IGCSE Maths: Work and Energy Key Points | IGCSE 数学：功和能量考点精讲

📚 IGCSE Maths: Work and Energy Key Points | IGCSE 数学：功和能量考点精讲

In IGCSE Maths, work and energy problems often appear in applied contexts, combining algebraic manipulation with physical principles. This guide covers the key formulas, concepts, and typical problem-solving strategies you need to master for your exams.

在 IGCSE 数学中，功和能量问题常以应用题形式出现，将代数运算与物理原理结合起来。本篇考点精讲涵盖了你必须掌握的核心公式、概念和典型解题策略，帮助你高效备考。

1. Definition of Work Done | 做功的定义

Work is done when a force moves an object in the direction of the force. It is calculated as the product of the force and the distance moved in that direction.

当力使物体沿力的方向移动时，力就对物体做了功。功的大小等于力与沿该方向移动的距离的乘积。

For a constant force F acting parallel to the displacement d, the work done W is given by:

对于平行于位移 d 作用的恒力 F，做功 W 的公式为：

W = F × d

If the force is not parallel to the displacement, we use the component of the force in the direction of motion. For an angle θ between the force and displacement:

如果力与位移不平行，我们需要使用力在运动方向上的分量。设力与位移之间的夹角为 θ，则：

W = Fd cos θ

In IGCSE Maths, you will mainly encounter forces acting horizontally or vertically, but questions may require you to resolve forces into components.

在 IGCSE 数学中，主要遇到的是水平或竖直方向的力，但题目可能要求你进行力的分解。

2. Units of Work and Energy | 功和能量的单位

Work and energy are both measured in joules (J). One joule is the work done when a force of 1 newton moves an object 1 metre in the direction of the force.

功和能量都以焦耳 (J) 为单位。1 焦耳等于 1 牛顿的力使物体沿力的方向移动 1 米所做的功。

In calculations, always ensure forces are in newtons (N) and distances in metres (m). Mass should be in kilograms (kg), and speed in metres per second (m/s).

计算时务必确保力的单位是牛顿 (N)，距离的单位是米 (m)。质量的单位是千克 (kg)，速度的单位是米每秒 (m/s)。

Remember: 1 J = 1 N m. This relationship is helpful for converting compound units.

记住：1 J = 1 N m。这个关系有助于进行复合单位换算。

3. Kinetic Energy | 动能

Kinetic energy (KE) is the energy an object possesses due to its motion. The formula for kinetic energy is:

动能 (KE) 是物体由于运动而具有的能量。动能的计算公式为：

KE = ½mv²

where m is the mass of the object and v is its speed. Notice that kinetic energy is proportional to the square of the speed – doubling the speed quadruples the kinetic energy.

其中 m 是物体的质量，v 是速度。注意，动能与速度的平方成正比——速度翻倍，动能变为原来的四倍。

IGCSE problems often ask you to calculate the speed from a given kinetic energy by rearranging the formula:

IGCSE 题目经常要求根据已知动能计算速度，需对公式变形：

v = √(2KE / m)

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy stored when an object is raised above the ground. The formula is:

重力势能 (GPE) 是物体被举高时储存的能量。公式为：

GPE = mgh

where m is mass, g is the acceleration due to gravity (usually 9.8 m/s² or 10 m/s² in IGCSE questions), and h is the vertical height above a reference level.

其中 m 为质量，g 为重力加速度（IGCSE 题目中通常取 9.8 m/s² 或 10 m/s²），h 是相对于参考平面的竖直高度。

The change in gravitational potential energy depends only on the vertical height change, not on the path taken.

重力势能的变化只与竖直高度的变化有关，与运动路径无关。

5. Conservation of Mechanical Energy | 机械能守恒

In a closed system where no external forces (like friction) do work, the total mechanical energy (KE + GPE) remains constant.

在没有外力（如摩擦力）做功的封闭系统中，总机械能（动能 + 重力势能）保持不变。

This principle is applied in many IGCSE problem-solving contexts, such as a falling object or a pendulum. The equation is:

这一原理广泛应用于 IGCSE 解题中，例如落体或单摆问题。公式为：

KE₁ + GPE₁ = KE₂ + GPE₂

If friction or air resistance is present, the work done against these forces reduces the mechanical energy, converting it into thermal energy.

如果有摩擦或空气阻力，克服这些力做功会使机械能减少，并转化为热能。

6. Power | 功率

Power is the rate at which work is done or energy is transferred. The formula is:

功率是做功或能量转移的速率。公式为：

P = W / t

where W is work done or energy transferred in joules, and t is time in seconds. The unit of power is the watt (W); 1 W = 1 J/s.

其中 W 为做功或能量转移的焦耳数，t 为时间秒数。功率的单位是瓦特 (W)；1 W = 1 J/s。

Another useful expression arises when a constant force moves at a constant speed v:

当恒力以恒定速度 v 移动时，还有一个常用的表达式：

P = Fv

You may be asked to calculate the power output of an engine or the time taken to perform a certain amount of work.

考试中可能要求计算发动机的功率输出，或完成一定量功所需的时间。

7. Efficiency | 效率

Efficiency measures how much of the input energy or power is converted into useful output. It is defined as:

效率衡量输入能量或功率中有多少转化为有用的输出。定义为：

Efficiency = (useful output energy / total input energy) × 100%

Alternatively, using power: Efficiency = (useful power output / total power input) × 100%.

