IGCSE OCR Maths: Kinematics Key Points | IGCSE OCR 数学：运动学考点精讲

📚 IGCSE OCR Maths: Kinematics Key Points | IGCSE OCR 数学：运动学考点精讲

Kinematics is the branch of mathematics that describes the motion of objects using graphs, equations, and key quantities such as displacement, velocity, and acceleration. In the IGCSE OCR Mathematics syllabus, you are expected to interpret distance–time and speed–time graphs, calculate acceleration, and apply the constant acceleration (SUVAT) equations to solve real-world problems. This guide breaks down every essential topic, explains the core ideas clearly, and highlights common pitfalls to help you master kinematics for your exams.

运动学是用图形、方程和位移、速度、加速度等关键量来描述物体运动的一个数学分支。在 IGCSE OCR 数学大纲中，你需要能够解读距离-时间图和速度-时间图，计算加速度，并运用匀加速运动（SUVAT）方程解决实际问题。本指南拆解每一个核心知识点，清晰地解释基本概念，并指出常见错误，帮助你彻底掌握运动学，轻松应对考试。

1. Introduction to Kinematics | 运动学简介

Kinematics deals with quantities like displacement (distance in a given direction), speed, velocity, and acceleration, without considering the forces causing the motion. In IGCSE OCR Maths, motion is usually restricted to straight-line motion with constant acceleration. The main tools you will use are graphical interpretation and four SUVAT equations linking initial velocity u, final velocity v, acceleration a, displacement s, and time t.

运动学处理的是位移（具有方向的距离）、速率、速度和加速度等量，而不考虑引起运动的力。在 IGCSE OCR 数学中，运动通常限制为匀加速直线运动。你将使用的主要工具是图形解读和四个 SUVAT 方程，它们将初速度 u、末速度 v、加速度 a、位移 s 和时间 t 联系起来。

2. Distance–Time Graphs | 距离–时间图

A distance–time graph plots distance travelled against time. The gradient of the graph at any point gives the speed. A straight diagonal line indicates constant speed, while a horizontal line indicates the object is stationary (zero gradient). A curved line shows changing speed; the gradient at a particular instant (tangent slope) gives the instantaneous speed. The total distance moved is simply the final value on the vertical axis, since distance never decreases.

距离–时间图描绘了行驶距离随时间的变化。图中任意一点的斜率表示速率。一条倾斜的直线代表匀速运动，水平线代表物体静止（斜率为零）。曲线代表速率在变化；某时刻的斜率（切线斜率）表示瞬时速率。总移动距离就是纵轴的最终值，因为距离从不减少。

3. Speed and Velocity | 速率与速度

Speed is a scalar quantity that measures how fast an object moves, while velocity is a vector that includes direction. In IGCSE motion problems, speed can be calculated as distance ÷ time, and velocity as displacement ÷ time. The magnitude of velocity is speed, but if motion reverses direction, velocity changes sign. Always check whether the question asks for speed or velocity in graph problems involving areas or gradients.

速率是标量，衡量物体运动的快慢；而速度是矢量，包含方向。在 IGCSE 运动问题中，速率可按距离 ÷ 时间计算，速度按位移 ÷ 时间计算。速度的大小就是速率，但如果运动方向反转，速度会改变符号。在涉及面积或斜率的图形问题中，一定要检查题目问的是速率还是速度。

4. Velocity–Time Graphs | 速度–时间图

The velocity–time graph is the most powerful graphical tool in kinematics. Its gradient gives acceleration, and the area under the graph represents the displacement (or distance if only speed is plotted). A horizontal line means constant velocity (zero acceleration), a sloping straight line means constant acceleration, and a curved line indicates changing acceleration (though usually not examined at IGCSE). The total area between the graph and the time axis gives the displacement, taking area below the axis as negative.

