Physical Quantities and Units in Mechanics | 力学中的量与单位

📚 Physical Quantities and Units in Mechanics | 力学中的量与单位

In A-Level Mechanics, every equation tells a story about motion, forces, and energy. But before we can solve any problem, we must first understand the language of measurement: physical quantities and their units. This article provides a complete and exam-focused guide to the quantities and units used throughout the mechanics syllabus.

在 A-Level 力学中，每一个方程都在描述运动、力和能量的故事。但在解任何题目之前，我们必须先理解测量的语言：物理量及其单位。本文为整个力学考纲中涉及的量和单位提供一份完整且紧扣考点的指南。

1. Base Quantities and Derived Quantities | 基本量与导出量

Physical quantities are divided into two categories. Base quantities are defined independently and cannot be expressed in terms of other quantities. In mechanics, the relevant base quantities are mass (kg), length (m), and time (s). Derived quantities, on the other hand, are combinations of base quantities through multiplication or division.

物理量分为两类。基本量是独立定义的，不能用其他量来表示。在力学中，相关的基本量包括质量 (kg)、长度 (m) 和时间 (s)。导出量则是通过基本量的乘除组合而成的量。

Examples of derived quantities include velocity, acceleration, force, momentum, and energy.

导出量的例子包括速度、加速度、力、动量和能量。
Derived units are built from base units using algebraic relationships, such as ms⁻¹ for velocity.

导出单位是利用代数关系从基本单位构建的，例如速度的单位 ms⁻¹。

2. The SI Unit System | 国际单位制

The International System of Units (SI) is the standard framework used in all scientific calculations. In A-Level mechanics, you should always convert all quantities to SI base units before substituting them into equations. The three most important base units for mechanics are the metre, the kilogram, and the second.

国际单位制 (SI) 是所有科学计算中使用的标准框架。在 A-Level 力学中，你应始终在代入方程前将所有量转换为 SI 基本单位。力学中最重要的三个基本单位是米、千克和秒。

Quantity	Base Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s

In addition, you may meet the ampere (A) for electric current in applied contexts, but pure mechanics questions rarely require it.

此外，在应用情境中你可能会遇到安培 (A) 作为电流单位，但纯力学题目通常不需要它。

3. Dimensional Analysis | 量纲分析

Dimensional analysis is a powerful tool for checking whether an equation is physically correct. Every physical quantity has a dimension, expressed in terms of mass M, length L, and time T. For example, velocity has dimensions LT⁻¹, and acceleration has dimensions LT⁻².

量纲分析是检查方程物理正确性的有力工具。每个物理量都有量纲，用质量 M、长度 L 和时间 T 表示。例如，速度的量纲是 LT⁻¹，加速度的量纲是 LT⁻²。

Force, from F = ma, has dimensions MLT⁻².

根据 F = ma，力的量纲为 MLT⁻²。
Work and energy have dimensions ML²T⁻².

功和能量的量纲为 ML²T⁻²。
Power, being energy per unit time, has dimensions ML²T⁻³.

功率是单位时间的能量，其量纲为 ML²T⁻³。

When adding terms in an equation, every term must have the same dimensions. This is a quick and reliable way to spot algebraic errors.

在方程中相加的每一项必须具有相同的量纲。这是发现代数错误的快速而可靠的方法。

4. Scalars and Vectors | 标量与矢量

Mechanics distinguishes between two types of quantities: scalars have magnitude only, while vectors have both magnitude and direction. This distinction affects how quantities are combined and how equations are applied.

力学区分两类量：标量只有大小，矢量既有大小又有方向。这一区别影响量的合成方式以及方程的应用方式。

Scalars in mechanics: mass, time, speed, distance, energy, work, power.

力学中的标量：质量、时间、速率、距离、能量、功、功率。
Vectors in mechanics: displacement, velocity, acceleration, force, momentum, impulse.

力学中的矢量：位移、速度、加速度、力、动量、冲量。

Remember that speed is the magnitude of velocity, and distance is the magnitude of displacement. Losing direction turns a vector into its scalar counterpart.

请记住，速率是速度的大小，距离是位移的大小。失去方向后，矢量就变成其对应的标量。

5. Kinematic Quantities: Displacement, Velocity, Acceleration | 运动学量：位移、速度、加速度

Kinematics describes motion without considering its causes. The three key quantities are displacement, velocity, and acceleration, each with a clear SI unit.

