IGCSE CCEA Mathematics: Kinematics – Key Points Revision | IGCSE CCEA 数学：运动学考点精讲

📚 IGCSE CCEA Mathematics: Kinematics – Key Points Revision | IGCSE CCEA 数学：运动学考点精讲

In IGCSE CCEA Mathematics, kinematics is the study of motion without considering its causes. You will need to understand key concepts such as speed, velocity, acceleration, distance-time graphs, and the equations of motion for constant acceleration. Mastering these topics is essential for both the non-calculator and calculator papers. This article breaks down the most important points and provides clear explanations in both English and Chinese to support your revision.

在 IGCSE CCEA 数学中，运动学研究物体运动而不涉及引起运动的原因。你需要理解速率、速度、加速度、距离-时间图以及匀加速运动方程等核心概念。掌握这些知识对于非计算器和计算器试卷都非常重要。本文以中英双语解析重要考点，助你高效复习。

1. Scalars and Vectors | 标量与矢量

In kinematics, quantities are classified as scalars or vectors. A scalar has magnitude only, such as distance and speed. A vector has both magnitude and direction, such as displacement and velocity.

在运动学中，物理量分为标量和矢量。标量只有大小，例如距离和速率；矢量既有大小又有方向，例如位移和速度。

Understanding the difference is crucial because many problems require vector treatment, especially when dealing with direction changes. For instance, if an object moves forward 10 m and then back 4 m, the total distance is 14 m, but the displacement is only 6 m in the forward direction.

理解其区别至关重要，许多问题都需要做矢量处理，尤其是涉及方向变化时。例如，一个物体向前运动 10 m，然后后退 4 m，总距离是 14 m，但位移只有向前 6 m。

2. Speed and Velocity | 速率与速度

Speed is the rate at which an object covers distance. It is a scalar and is always positive. Average speed is calculated as total distance divided by total time.

速率是物体移动距离的快慢，是标量，总是正值。平均速率等于总距离除以总时间。

average speed = total distance / total time

Velocity is the rate of change of displacement. It is a vector and can be positive or negative depending on direction. Average velocity is displacement divided by time.

速度是位移变化的快慢，是矢量，根据方向可取正或负。平均速度等于位移除以时间。

average velocity = displacement / time

In exam questions, be careful to distinguish whether they ask for speed or velocity, as this affects whether you use distance or displacement.

在考试题目中，务必区分题目问的是速率还是速度，这决定了你是用距离还是位移。

3. Acceleration | 加速度

Acceleration is the rate of change of velocity. It is a vector quantity. If an object’s velocity increases, acceleration is positive in that direction; if it decreases, the acceleration is negative (deceleration).

加速度是速度变化的快慢，是矢量。若物体速度增加，则沿该方向加速度为正；若速度减小，加速度为负（减速）。

a = (v − u) / t

where u is initial velocity, v is final velocity and t is the time taken. The unit of acceleration is metres per second squared (m s⁻²).

其中 u 为初速度，v 为末速度，t 为所用时间。加速度的单位是米每二次方秒 (m s⁻²)。

Remember that a negative acceleration does not always mean slowing down — it depends on the direction of motion. If velocity is negative and acceleration is also negative, the object speeds up in the negative direction.

注意，负加速度不总意味着减速——这取决于运动的方向。如果速度为负，加速度也为负，那么物体沿负方向加速。

4. Distance-Time Graphs | 距离-时间图

A distance-time graph shows how distance changes over time. Time is on the x-axis and distance on the y-axis. The gradient of the graph gives the speed.

距离-时间图展示距离随时间的变化。横轴为时间，纵轴为距离。图形的斜率表示速率。

A straight line sloping upwards indicates constant speed. A horizontal line means the object is stationary. A curved line indicates changing speed (acceleration or deceleration).

向上倾斜的直线表示匀速率运动；水平线表示物体静止；曲线则表示速率在变化（加速或减速）。

speed = gradient = (change in distance) / (change in time)

In CCEA questions, you may be asked to calculate speed from a tangent on a curved graph, or to describe the motion shown by the graph in words.

在 CCEA 考题中，可能会要求你通过曲线上某点的切线求速率，或用语言描述图形展示的运动。

5. Speed-Time Graphs | 速度-时间图

A speed-time graph plots speed against time. The gradient of the line gives the acceleration. A horizontal line means constant speed, a sloping line indicates acceleration or deceleration, and a curved line shows changing acceleration.

速度-时间图以速度为纵轴、时间为横轴。线的斜率表示加速度。水平线表示匀速，倾斜线表示加速或减速，曲线表示加速度在变化。

acceleration = gradient = (v − u) / t

If the graph slopes downwards, acceleration is negative, meaning the object is decelerating or accelerating in the opposite direction.

若图形向下倾斜，加速度为负，意味着物体在减速或朝相反方向加速。

Always check the units on the axes — some graphs use velocity rather than speed, so the sign may indicate direction. With speed-time graphs, speed is always positive.

务必注意坐标轴单位——有些图用的是速度（velocity）而不是速率，符号表明方向。而速度-时间图（speed-time graph）的速率总是正的。

6. Area Under a Speed-Time Graph | 速度-时间图下的面积

The area between the line and the time axis on a speed-time graph represents the distance travelled. If the graph drops below the axis (rare in speed-time graphs, but possible in velocity-time graphs) the area counts as negative displacement, so treat with care.

速度-时间图中，线与时间轴之间的面积代表运动所经过的距离。如果图形出现在时间轴下方（速度-时间图少见，但速度矢量图可能出现），该面积算作负位移，需小心处理。

To find the total distance, you can split the area into simple shapes such as rectangles and triangles. Add the areas together, respecting the scale of the axes.