速度–时间图是运动学中最强大的图形工具。它的斜率表示加速度，而图线与时间轴围成的面积代表位移（如果只画速率图，则面积代表距离）。水平线表示匀速（加速度为零），倾斜直线表示匀加速，曲线表示加速度在变化（但 IGCSE 通常不考查）。图线与时间轴之间的总面积等于位移，其中时间轴下方的面积取负值。

5. Acceleration | 加速度

Acceleration is the rate of change of velocity, calculated as a = (v – u) / t. Its unit is m/s². If velocity increases, acceleration is positive; if velocity decreases (deceleration), acceleration is negative. For a velocity–time graph, the acceleration between two points is the slope of the line connecting them. When dealing with constant acceleration, you can use any pair of velocity and time values to find a. Remember: negative acceleration means slowing down if the velocity is positive.

加速度是速度的变化率，计算式为 a = (v – u) / t，单位是 m/s²。如果速度增加，加速度为正；速度减小（减速）时，加速度为负。在速度–时间图中，两点之间的加速度是连接它们的直线的斜率。处理匀加速运动时，你可以用任意一组速度和时间值求 a。请记住：如果速度为正，负加速度意味着减速。

6. Constant Acceleration Equations (SUVAT) | 匀加速运动方程

When acceleration is constant, four equations relate the five variables s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). You must learn these by heart:

当加速度恒定时，有四个方程将 s（位移）、u（初速度）、v（末速度）、a（加速度）和 t（时间）五个变量联系起来。你必须熟记这些方程：

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

Each equation misses one of the five quantities, so choose the one that does not involve the quantity you don’t know and don’t need to find. Generally, list the known and unknown variables, then select the matching SUVAT equation. Always substitute values with correct units and check that signs (especially for deceleration) are consistent.

每个方程都缺少五个量中的一个，因此要选择那个不包含未知且不需要求的量的方程。一般做法是列出已知量和未知量，然后选择匹配的 SUVAT 方程。始终以正确的单位代入数值，并确保符号（尤其是减速的负号）一致。

7. Using SUVAT: Worked Example | SUVAT 应用：典型例题

A car accelerates uniformly from rest (u = 0) at 3 m/s² for 5 seconds. Find the final velocity and the distance covered. First, for v: v = u + at = 0 + 3×5 = 15 m/s. Then, for s: s = ut + ½at² = 0×5 + ½×3×25 = 37.5 m. Alternatively, you could use s = ½(u+v)t = ½×(0+15)×5 = 37.5 m. Always double-check that the equation matches the given data and that you haven’t missed a conversion (like km/h to m/s).

一辆汽车从静止（u = 0）以 3 m/s² 匀加速行驶 5 秒钟。求末速度和行驶距离。首先求 v：v = u + at = 0 + 3×5 = 15 m/s。然后求 s：s = ut + ½at² = 0×5 + ½×3×25 = 37.5 m。你也可以用 s = ½(u+v)t = ½×(0+15)×5 = 37.5 m。务必再次检查所选方程与给定数据是否匹配，以及你是否遗漏了单位转换（例如 km/h 转 m/s）。

8. Acceleration Due to Gravity | 重力加速度

For objects falling freely near the Earth’s surface, the acceleration is constant and approximately 9.8 m/s² downwards (often rounded to 10 m/s² in exam questions). In SUVAT problems involving vertical motion, you can use the same equations, but you must take a = g (or –g depending on the sign convention). If an object is thrown upwards, its initial velocity is positive, acceleration due to gravity is negative, and at the highest point its velocity v = 0. Always define a positive direction clearly to avoid sign errors.

