KS3 Maths: Work and Energy | 功和能量考点精讲

📚 KS3 Maths: Work and Energy | 功和能量考点精讲

In KS3 maths, you will often encounter questions that use real-world science contexts, and one of the most common is work and energy. These questions test your ability to substitute numbers into formulas, rearrange equations, convert units, and calculate percentages. Understanding the mathematical side of work, energy, and power helps you solve problems accurately and builds a strong foundation for GCSE physics and maths.

在KS3数学中，你经常会遇到以真实科学情境为背景的题目，其中最常见的就是功和能量。这些问题考查你代入数值到公式、变形方程、单位换算以及计算百分比的能力。理解功、能量和功率的数学计算方法，能帮助你准确解题，并为GCSE物理和数学打下坚实的基础。

1. Introduction to Work and Energy in Maths | 数学中的功和能量简介

Work and energy are concepts from physics, but the calculations behind them are pure maths. In KS3, you will learn to work with simple formula triangles, rearrange three-term equations, and apply percentage change to efficiency. This topic helps you practise multiplication, division, and using standard units.

功和能量是物理概念，但背后的计算纯粹是数学。在KS3阶段，你将学习使用简单的公式三角形、整理三项方程，并将百分比变化应用于效率。这个专题帮助你练习乘法、除法以及标准单位的使用。

When a force moves an object, work is done. Energy is the ability to do work, and power is the rate at which work is done. All of these quantities are linked by mathematical formulas that you must be able to use confidently.

当一个力使物体移动时，就做了功。能量是做功的能力，功率则是做功的快慢。所有这些物理量都通过数学公式联系在一起，你需要能够自信地运用它们。

2. The Work Formula: W = F x d | 功的公式：W = F x d

The most fundamental equation in this topic is work done equals force multiplied by distance moved in the direction of the force. In symbols, this is written as W = F x d, where W is work done in joules (J), F is force in newtons (N), and d is distance in metres (m).

这个专题中最基本的方程是：功 = 力 x 沿力方向移动的距离。用符号表示为W = F x d，其中W表示功，单位是焦耳(J)；F表示力，单位是牛顿(N)；d表示距离，单位是米(m)。

From this formula, you can find any unknown if the other two are known. Use division to rearrange: F = W / d and d = W / F. Always check that the units match before substituting numbers.

通过这个公式，如果已知其中两个量，就可以求出第三个量。使用除法进行变形：F = W / d 和 d = W / F。代入数值前一定要检查单位是否一致。

3. Calculating Force, Distance, or Work | 计算力、距离或功

In a typical KS3 maths problem, you might be asked to calculate how much work is done when a trolley is pushed with a force of 15 N over a distance of 4 m. Simply multiply: W = 15 x 4 = 60 J. Remember that the direction of the force and motion must be the same.

在典型的KS3数学问题中，你可能会遇到这样的题：用15 N的力推动小推车移动了4 m，求做了多少功。只需相乘：W = 15 x 4 = 60 J。要记住力的方向和运动方向必须一致。

If the question gives work and force, divide to find distance. For example, 120 J of work is done with a constant force of 30 N. The distance moved is d = 120 / 30 = 4 m. Always write down the formula first and show your substitution step by step.

如果题目给出功和力，通过除法求距离。例如，用30 N的恒力做了120 J的功，移动的距离就是d = 120 / 30 = 4 m。一定要先写出公式，并逐步展示代入过程。

4. Units of Work and Energy: Joules | 功和能量的单位：焦耳

Work and energy are both measured in joules (J). One joule is the work done when a force of one newton moves an object one metre. In maths questions, you may need to convert units such as kilojoules (kJ) to joules, where 1 kJ = 1000 J.

功和能量都以焦耳(J)为单位。1焦耳等于1牛顿的力使物体移动1米所做的功。在数学题中，你可能需要换算单位，例如将千焦(kJ)转换为焦耳，1 kJ = 1000 J。

Sometimes distance is given in centimetres (cm). You must change it to metres before using the formula, because the newton-metre relationship is built on standard SI units. Divide the number of centimetres by 100 to convert to metres.

