时速率。一段时间间隔内的平均速率就是总移动距离除以总用时。

Example: if an object travels 40 m in 5 seconds at constant speed, then rests for 10 seconds, the graph shows a rise, then a plateau. During the rest period, gradient = 0, speed = 0.

例题：如果一个物体以恒定速度在 5 秒内移动 40 米，然后静止 10 秒，图形先上升，然后平坦。休息期间，斜率为 0，速度为 0。

5. Velocity and Acceleration | 速度与加速度

Acceleration measures how quickly velocity changes. The formula is: acceleration = change in velocity ÷ time. In IGCSE Maths, we often use ‘speed’ instead of ‘velocity’ for simplicity, but the concept is similar when motion is in a straight line. The standard equation is a = (v – u) ÷ t, where u is initial velocity, v is final velocity, and t is time.

加速度衡量速度变化的快慢。公式为：加速度 = 速度变化量 ÷ 时间。在 IGCSE 数学中，为简单起见，我们常用“速率”代替“速度”，但当运动沿直线时，概念类似。标准方程为 a = (v – u) ÷ t，其中 u 为初速度，v 为末速度，t 为时间。

a = (v − u) ÷ t

Acceleration can be positive (speeding up) or negative (slowing down, often called deceleration). The unit of acceleration is typically m/s².

加速度可为正（加速）或负（减速，常称为减速度）。加速度的单位通常是 米/秒²（m/s²）。

6. Velocity-Time Graphs | 速度-时间图

A velocity-time graph plots velocity on the vertical axis and time on the horizontal axis. Its gradient gives the acceleration. A straight sloping line indicates constant acceleration; a horizontal line shows constant velocity. The area under the graph between two time values gives the distance travelled during that interval.

速度-时间图将速度标在纵轴，时间标在横轴。其斜率给出加速度。倾斜直线表示匀加速；水平线表示匀速。两个时间值之间的图形下方的面积代表该时间段内行驶的距离。

It is essential to distinguish between distance and displacement when the velocity becomes negative. In a velocity-time graph, area below the time axis represents motion in the opposite direction and reduces displacement, but for total distance travelled, you take absolute areas.

当速度为负时，区分距离和位移很重要。在速度-时间图中，时间轴下方的面积代表相反方向的运动，会减少位移，但对于总行驶距离，需要取面积的绝对值。

7. Finding Distance from a Velocity-Time Graph | 从速度-时间图求距离

To calculate the distance travelled, you split the area under the graph into simple shapes such as rectangles, triangles, and trapeziums. Use area formulas: area of rectangle = base × height; area of triangle = ½ × base × height; area of trapezium = ½ × (sum of parallel sides) × height. Sum the areas, remembering to treat any negative velocity areas as positive if you need total distance.

要计算行驶距离，将图形下方面积分解为简单形状，如矩形、三角形和梯形。使用面积公式：矩形面积 = 底 × 高；三角形面积 = ½ × 底 × 高；梯形面积 = ½ × (平行边之和) × 高。对面积求和，注意如果需要总距离，应将任何负速度区域视为正面积。

Example: a car accelerates from rest to 20 m/s in 10 s, maintains that speed for 30 s, then decelerates to rest in 20 s. The graph forms a trapezium and a triangle (or a combination). Total distance = area of first triangle (½×10×20) + rectangle (30×20) + area of second triangle (½×20×20) = 100 + 600 + 200 = 900 m.

例题：一辆车从静止加速到 20 m/s 用时 10 秒，保持该速度 30 秒，然后在 20 秒内减速至静止。图形由一个梯形和一个三角形（或组合）构成。总距离 = 第一个三角形面积 (½×10×20) + 矩形面积 (30×20) + 第二个三角形面积 (½×20×20) = 100 + 600 + 200 = 900 米。

8. Acceleration from Velocity-Time Graphs | 从速度-时间图求加速度

The gradient of a velocity-time graph is calculated as rise ÷ run, which gives acceleration. For straight line segments, pick two points on the line and use a = (v₂ − v₁) ÷ (t₂ − t₁). A positive gradient means acceleration, a negative gradient means deceleration. Zero gradient means constant velocity.