或者用功率表示：效率 = (有用输出功率 / 总输入功率) × 100%。

Since energy is always conserved, the “lost” energy is usually dissipated as heat or sound. Efficiency is always less than 1 (or less than 100%) for real machines.

由于能量总是守恒的，“损失”的能量通常以热或声音的形式耗散。对于真实的机器，效率永远小于 1（或小于 100%）。

8. Work Done Against Friction | 克服摩擦做功

When an object moves against friction, work is done to overcome the frictional force. This work is converted into thermal energy. The formula is the same: W = F_f × d, where F_f is the frictional force.

当物体移动时克服摩擦做功，这部分功转化为热能。公式相同：W = F_f × d，其中 F_f 为摩擦力。

In problems involving inclined planes, the work done against friction reduces the gain in mechanical energy. The work–energy principle then states:

在涉及斜面的问题中，克服摩擦做功会减少机械能的增加量。此时功能原理表示为：

Net work done = change in kinetic energy + change in gravitational potential energy

This is often simplified in IGCSE Maths to: Work input = useful work output + work against friction.

在 IGCSE 数学中，这通常简化为：输入功 = 有用输出功 + 克服摩擦做功。

9. Work Done by a Force at an Angle | 成角度的力做功

Although many IGCSE problems assume forces are parallel to displacement, some extended questions involve a force pulling at an angle. The effective component is the horizontal part of the force.

尽管许多 IGCSE 问题假设力与位移平行，某些扩展题目会涉及一个成角度的拉力。有效分量是力的水平分量。

For a force F applied at an angle θ to the horizontal, the work done over a horizontal distance d is:

对于与水平方向成 θ 角作用的力 F，在水平距离 d 上做的功为：

W = F cos θ × d

The vertical component does no work if the motion is horizontal, as it does not move the object in its direction.

如果运动是水平的，竖直分量不做功，因为它没有使物体沿其方向移动。

Always check whether the force acts in exactly the same direction as the movement before applying W = Fd directly.

在直接应用 W = Fd 之前，务必检查力的方向是否与运动方向完全一致。

10. Typical IGCSE Problem Types | IGCSE 典型题型

IGCSE Maths exam questions on work and energy often combine several concepts. Below are typical problem types with strategies:

IGCSE 数学中关于功和能量的考题常常结合多个概念。以下是典型题型和解题策略：

Finding speed from work or energy: Use KE = ½mv² or energy conservation to find v after a given change in height or work done.

从功或能量求速度： 利用 KE = ½mv² 或能量守恒，在给定高度变化或做功后求 v。
Power and time: Use P = W/t or P = Fv to find unknown time, force, or speed.

功率和时间： 使用 P = W/t 或 P = Fv 求未知的时间、力或速度。
Inclined planes: Calculate GPE change using vertical height h = L sin θ, where L is the slope length.

斜面问题： 利用竖直高度 h = L sin θ 计算重力势能变化，其中 L 为斜面长度。
Braking distances: Use work done by friction = change in KE to relate braking force and distance.

制动距离： 利用克服摩擦力做功 = 动能变化量，建立制动力与距离的关系。
Efficiency of machines: Set up ratios and solve for unknown energy amounts using the efficiency formula.

机械效率： 利用效率公式建立比例关系，求解未知能量值。

11. Graphs and Work | 图像与功

The work done by a varying force can be found using a force–distance graph. For a constant force, the graph is a horizontal line, and the area under the graph is a rectangle representing F × d.

变力做的功可以利用力–距离图像求出。对于恒力，图像是一条水平线，图线下方的面积是一个矩形，表示 F × d。

More generally, work done equals the area under the force–distance graph. This is important in Extension Maths where integration might be introduced, but in IGCSE Core, you may simply be asked to calculate the area of simple shapes.

更普遍地说，做功等于力–距离图线下方的面积。这在 Extension Maths 中可能引入积分，但在 IGCSE Core 中，你可能只需计算简单图形的面积。

A common question: “The graph shows how the force acting on an object varies with distance. Calculate the total work done.” You find the area of rectangles, triangles, or trapeziums beneath the curve.

常见题目：“图像显示了作用在物体上的力随距离的变化。计算所做的总功。”你需要计算曲线下方矩形、三角形或梯形的面积。

12. Summary of Key Formulas | 核心公式总结

Here is a summary table of the essential formulas you need to remember for IGCSE Maths Work and Energy problems.

以下表格总结了 IGCSE 数学功和能量问题中必须记忆的基本公式。

Quantity	Formula	Units
Work done	W = Fd (parallel) W = Fd cos θ (non‑parallel)	J (joule)
Kinetic Energy	KE = ½mv²	J
Gravitational Potential Energy	GPE = mgh	J
Power	P = W/t P = Fv	W (watt)
Efficiency	η = (useful output / input) × 100%	%

Make sure you can substitute values correctly and rearrange these equations for any variable.

确保你能正确代入数值，并能对这些公式中的任意变量进行变形求解。

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IGCSE Maths: Work and Energy Key Points | IGCSE 数学：功和能量 考点精讲