IGCSE CCEA Maths: Mechanics Key Points | IGCSE CCEA 数学：力学考点精讲

📚 IGCSE CCEA Maths: Mechanics Key Points | IGCSE CCEA 数学：力学考点精讲

This article covers the essential mechanics topics in the IGCSE CCEA Mathematics specification. You will find clear explanations, key formulas, and practical tips to help you master motion, forces, momentum, and moments. Each concept is presented in both English and Chinese to support bilingual learning.

本文涵盖 IGCSE CCEA 数学大纲中的力学核心考点，提供清晰的解释、关键公式和实用技巧，帮助你掌握运动、力、动量和力矩。每个概念均以中英双语呈现，辅助学习。

1. Scalars and Vectors | 标量与向量

In mechanics, quantities are divided into scalars and vectors. A scalar has only magnitude (size), while a vector has both magnitude and direction. Examples of scalars include speed, distance, mass, and time. Vectors include displacement, velocity, acceleration, and force. When solving problems, always note whether direction matters.

力学中的物理量分为标量与向量。标量仅有大小，而向量既有大小又有方向。标量的例子包括速率、路程、质量与时间。向量则包括位移、速度、加速度与力。解题时务必留意方向是否起作用。

Vectors can be represented by arrows, where the length shows magnitude and the arrowhead shows direction. Adding vectors requires considering their directions, either by tip-to-tail drawing or by resolving into components. For one-dimensional motion along a straight line, you can use positive and negative signs to indicate opposite directions, such as taking right as positive and left as negative.

向量可用箭头表示，长度代表大小，箭头代表方向。向量相加需考虑方向，可采用三角形法则或分解为分量。对于一维直线运动，可用正负号表示相反方向，例如取向右为正、向左为负。

2. SUVAT Equations | 匀加速运动方程

The equations of motion for constant acceleration in a straight line are often called the SUVAT equations. They link displacement s, initial velocity u, final velocity v, acceleration a, and time t. All five quantities are vectors, so in one-dimensional problems you must assign a positive direction.

匀加速直线运动的方程常被称为 SUVAT 方程，关联位移 s、初速度 u、末速度 v、加速度 a 与时间 t。这五个量都是向量，因此在一维问题中必须设定正方向。

The five key equations are:

五个关键方程如下：

Name	Equation	Missing quantity
First equation	v = u + at	s
Second equation	s = ut + ½ at²	v
Third equation	s = ½ (u + v) t	a
Fourth equation	v² = u² + 2as	t
Fifth equation	s = vt − ½ at²	u

You can identify which equation to use by listing the known quantities and the one you need. The equation that does not contain the unwanted quantity is the correct choice. Always check that units are consistent, for example converting km/h to m/s before substituting into the formulas.

你可以通过列出已知量和待求量来选择方程：不含多余量的那个方程即为正确选择。务必确保单位一致，例如题干给出 km/h 需先转换为 m/s 再代入公式。

3. Free Fall and Gravity | 自由落体与重力

When an object falls freely under gravity alone, it moves with a constant downward acceleration of g = 9.8 m s⁻² (on Earth, ignoring air resistance). This is a special case of uniform acceleration, so you can apply the SUVAT equations with a = g. The direction of g is always vertically downwards.

物体仅受重力作用自由下落时，以恒定的向下加速度 g = 9.8 m s⁻² 运动（地球表面，忽略空气阻力）。这是匀加速运动的特殊情况，可将 a = g 代入 SUVAT 方程。g 的方向总是竖直向下。

In vertical motion problems, you must choose a positive direction (usually upwards or downwards). If you take upwards as positive, then a = −9.8 m s⁻². If you take downwards as positive, then a = +9.8 m s⁻². An object thrown upwards will have an initial positive velocity, accelerate negatively, stop momentarily at the top, and then fall back down.