速度-时间图的梯度用 纵轴变化 ÷ 横轴变化 计算，得到加速度。对于直线段，在线上取两点，用 a = (v₂ − v₁) ÷ (t₂ − t₁) 计算。正梯度表示加速，负梯度表示减速。零梯度表示匀速。

If the graph is curved, you must draw a tangent to find instantaneous acceleration, just as with distance-time graphs for speed. This is a common higher-tier skill.

如果图形是弯曲的，必须作切线来求瞬时加速度，这与在距离-时间图中求速度的方法类似。这是较高层级考卷中常见的技能。

9. Variable Acceleration and Tangents | 变加速度与切线

When acceleration is not constant, the velocity-time graph is a curve. To find the acceleration at a specific time, carefully draw a tangent at the point and find its gradient. Use a ruler to draw the tangent, extend it to form a large triangle, and calculate the gradient accurately. This gives the instantaneous acceleration.

当加速度不恒定时，速度-时间图是一条曲线。要找到某一特定时刻的加速度，在该点仔细作切线，并求其斜率。用直尺画切线，将其延伸形成一个大三角形，准确计算斜率。这就得到了瞬时加速度。

Likewise, for distance-time curves, the tangent’s gradient gives instantaneous speed. Practice these skills with different curves to ensure you can estimate gradients with precision.

同样，对于距离-时间曲线，切线的斜率给出瞬时速率。用不同的曲线练习这些技能，以确保你能精确估计斜率。

10. Common Graphs Comparison | 常见图形对比

It is vital not to confuse distance-time graphs with velocity-time graphs. The table below summarises the differences in interpretation. In an exam, always check the labels on the axes before answering.

不要混淆距离-时间图和速度-时间图，这一点至关重要。下表总结了理解上的差异。考试时，在回答问题之前务必检查坐标轴的标签。

Graph Type	Gradient gives	Area under graph	Horizontal line means
Distance-Time	Speed	Not meaningful (unless special cases)	Stationary
Velocity-Time	Acceleration	Distance travelled	Constant velocity

11. Typical Exam Question Walkthrough | 典型考题演练

A common exam problem: ‘The velocity-time graph of a car consists of three straight line segments: accelerates uniformly from 0 to 15 m/s in 5 s, travels at constant velocity for 12 s, then decelerates uniformly to rest in 8 s. (a) Find the acceleration during the first 5 s. (b) Find the total distance travelled. (c) Sketch the corresponding distance-time graph.’

常见考题：“一辆车的速度-时间图由三段直线段组成：0 至 5 秒匀加速至 15 m/s，匀速行驶 12 秒，然后 8 秒内匀减速至静止。(a) 求前 5 秒的加速度。(b) 求总行驶距离。(c) 画出相应的距离-时间图。”

Solution: (a) a = (15 − 0) ÷ 5 = 3 m/s². (b) Distance = area of triangle (½×5×15) + rectangle (12×15) + area of second triangle (½×8×15) = 37.5 + 180 + 60 = 277.5 m. (c) The distance-time graph during acceleration is a curve with increasing gradient, then a straight line during constant speed, and finally a curve with decreasing gradient until flat.

解答：(a) a = (15 − 0) ÷ 5 = 3 m/s²。(b) 距离 = 三角形面积 (½×5×15) + 矩形面积 (12×15) + 第二个三角形面积 (½×8×15) = 37.5 + 180 + 60 = 277.5 m。(c) 加速段的距离-时间图是斜率递增的曲线，匀速段是直线，最后减速段是斜率递减的曲线直至水平。

12. Top Tips for Exam Success | 考试成功的重要提示

Always double-check the units given in the question. Convert minutes to seconds, km to m when necessary. For tangent questions, use a sharp pencil and a ruler, and make your triangle as large as possible to reduce errors. Show all your working clearly; even if the final answer is wrong, you can earn method marks.

务必再次检查题目给出的单位。必要时将分钟转换为秒，公里转换为米。对于切线问题，使用削尖的铅笔和直尺，并让三角形尽可能大以减少误差。清晰地展示所有解题步骤；即使最终答案错误，也能获得过程分。

Memorise the formula triangle for speed, distance, time, and the equation for acceleration. Understand the graph interpretations in the table above. Practice with past papers to become familiar with how AQA phrases its kinematics questions.

记住速度、距离、时间的公式三角形和加速度方程。理解上表中的图形解读。通过真题练习熟悉 AQA 命题人如何表述运动学问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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