对于在地球表面附近自由下落的物体，加速度恒定，约为 9.8 m/s² 向下（考试中常取 10 m/s²）。在涉及竖直运动的 SUVAT 问题中，你可以使用相同的方程，但必须取 a = g（或根据正方向约定取 –g）。如果物体向上抛出，其初速度为正，重力加速度为负，在最高点速度 v = 0。一定要清晰地定义正方向，以免出现符号错误。

9. Interpreting Graphs: Area and Gradient | 图像解读：面积与斜率

Exam questions frequently feature velocity–time graphs requiring you to find acceleration (gradient) and distance (area). The area under a velocity–time graph can be found by counting squares or using geometry (rectangles, triangles, trapeziums). If the graph dips below the time axis, the object is moving in the opposite direction; the area below the axis represents negative displacement (or distance if just speed is considered). Remember that the gradient of a distance–time graph gives speed, while the gradient of a displacement–time graph gives velocity.

考试题目经常出现速度–时间图，要求你求加速度（斜率）和距离（面积）。速度–时间图下的面积可以用数方格或几何方法（矩形、三角形、梯形）计算。如果图线低于时间轴，说明物体在反向运动；轴下方的面积表示负位移（若只考虑速率则代表距离）。请记住，距离–时间图的斜率给出速率，而位移–时间图的斜率给出速度。

10. Common Mistakes | 常见错误

One frequent error is confusing distance–time and velocity–time graphs. In a distance–time graph, a straight sloping line means constant speed, not acceleration. In a velocity–time graph, the same shape means constant acceleration. Another mistake is using area to find speed or distance on a distance–time graph – that does not work. Also, forgetting to check if the question asks for speed or velocity can cost marks. When using SUVAT, mixing up positive and negative signs for acceleration and velocity is a common source of error, especially in vertical motion problems. Always write down the known quantities and the required one before choosing an equation.

一个常见错误是混淆距离–时间图和速度–时间图。在距离–时间图中，一条倾斜直线代表匀速，而不是加速；在速度–时间图中，同样的形状代表匀加速。另一个错误是试图在距离–时间图上用面积求速率或距离——这行不通。此外，忘记检查题目问的是速率还是速度可能导致失分。使用 SUVAT 时，将加速度和速度的正负号搞混是一个常见的错误源头，尤其是在竖直运动问题中。在选择方程之前，一定要先写下已知量和待求量。

11. Exam Tips | 考试技巧

Always show your working when calculating areas under graphs, and label areas clearly if you split shapes. Where SUVAT equations are needed, write down the equation in symbolic form before substituting values. If acceleration is constant, any convenient points on the graph may be used to find it. When finding the gradient of a curve, draw a tangent carefully and use a large triangle to improve accuracy. Double-check unit conversions: speeds are often given in km/h and need to be changed to m/s (÷3.6). For multi‑part questions, carry exact values through and only round your final answer.

在计算图下面积时，务必展示你的计算过程；如果你将图形拆分，要清楚地标示各部分面积。需要使用 SUVAT 方程时，先写出符号形式的方程，再代入数值。如果加速度恒定，图上的任意合适点都可用来求加速度。求曲线斜率时，仔细画出切线，并使用较大的三角形以提高准确性。再次检查单位换算：速度常以 km/h 给出，需要转换为 m/s（除以 3.6）。对于多步骤问题，中间步骤保留精确值，只在最终答案处四舍五入。

12. Summary and Practice | 总结与练习

Kinematics is a highly structured topic: master the graphs, memorise the four SUVAT equations, and practise linking numerical data with visual representations. When you practise past papers, focus on sketching quick diagrams to visualise motion, identifying positive directions, and systematically listing known variables. With consistent practice, you will be able to handle any kinematics question confidently. Remember, the key is interpreting the gradient and area under a velocity–time graph, and applying the correct SUVAT equation based on the known quantities.

运动学是一个结构非常清晰的主题：掌握图形，熟记四个 SUVAT 方程，并多加练习将数值数据与视觉图形结合起来。做往年真题时，要重点关注画出快速示意图来将运动可视化、确定正方向、系统地列出已知变量。通过持续练习，你将能够从容应对任何运动学问题。请记住，关键在于解读速度–时间图的斜率和面积，并根据已知量选择正确的 SUVAT 方程。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE OCR Maths: Kinematics Key Points | IGCSE OCR 数学：运动学考点精讲