有时距离会以厘米(cm)给出。在使用公式前，必须将其转换为米，因为牛顿-米的关系建立在标准国际单位之上。将厘米数除以100即可转换为米。

5. Kinetic Energy and Gravitational Potential Energy | 动能与重力势能

Although these are typically physics formulas, you will use maths to calculate stored energy. Kinetic energy (KE) depends on mass and speed: KE = ½ x m x v², where m is mass in kilograms (kg) and v is speed in metres per second (m/s).

尽管这些通常是物理公式，但你会用数学来计算储存的能量。动能(KE)取决于质量和速度：KE = ½ x m x v²，其中m是质量，单位千克(kg)；v是速度，单位米每秒(m/s)。

Gravitational potential energy (GPE) is given by GPE = m x g x h. Here m is mass in kg, g is the gravitational field strength (usually 10 N/kg on Earth), and h is height in metres. You must multiply carefully and follow the order of operations.

重力势能(GPE)的计算公式为GPE = m x g x h。这里m是质量(kg)，g是重力场强度(地球上通常取10 N/kg)，h是高度(m)。你需要仔细相乘并遵循运算顺序。

6. Efficiency Calculations | 效率计算

Efficiency is a key percentage topic in KS3 maths. It tells you how much input energy is converted into useful output. The formula is Efficiency = (Useful output energy / Total input energy) x 100%. Always multiply the ratio by 100 to express it as a percentage.

效率是KS3数学中一个重要的百分比专题。它告诉你输入能量有多少转化为有用的输出。计算公式为效率 = (有用的输出能量 / 总输入能量) x 100%。一定要将比值乘以100，以百分数表示。

For example, a motor uses 80 J of electrical energy and does 60 J of useful work. The efficiency is (60 / 80) x 100 = 75%. You can also use the formula to find missing energy values by rearranging the proportion.

例如，一个电动机消耗了80 J的电能，做了60 J的有用功。效率为(60 / 80) x 100 = 75%。你也可以通过比例变形来求出缺失的能量值。

7. Power: P = E / t | 功率：P = E / t

Power is the rate of energy transfer or the rate of doing work. The equation you need is P = E / t, where P is power in watts (W), E is energy transferred or work done in joules (J), and t is time in seconds (s).

功率是能量转移的速率或做功的速率。你需要掌握的方程为P = E / t，其中P代表功率，单位瓦特(W)；E代表转移的能量或做的功，单位焦耳(J)；t代表时间，单位秒(s)。

If time is given in minutes, always convert to seconds by multiplying by 60. For instance, a lamp transfers 600 J of energy in 2 minutes. Power = 600 / (2 x 60) = 600 / 120 = 5 W.

如果时间以分钟给出，一定要乘以60转换为秒。例如，一盏灯在2分钟内转移了600 J的能量。功率 = 600 / (2 x 60) = 600 / 120 = 5 W。

8. Energy Transfers and Sankey Diagrams | 能量转移与桑基图

Sankey diagrams show energy transfers using arrows. The width of each arrow represents the amount of energy. You can use proportional reasoning to calculate wasted or useful energy. If the total input is 100 J and the useful output arrow is three times as wide as the wasted arrow, you can set up a ratio to find both amounts.

桑基图用箭头表示能量转移，箭头的宽度代表能量的多少。你可以使用比例推理来计算浪费的能量或有用能量。如果总输入是100 J，且有用输出箭头宽度是浪费箭头宽度的三倍，你可以设立比例来求出两者的大小。

These questions often involve simple algebra. Let wasted energy = x, then useful energy = 3x, and total input energy x + 3x = 100 J. Solving gives x = 25 J wasted, useful = 75 J. This is an excellent opportunity to practise forming and solving equations.

