Compound Measures: Speed, Density and Pressure | 复合单位度量：速度、密度与压强

📚 Compound Measures: Speed, Density and Pressure | 复合单位度量：速度、密度与压强

Compound measures are quantities that combine two or more different units of measurement. In the Edexcel IGCSE Mathematics syllabus, the three most important compound measures are speed, density and pressure. Each one is defined as a ratio between two fundamental quantities, and mastering them requires confidence in unit conversion, formula rearrangement and proportional reasoning.

复合单位度量是指将两种或更多不同计量单位组合而成的量。在 Edexcel IGCSE 数学考纲中，最重要的三个复合单位是速度、密度和压强。它们各自定义为两个基本量之间的比值，掌握它们需要对单位换算、公式变形和比例推理有充分的自信。

1. The Concept of a Compound Measure | 复合单位的概念

A compound measure is formed when one quantity is divided by another. For example, speed is distance divided by time; density is mass divided by volume; pressure is force divided by area. In every case, the resulting unit tells you which two base quantities are involved and how they relate to one another.

复合单位是将一个量除以另一个量所得到的度量。例如，速度是距离除以时间；密度是质量除以体积；压强是力除以面积。在每种情况下，最终得到的单位都能告诉你涉及哪两个基本量以及它们之间的关系。

Recognising the structure of a compound measure is essential for solving problems. Once you identify that a quantity is a rate or a ratio of two units, you can decide which formula to use and how to manipulate it. Most exam questions test this recognition skill directly.

识别复合单位的结构对于解题至关重要。一旦你确定某个量是两个单位之间的比率，你就可以决定使用哪个公式以及如何变形。多数考题直接考查这种识别能力。

The table below summarises the three key compound measures covered in this article.

下表总结了本文涉及的三个关键复合单位。

Measure	Definition	Typical Units
Speed	Distance ÷ Time	km/h, m/s, mph
Density	Mass ÷ Volume	g/cm³, kg/m³
Pressure	Force ÷ Area	N/m², Pa, N/cm²

2. Speed: Distance Divided by Time | 速度：距离除以时间

Speed is the most familiar compound measure. It tells you how much distance is covered in each unit of time. The standard formula is:

速度是最熟悉的复合单位。它告诉你每单位时间内覆盖多少距离。标准公式为：

Speed = Distance ÷ Time

From this single equation, two rearrangements follow immediately:

从这个单一方程可以立即得到两个变形公式：

Distance = Speed × Time

Time = Distance ÷ Speed

You should be able to apply all three forms fluently. In exam questions, the required rearrangement is rarely stated explicitly; you must decide which form fits the information given.

你应该能够熟练运用这三种形式。在考题中，所需的变形很少被明确说明；你必须根据已知信息决定使用哪种形式。

For example, if a car travels 240 kilometres in 3 hours, its average speed is found by dividing distance by time:

例如，一辆汽车在3小时内行驶240公里，其平均速度通过距离除以时间求得：

Speed = 240 ÷ 3 = 80 km/h

Notice the unit of the answer: kilometres per hour (km/h). The slash or “per” indicates a division of the distance unit by the time unit.

注意答案的单位：千米每小时（km/h）。斜杠或”每”表示距离单位除以时间单位。

3. Converting Between Speed Units | 速度单位之间的换算

Unit conversion is one of the most frequently tested skills in this topic. The most common conversion is between kilometres per hour (km/h) and metres per second (m/s). To perform this conversion, you must convert both the numerator and the denominator.

单位换算是本主题中最常考查的技能之一。最常见的换算是千米每小时（km/h）与米每秒（m/s）之间的转换。要进行此换算，你必须同时转换分子和分母。

Start with 1 km/h. Convert kilometres to metres: 1 km = 1000 m. Convert hours to seconds: 1 hour = 3600 s. Therefore:

从1 km/h 开始。将千米转换为米：1 km = 1000 m。将小时转换为秒：1 小时 = 3600 s。因此：

1 km/h = 1000 ÷ 3600 m/s = 5/18 m/s ≈ 0.278 m/s

So to convert from km/h to m/s, multiply by 5/18. To convert from m/s to km/h, multiply by 18/5, which is 3.6.

因此从 km/h 转换为 m/s，乘以 5/18。从 m/s 转换为 km/h，乘以 18/5，即 3.6。

Worked Example 1

示例1

Convert 72 km/h to m/s.

将 72 km/h 转换为 m/s。

72 × 5/18 = 20 m/s

Convert 30 m/s to km/h.

将 30 m/s 转换为 km/h。

30 × 18/5 = 108 km/h

A useful shortcut: when converting speed units, remember that m/s is always a smaller number than km/h for the same physical speed, because each metre is shorter than a kilometre and each second is shorter than an hour in a way that makes the overall factor less than 1.