在竖直运动问题中，必须选定正方向（通常向上或向下）。若取向上为正，则 a = −9.8 m s⁻²；若取向下为正，则 a = +9.8 m s⁻²。向上抛出的物体初速度为正、加速度为负，在最高点瞬间速度为零，随后下落。

4. Velocity-Time Graphs | 速度-时间图

A velocity-time graph (v-t graph) shows how velocity changes with time. The gradient of the graph gives acceleration, and the area under the graph between two time points gives the displacement. If the graph is a straight line, acceleration is constant; a horizontal line indicates zero acceleration (constant velocity).

速度-时间图（v-t 图）展示速度随时间的变化规律。图线的斜率（梯度）代表加速度，图线与时间轴之间所围面积代表位移。若图线为直线，则加速度恒定；水平线表示加速度为零（匀速运动）。

You can calculate displacement by finding the area of rectangles, triangles, or trapeziums under the graph. For motion with changing acceleration, the gradient at a point (the tangent) gives instantaneous acceleration. Many exam questions ask you to determine total distance travelled, which is the sum of all areas, taking absolute values if velocity changes sign.

计算位移时，可求出图线下方矩形、三角形或梯形的面积。若加速度变化，某点切线的斜率表示瞬时加速度。许多考题要求计算总路程，此时需将所有面积的绝对值相加（速度变号时尤为注意）。

5. Newton’s Laws of Motion | 牛顿运动定律

Newton’s first law states that an object at rest remains at rest, and an object in motion remains in motion with constant velocity, unless acted upon by a net external force. This property is called inertia. The first law helps you identify situations where forces are balanced and acceleration is zero.

牛顿第一定律指出：除非受到净外力，静止物体保持静止，运动物体保持匀速直线运动。这一性质称为惯性。第一定律可用于识别受力平衡、加速度为零的情形。

Newton’s second law gives the relationship F = ma, where F is the resultant force, m is the mass, and a is the acceleration. The acceleration is in the same direction as the resultant force. In IGCSE mechanics, you will apply this law repeatedly to single particles and connected objects, always resolving forces along the direction of motion.

牛顿第二定律给出 F = ma，其中 F 为合力，m 为质量，a 为加速度。加速度方向与合力方向一致。在 IGCSE 力学中，你会反复应用此定律处理单个物体或连接体问题，并始终沿运动方向分解力。

Newton’s third law says that for every action force there is an equal and opposite reaction force. These two forces act on different bodies, so they do not cancel each other out for a single object. A classic example is a book on a table: the book pushes down on the table, and the table pushes up on the book with an equal force.

牛顿第三定律指出，每一个作用力都有一个等大反向的反作用力。这两个力作用在不同物体上，因此对单个物体而言不会相互抵消。典型例子：放在桌上的书向下压桌面，桌面向书施加等大向上的支持力。

6. Momentum and Impulse | 动量与冲量

Momentum p is defined as the product of mass and velocity: p = mv. It is a vector quantity and its unit is kg m s⁻¹. The impulse of a force is the product of force and the time for which it acts: Impulse = FΔt. Impulse equals the change in momentum: FΔt = Δp = m(v − u).

动量 p 定义为质量与速度的乘积：p = mv。动量是向量，单位为 kg m s⁻¹。冲量等于力与其作用时间的乘积：冲量 = FΔt。冲量与动量的变化量相等：FΔt = Δp = m(v − u)。

In a closed system with no external forces, the total momentum before a collision or explosion is equal to the total momentum after. This principle of conservation of momentum is applied in problems involving two objects colliding or pushing apart. Always remember to assign positive and negative directions when calculating total momentum.

在没有外力的封闭系统中，碰撞或爆炸前的总动量等于总动量之后。这一动量守恒原理应用于两物体碰撞或分离的问题。计算总动量时务必设定正负方向。

7. Forces in Equilibrium | 力的平衡

A body is in equilibrium when the net force acting on it is zero and it has no net moment (no turning effect). For forces acting at a point, equilibrium means the vector sum of all forces is zero. In two dimensions, you can resolve forces into horizontal and vertical components and check that both sums are zero.