这类题目常常涉及简单的代数。设浪费的能量为x，则有用能量为3x，总输入能量为x + 3x = 100 J。解得x = 25 J（浪费），有用能量为75 J。这是练习建立方程和求解方程的好机会。

9. Solving Word Problems Involving Work and Power | 解功和功率的应用题

Many KS3 maths papers include word problems that combine work, energy, and power. Start by highlighting the numerical information and the units. Identify which formula fits the situation, substitute the numbers, and solve step by step.

许多KS3数学试卷包含结合功、能量和功率的应用题。首先要标出数值信息和单位。确定哪个公式适用于该情况，代入数字，然后逐步求解。

For example: ‘A crane lifts a 200 kg mass through a height of 6 m in 5 seconds. Calculate the power developed, taking g = 10 N/kg.’ First, find the weight (force) = mass x g = 200 x 10 = 2000 N. Then work done = force x distance = 2000 x 6 = 12000 J. Finally, power = work / time = 12000 / 5 = 2400 W.

例如：“一台起重机在5秒内将200 kg的物体提升6 m。取g = 10 N/kg，求产生的功率。” 首先，求重量(力) = 质量 x g = 200 x 10 = 2000 N。然后求做功 = 力 x 距离 = 2000 x 6 = 12000 J。最后，功率 = 功 / 时间 = 12000 / 5 = 2400 W。

10. Unit Conversions and Standard Form | 单位转换与标准形式

Correct unit conversion is vital. You must be comfortable switching between joules and kilojoules, metres and centimetres, grams and kilograms, and minutes and seconds. The table below summarises the most common conversions.

正确的单位换算至关重要。你必须熟悉焦耳与千焦、米与厘米、克与千克、分钟与秒之间的转换。下表总结了最常见的换算关系。

Conversion	Factor
kJ to J	x 1000
cm to m	divide by 100
g to kg	divide by 1000
min to s	x 60

In more advanced problems, you may be asked to present very small or large answers in standard form. For instance, 0.005 J is 5 x 10⁻³ J. Practise moving the decimal point and counting the powers of ten correctly.

在更高级的题目中，你可能会被要求用标准形式表示非常小或非常大的答案。例如，0.005 J 是 5 x 10⁻³ J。练习正确移动小数点并数出10的幂次。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

One frequent error is forgetting to square the velocity in the kinetic energy formula. Remember that v² means v multiplied by itself, not v x 2. Also, always check that mass is in kilograms. If a problem gives mass in grams, divide by 1000 first.

一个常见错误是在动能公式中忘记将速度平方。记住v²是v乘以自身，不是v乘以2。此外，务必检查质量是否用千克表示。如果题目给出的质量单位是克，先除以1000。

Another tip: write down the formula triangle or the rearranged equation before plugging in numbers. This reduces the chance of mixing up division and multiplication. During exams, underline the quantities given and what you need to find.

另一个技巧：在代入数字之前，先写出公式三角形或整理后的方程。这样可以减少混淆除法和乘法的可能性。考试时，在给出的量和需要求解的量下画线。

Finally, always include the correct unit in your answer. A number without a unit loses marks in most marking schemes. J, W, N, m, s are essential.

最后，一定要在答案中写上正确的单位。在大多数评分标准中，没有单位的数字会被扣分。J、W、N、m、s这些单位是必不可少的。

12. Summary and Practice Ideas | 总结与练习建议

Mastering work and energy calculations in KS3 maths means you can confidently use formulas, convert units, and solve multi-step problems. The key equations to remember are W = F x d, P = E / t, and Efficiency = (useful output / total input) x 100%.

掌握KS3数学中功和能量的计算，意味着你可以自信地使用公式、转换单位并解决多步骤问题。需要记住的关键方程有W = F x d，P = E / t，以及效率 = (有用输出 / 总输入) x 100%。

Practice by creating your own questions from everyday situations, such as lifting school bags, climbing stairs, or charging devices. The more you practise substituting and rearranging, the more automatic these skills will become.

你可以从日常生活中自行出题练习，比如提起书包、爬楼梯或给设备充电。你越多练习代入和变形，这些技能就会越熟练、越自然。

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KS3 Maths: Work and Energy | 功和能量考点精讲