一个有用的捷径：在换算速度单位时，记住对于相同的物理速度，m/s 的数值总是小于 km/h，因为每米比每千米短，每秒比每小时短，综合因素使得整体换算系数小于1。

4. Average Speed and Real-World Problems | 平均速度与实际应用问题

Average speed is defined as the total distance travelled divided by the total time taken. This is different from the mean of several different speeds. If a journey consists of multiple stages at different speeds, you cannot simply average the speeds unless the time spent in each stage is identical.

平均速度定义为总行驶距离除以总耗时。这与多个不同速度的算术平均值不同。如果一段旅程由多个不同速度的阶段组成，除非每个阶段所用时间相同，否则不能简单地对速度求平均。

Worked Example 2

示例2

A cyclist travels 10 km at 20 km/h, then another 15 km at 30 km/h. Find the average speed for the whole journey.

一名骑行者以20 km/h 骑行10 km，再以30 km/h 骑行15 km。求全程平均速度。

First, find the time for each stage.

首先求每个阶段的时间。

Time₁ = 10 ÷ 20 = 0.5 hours

Time₂ = 15 ÷ 30 = 0.5 hours

Total time = 1.0 hour. Total distance = 25 km.

总时间 = 1.0 小时。总距离 = 25 km。

Average speed = 25 ÷ 1.0 = 25 km/h

In this case, because the times are equal, the arithmetic mean of 20 and 30 would give the same answer. But consider a different scenario where the times are not equal.

在这个例子中，因为时间相等，20和30的算术平均数会给出同样的答案。但考虑一个时间不等的情景。

Worked Example 3

示例3

A car travels 60 km at 60 km/h, then 60 km at 30 km/h. Find the average speed.

一辆汽车以60 km/h行驶60 km，再以30 km/h行驶60 km。求平均速度。

Time₁ = 60 ÷ 60 = 1 hour. Time₂ = 60 ÷ 30 = 2 hours.

时间₁ = 60 ÷ 60 = 1 小时。时间₂ = 60 ÷ 30 = 2 小时。

Total time = 3 hours. Total distance = 120 km.

总时间 = 3 小时。总距离 = 120 km。

Average speed = 120 ÷ 3 = 40 km/h

Notice that the arithmetic mean of 60 and 30 is 45 km/h, which is incorrect. The correct harmonic-type averaging accounts for the unequal time spent at each speed.

注意60和30的算术平均数是 45 km/h，这是错误的。正确的调和式平均考虑了每种速度下花费的不相等时间。

5. Density: Mass Divided by Volume | 密度：质量除以体积

Density measures how much mass is packed into a given volume. The denser a material is, the more mass it contains per unit volume. The formula is:

密度衡量在给定体积内包含了多少质量。材料密度越大，每单位体积所含质量就越多。公式为：

Density = Mass ÷ Volume

Rearranged forms:

变形形式：

Mass = Density × Volume

Volume = Mass ÷ Density

Typical units for density in IGCSE Mathematics are g/cm³ and kg/m³. Water has a density of approximately 1 g/cm³, which is equivalent to 1000 kg/m³.

IGCSE数学中密度的典型单位是 g/cm³ 和 kg/m³。水的密度约为 1 g/cm³，即 1000 kg/m³。

Worked Example 4

示例4

A metal block has mass 450 g and volume 50 cm³. Find its density.

一块金属的质量为 450 g，体积为 50 cm³。求其密度。

Density = 450 ÷ 50 = 9 g/cm³

This tells us each cubic centimetre of this metal contains 9 grams of mass.

这告诉我们这种金属每立方厘米包含9克质量。

6. Density Problems Involving Volume Calculation | 涉及体积计算的密度问题

Some questions give you density and mass, and ask you to determine the volume. These problems require you to rearrange the density formula and often involve compound shapes or unit conversion.

有些问题给出密度和质量，要求你确定体积。这些问题需要你变形密度公式，并常涉及复合图形或单位换算。

Worked Example 5

示例5

A liquid has a density of 0.8 g/cm³. What is the mass of 2 litres of this liquid? (Recall that 1 litre = 1000 cm³.)

一种液体的密度为 0.8 g/cm³。2升这种液体的质量是多少？（回顾1升 = 1000 cm³。）

First convert the volume to cm³:

首先将体积转换为 cm³：

Volume = 2 × 1000 = 2000 cm³

Then use the mass formula:

然后使用质量公式：

Mass = 0.8 × 2000 = 1600 g = 1.6 kg

Pay close attention to whether the question expects the answer in grams or kilograms. Checking the units at the beginning of your working can save you from careless errors.