求总距离时，可将面积分成矩形和三角形等简单图形，分别计算面积再求和，注意坐标轴的比例尺。

distance = area under the speed-time graph

This is a very common exam question. You may also need to find the distance travelled in a given time interval, or calculate the average speed from the total area.

这是非常常见的考题。你也可能需要求某段时间内运动的距离，或根据总面积计算平均速率。

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

When acceleration is constant, there are five key variables linking displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). The four equations of motion, often called the SUVAT equations, are:

当加速度恒定时，有五个关键变量：位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。四个运动方程，通常称为 SUVAT 方程，分别是：

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

All of these equations apply only when acceleration is uniform. You must choose the equation that uses the variables you know and the one you need to find.

所有方程仅适用于匀加速运动。你需要选择含有已知量和待求量的方程。

8. Applying SUVAT – Worked Examples | 应用 SUVAT 解题示例

Let’s look at a typical IGCSE CCEA problem. A car accelerates uniformly from rest and reaches a velocity of 20 m s⁻¹ in 8 seconds. Find the acceleration and the distance travelled.

我们来看一道典型的 IGCSE CCEA 例题。一辆汽车从静止开始匀加速，8 秒后速度达到 20 m s⁻¹。求加速度和行驶距离。

Solution: Given u = 0, v = 20, t = 8. First, find acceleration using v = u + at:

解：已知 u = 0, v = 20, t = 8。首先用 v = u + at 求加速度：

20 = 0 + a × 8 ⇒ a = 20 ÷ 8 = 2.5 m s⁻²

Then find distance using s = ut + ½at²:

然后用 s = ut + ½at² 求距离：

s = 0 × 8 + ½ × 2.5 × 8² = ½ × 2.5 × 64 = 80 m

Always check the units and ensure you have substituted correctly. It is helpful to list the known quantities first.

务必检查单位，确保代入无误。列出已知量是个好习惯。

Another example: A cyclist travelling at 12 m s⁻¹ brakes with constant deceleration and stops after covering 36 m. Find the deceleration.

另一个例子：一名自行车手以 12 m s⁻¹ 的速度行驶，刹车后匀减速，滑行 36 m 后停下。求减速度。

Here u = 12, v = 0, s = 36. Use v² = u² + 2as:

这里 u = 12, v = 0, s = 36。使用 v² = u² + 2as：

0² = 12² + 2 × a × 36 ⇒ 0 = 144 + 72a ⇒ a = −2 m s⁻²

The negative sign indicates deceleration.

负号表示减速。

9. Free Fall and Acceleration Due to Gravity | 自由落体与重力加速度

Near the Earth’s surface, all objects fall with the same constant acceleration due to gravity, denoted by g. This is approximately 9.8 m s⁻², though some CCEA questions may use 10 m s⁻² for simplicity.

在地球表面附近，所有物体都以相同的恒定加速度下落，称为重力加速度，记作 g。其值约为 9.8 m s⁻²，不过部分 CCEA 题目可能简化取 10 m s⁻²。

When applying SUVAT equations to vertical motion, you often choose upward as the positive direction. In that case, acceleration becomes a = −g (since gravity acts downwards). If you drop an object from rest, u = 0.

用 SUVAT 方程处理竖直运动时，常选取向上为正方向。此时加速度 a = −g（因为重力向下）。如果从静止释放物体，u = 0。

Example: A stone is dropped from a cliff. How far does it fall in 3 seconds? (Take g = 9.8 m s⁻²).

例题：一块石头从悬崖掉落。3 秒内它下落多远？（取 g = 9.8 m s⁻²）。

Solution: u = 0, a = g = 9.8, t = 3. Displacement s will be downward, but we can calculate distance as a positive value using s = ut + ½at²:

解：u = 0, a = g = 9.8, t = 3。位移向下，我们可以用 s = ut + ½at² 计算距离（取正值）：

s = 0 + ½ × 9.8 × 3² = 44.1 m

Be careful with sign conventions in vertical motion problems.

解竖直运动问题时要注意符号约定。

10. Interpreting Kinematics Graphs in Context | 在实际情境中解读运动学图形

CCEA often places kinematics graphs in real-world settings, such as journeys of cars, trains or athletes. You need to interpret what each section of the graph means.

CCEA 常将运动学图形置于真实场景中，如汽车、火车或运动员的行程。你需要解释图形各部分所代表的意义。

For a distance-time graph: a steep gradient indicates a fast speed; a flat section shows the object has stopped. A curve becoming steeper means acceleration, while a curve levelling off indicates deceleration.

对于距离-时间图：陡峭的斜率表示高速；平坦段表示物体停止。曲线越来越陡说明加速，曲线趋于平缓说明减速。

For a speed-time graph: the slope indicates how quickly speed changes. A line with negative slope shows deceleration. The area under the graph gives distance.

对于速度-时间图：斜率表明速度变化的快慢。负斜率的线表示减速。图形下的面积表示距离。

Sometimes you must calculate average speed for a whole journey by dividing total distance (total area under the speed-time graph) by total time.

有时你需要计算整个行程的平均速率，用总距离（速度-时间图下的总面积）除以总时间。

Make sure you can sketch graphs from a description of motion and describe motion from a given graph. These skills are regularly tested.

务必做到能根据运动描述画草图，并根据给定的图形描述运动。这些技能经常被考查。

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IGCSE CCEA Mathematics: Kinematics – Key Points Revision | IGCSE CCEA 数学：运动学 考点精讲