物体所受合外力为零且合力矩为零（无转动效应）时，物体处于平衡状态。对于共点力，平衡意味着所有力的向量和为零。在二维问题中，可将各力分解为水平分量与竖直分量，分别核对两方向的合力是否为零。

Common exam questions involve a particle held in equilibrium by strings, or an object on a rough inclined plane. You need to draw a free-body diagram showing weight, normal reaction, tension, and friction. Then apply the conditions for equilibrium: upward forces equal downward forces, and forces to the left equal forces to the right.

常见考题包括用绳子悬挂的质点或粗糙斜面上的物体。你需要画出受力分析图，标示重力、法向反作用力、绳拉力和摩擦力，然后应用平衡条件：向上力等于向下力，向左力等于向右力。

8. Moments and Turning Effect | 力矩与转动效应

The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force: Moment = F × d. The unit is newton-metre (N m). Moments can cause rotation clockwise or anticlockwise; by convention, one direction is taken as positive.

力对某点的力矩等于力的大小乘以该点到力作用线的垂直距离：力矩 = F × d。单位为牛顿·米 (N m)。力矩可使物体顺时针或逆时针转动；通常约定一个转向为正。

For a body to be in rotational equilibrium, the sum of clockwise moments about any pivot must equal the sum of anticlockwise moments. This principle is used extensively in lever and beam problems, such as a uniform rod supported at a pivot with weights hung on either side. Always show your working with a clear moment equation.

物体处于转动平衡时，对任意支点，顺时针力矩之和必须等于逆时针力矩之和。这一原理广泛应用于杠杆和横梁问题，如均匀杆在支点支撑且两端悬挂重物。解答时务必清晰列出力矩平衡方程。

9. Common Problem-Solving Strategies | 常见解题策略

Start every mechanics problem by drawing a clear, labelled diagram. For motion questions, list the given s, u, v, a, t values and decide on the positive direction. For force problems, sketch a free-body diagram showing all forces. This visual step helps prevent sign errors and clarifies which direction to take as positive.

每道力学题都应从绘制清晰标记的示意图开始。运动问题要列出已知的 s, u, v, a, t 值并选定正方向。力的问题要画出受力分析图，标明所有作用力。这一可视化步骤有助于避免符号错误，并明确正方向的选取。

When objects are connected by a string over a pulley, the tension in the string is usually the same throughout, and the acceleration of the connected objects has the same magnitude. Apply F = ma to each object separately and solve the resulting simultaneous equations. Remember that if the string is light and the pulley is smooth, the tension is constant.

对于绕过滑轮的绳子连接体，绳中张力通常处处相等，且各物体的加速度大小相同。分别对每个物体应用 F = ma，然后解联立方程。记住，若绳子轻质、滑轮光滑，张力大小不变。

10. Exam Tips | 考试技巧

Always check units: convert grams to kilograms, kilometres per hour to metres per second, and minutes to seconds before plugging numbers into equations. Write down the relevant formula first, then substitute, and finally show your calculation step by step to gain method marks.

务必检查单位：代入方程前，将克转为千克，千米/小时转为米/秒，分钟转为秒。先写出相关公式，再代入数值，并逐步展示计算过程以获取步骤分。

Where a direction is required, state it clearly (e.g. ‘to the right’ or ‘upwards’). Use g = 9.8 m s⁻² unless the question specifies otherwise. For momentum and impulse problems, positive and negative signs are essential to reflect direction, so assign a positive direction at the start and stick to it throughout.

当题目要求给出方向时，应明确说明（如“向右”或“向上”）。除题目特别说明外，使用 g = 9.8 m s⁻²。在动量和冲量问题中，必须用正负号表示方向，因此在解题开始时设定正方向并始终遵循。

Finally, make good use of past paper questions under timed conditions. Practice both conceptual understanding (e.g. explaining why a skydiver reaches a terminal velocity) and numerical problem-solving to build confidence and speed for the real exam.

最后，在定时条件下充分利用历年真题进行练习。既要训练概念理解（例如解释跳伞者为何达到终极速度），也要加强数值计算类题目，以积累信心并提升实战速度。

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IGCSE CCEA Maths: Mechanics Key Points | IGCSE CCEA 数学：力学 考点精讲