特别注意题目要求以克还是千克为单位。在计算开始时检查单位可以避免粗心错误。

7. Pressure: Force Divided by Area | 压强：力除以面积

Pressure describes how concentrated a force is over a surface. The same force applied over a smaller area produces a greater pressure. The formula is:

压强描述的是力在表面上集中的程度。相同的力作用在较小的面积上会产生更大的压强。公式为：

Pressure = Force ÷ Area

Rearranged forms:

变形形式：

Force = Pressure × Area

Area = Force ÷ Pressure

The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m². In IGCSE problems, force is usually given in newtons (N) and area in m², cm² or mm².

压强的SI单位是帕斯卡（Pa），其中 1 Pa = 1 N/m²。在IGCSE问题中，力通常以牛顿（N）为单位，面积以 m²、cm² 或 mm² 为单位。

Worked Example 6

示例6

A force of 500 N is applied to an area of 2 m². Calculate the pressure.

500 N 的力作用于 2 m² 的面积上。计算压强。

Pressure = 500 ÷ 2 = 250 N/m² = 250 Pa

Now consider the same force applied to an area of 0.5 m².

现在考虑同样的力作用于 0.5 m² 的面积上。

Pressure = 500 ÷ 0.5 = 1000 N/m² = 1000 Pa

Halving the area doubles the pressure, showing the inverse relationship between pressure and area.

面积减半使压强加倍，表明压强与面积呈反比关系。

8. Pressure in Real-Life Contexts | 压强在实际生活中的应用

Examiners often frame pressure questions in real-world contexts such as snowshoes, high-heeled shoes, knives or hydraulic systems. The key is to identify the force and the area correctly.

考官经常将压强问题置于真实生活场景中，例如雪鞋、高跟鞋、刀具或液压系统。关键是正确识别力和面积。

Worked Example 7

示例7

A woman of mass 60 kg wears high-heeled shoes. The total area of the heels in contact with the ground is 4 cm². Taking gravitational field strength as 10 N/kg, calculate the pressure exerted on the ground.

一位质量为60 kg的女性穿着高跟鞋。鞋跟与地面接触的总面积为 4 cm²。取重力场强度为 10 N/kg，计算对地面施加的压强。

First, find the force (weight) exerted:

首先，求施加的力（重量）：

Force = mass × gravitational field strength = 60 × 10 = 600 N

Convert the area to m², since pressure in Pa requires the area in m²:

将面积转换为 m²，因为以Pa为单位的压强要求面积以m²表示：

4 cm² = 4 ÷ 10000 = 0.0004 m²

Now calculate the pressure:

现在计算压强：

Pressure = 600 ÷ 0.0004 = 1,500,000 Pa = 1500 kPa

This enormously high pressure explains why high heels can damage wooden floors. The same concept explains why snowshoes distribute weight over a larger area to reduce pressure and prevent sinking into the snow.

如此高的压强解释了为什么高跟鞋会损坏木地板。同样的概念解释了为什么雪鞋将重量分布到更大的面积上以降低压强，防止陷入雪中。

9. Converting Area and Volume Units | 面积和体积单位的换算

When working with density and pressure, you will frequently need to convert between units of area and volume. These conversions are non-intuitive and require care.

在处理密度和压强时，你经常需要在面积和体积单位之间进行换算。这些换算不直观，需要特别小心。

For area:

对于面积：

1 m² = 100 cm × 100 cm = 10,000 cm²

1 cm² = 10 mm × 10 mm = 100 mm²

For volume:

对于体积：

1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³

1 cm³ = 10 mm × 10 mm × 10 mm = 1000 mm³

Many students incorrectly multiply by 100 when converting cubic units. Remember: the conversion factor for cubic units is the cube of the linear conversion factor.

许多学生在转换立方单位时错误地乘以100。记住：立方单位的换算系数是线性换算系数的立方。

The same principle applies to square units: the conversion factor for square units is the square of the linear conversion factor.

同样的原理适用于平方单位：平方单位的换算系数是线性换算系数的平方。

10. Multi-Step Problems Combining Compound Measures | 结合复合单位的多步问题

The most challenging problems in this topic combine two or more skills. For instance, a question might require you to convert units, then find a volume, then calculate a density. You should approach such problems by listing what is given, what is needed, and what intermediate steps are required.

本主题中最具挑战性的问题往往结合两种或多种技能。例如，一个问题可能要求你先换算单位，再求体积，再计算密度。你应该通过列出已知条件、需求量以及所需中间步骤来逐步解决这类问题。

Worked Example 8

示例8

A cube has sides of length 20 cm and a mass of 7.2 kg. Calculate its density in g/cm³.

一个立方体的边长为 20 cm，质量为 7.2 kg。计算其密度（以 g/cm³ 为单位）。

First, find the volume of the cube:

首先，求立方体的体积：

Volume = 20 × 20 × 20 = 8000 cm³

The formula for density requires mass in grams. Convert 7.2 kg to grams:

密度公式要求质量以克为单位。将 7.2 kg 转换为克：

Mass = 7.2 × 1000 = 7200 g

Now apply the density formula:

现在应用密度公式：

Density = 7200 ÷ 8000 = 0.9 g/cm³

This value is less than the density of water (1 g/cm³), so the cube would float in water. Checking your answer against such physical intuition is a good habit in this topic.

该值小于水的密度（1 g/cm³），因此这个立方体会浮在水面上。在本主题中，用这种物理直觉来检查答案是一个好习惯。

11. Errors to Avoid | 需要避免的错误

Several common mistakes recur in this topic. Being aware of them can help you avoid losing marks in the exam.

在本主题中有几个常见的重复错误。意识到这些可以帮助你在考试中避免失分。

Incorrect unit conversion for squared and cubed units: Always square or cube the linear conversion factor. For example, 1 m² = 10,000 cm², not 100 cm².

平方和立方单位换算错误：始终将线性换算系数平方或立方。例如，1 m² = 10,000 cm²，而不是100 cm²。
Forgetting to convert mass units: The density formula requires the mass unit in the formula to match the numerator unit in the density. Always check whether grams or kilograms are required.

忘记转换质量单位：密度公式要求公式中的质量单位与密度分子单位一致。始终检查题目要求克还是千克。
Using the wrong rearrangement: Make sure you identify whether you are finding speed, distance or time before substituting values. The formula triangle, if taught, should be used with caution.

使用错误的公式变形：在代入数值之前，先确认你是在求速度、距离还是时间。如果使用公式三角形，务必谨慎。
Adding speeds instead of using total distance over total time: The average speed is not the arithmetic mean of the speeds, especially when the time intervals differ.

将速度相加而不是用总距离除以总时间：平均速度不是速度的算术平均数，特别是当时间间隔不等时。
Ignoring the units in the final answer: An answer without a correct unit may be marked wrong, even if the numerical value is correct.

忽略最终答案中的单位：即使数值正确，没有正确单位的答案也可能被扣分。

12. Exam-Style Practice Questions | 考试风格练习题

Now apply your understanding to these exam-style questions. Attempt them before checking the solutions.

现在通过以下考试风格练习题来应用你的理解。先尝试作答，再对照解答。

Practice Question 1

练习1

A train travels 300 km in 2.5 hours. Calculate its average speed in km/h and then convert this speed to m/s.

一列火车在2.5小时内行驶300 km。计算其平均速度（km/h），然后将该速度转换为 m/s。

Speed = 300 ÷ 2.5 = 120 km/h

120 × 5/18 = 33.33… m/s ≈ 33.3 m/s

Practice Question 2

练习2

An object has a mass of 250 g and a density of 5 g/cm³. Find its volume.

一个物体的质量为 250 g，密度为 5 g/cm³。求其体积。

Volume = Mass ÷ Density = 250 ÷ 5 = 50 cm³

Practice Question 3

练习3

A rectangular box measures 40 cm by 30 cm. A force of 600 N is applied evenly over the top face of the box. Calculate the pressure in N/cm² and then convert to Pa.

一个长方体盒子的底面为40 cm × 30 cm。600 N 的力均匀分布在盒子顶面上。计算以 N/cm² 为单位的压强，然后转换为 Pa。

Area = 40 × 30 = 1200 cm²

Pressure = 600 ÷ 1200 = 0.5 N/cm²

To convert to Pa, note that 1 cm² = 0.0001 m², so 1 N/cm² = 10,000 N/m² = 10,000 Pa:

要转换为 Pa，注意 1 cm² = 0.0001 m²，因此 1 N/cm² = 10,000 N/m² = 10,000 Pa：

Pressure = 0.5 × 10,000 = 5000 Pa

Compound measures unite the separate ideas of distance, time, mass, volume and force through division. By repeatedly practising the three central formulas and their rearrangements, and by becoming fluent in unit conversion, you can confidently tackle any IGCSE question on speed, density or pressure. Always write down the formula you are using, show each step clearly, and attach the correct units to every final answer.

复合度量通过除法将距离、时间、质量、体积和力这些独立的概念联系起来。通过反复练习三个核心公式及其变形，并熟练进行单位换算，你可以自信地应对IGCSE中任何关于速度、密度或压强的问题。始终写下你所使用的公式，清晰地展示每一步，并为每个最终答案附上正确的单位。

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Compound Measures: Speed, Density and Pressure | 复合单位度量：速度、密度与压强