运动学描述运动而不考虑其原因。三个关键量是位移、速度和加速度，每个都有明确的 SI 单位。

Displacement s is the straight-line distance from the start point to the end point in a specified direction. Its unit is the metre (m).

位移 s 是从起点到终点沿指定方向的直线距离。其单位是米 (m)。

Velocity v is the rate of change of displacement. Its SI unit is metres per second, written as ms⁻¹ or m/s.

速度 v 是位移的变化率。其 SI 单位是米每秒，写作 ms⁻¹ 或 m/s。

Acceleration a is the rate of change of velocity. Its SI unit is metres per second squared, written as ms⁻².

加速度 a 是速度的变化率。其 SI 单位是米每二次方秒，写作 ms⁻²。

v = Δs / Δt and a = Δv / Δt

Always use displacement rather than distance when applying the constant acceleration equations, because those equations are vector-based.

在应用匀加速运动方程时，务必使用位移而非路程，因为这些方程是基于矢量的。

6. Dynamics: Mass, Force and Weight | 动力学量：质量、力与重量

Dynamics explains motion through the action of forces. The central quantity is force, whose unit is the newton (N). One newton is the force required to accelerate a mass of 1 kg at 1 ms⁻².

动力学通过力的作用解释运动。核心量是力，其单位是牛顿 (N)。1 牛顿是使 1 kg 质量产生 1 ms⁻² 加速度所需的力。

Mass m is a scalar quantity measured in kilograms (kg). It measures the amount of matter in an object and is invariant regardless of location.

质量 m 是以千克 (kg) 为单位测量的标量。它衡量物体中物质的多少，不随位置改变而变化。

Weight W is the gravitational force acting on an object. It is calculated as W = mg, where g is the gravitational field strength, approximately 9.8 Nkg⁻¹ or ms⁻² on Earth.

重量 W 是作用在物体上的重力。计算公式为 W = mg，其中 g 是重力场强度，在地球上约为 9.8 Nkg⁻¹ 或 ms⁻²。

Mass is a scalar; weight is a vector pointing toward the centre of the Earth.

质量是标量；重量是指向地心的矢量。
Weight changes with location, but mass does not.

重量随位置而变化，但质量不变。

7. Momentum and Impulse | 动量与冲量

Momentum is a vector quantity defined as the product of mass and velocity. Its SI unit is kg ms⁻¹, which is equivalent to Ns.

动量是定义为质量与速度乘积的矢量。其 SI 单位是 kg ms⁻¹，等价于 Ns。

p = mv

Impulse is the product of force and the time interval over which it acts. Its unit is the newton-second (Ns), and it equals the change in momentum.

冲量是力与作用时间间隔的乘积。其单位是牛顿秒 (Ns)，等于动量的变化量。

Impulse = F × Δt = Δp

In collisions, total momentum is conserved in the absence of external forces. This principle is known as conservation of momentum and is fundamental to solving collision problems.

在碰撞中，若没有外力作用，总动量守恒。这一原理称为动量守恒定律，是解决碰撞问题的基础。

8. Work, Energy and Power | 功、能量与功率

Work done by a constant force is the product of the force and the displacement in the direction of the force. Its unit is the joule (J), where 1 J = 1 Nm.

恒力所做的功是力与沿力方向位移的乘积。其单位是焦耳 (J)，其中 1 J = 1 Nm。

W = F × s × cos θ

Energy is the capacity to do work. Mechanical energy exists in two main forms: kinetic energy (due to motion) and potential energy (due to position). Both are measured in joules.

能量是做功的能力。机械能以两种主要形式存在：动能（由于运动）和势能（由于位置）。两者都以焦耳为单位。

Kinetic Energy Eₖ = ½mv²

Gravitational Potential Energy Eₚ = mgh

Power is the rate at which work is done or energy is transferred. Its unit is the watt (W), where 1 W = 1 Js⁻¹. Power is a scalar quantity.

功率是做功或能量传递的速率。其单位是瓦特 (W)，其中 1 W = 1 Js⁻¹。功率是标量。

P = W / t = F × v

9. Common Unit Conversions | 常见单位换算

Examiners expect you to convert non-SI units into SI units before calculations. The most frequent conversions in mechanics are:

考官期望你在计算前将非 SI 单位转换为 SI 单位。力学中最常见的换算包括：

Given	Convert to	Multiply by
1 km	m	1000
1 hour	seconds	3600
1 g	kg	0.001
1 tonne	kg	1000
1 cm²	m²	1 × 10⁻⁴
1 cm³	m³	1 × 10⁻⁶

Pay special attention to squared and cubed conversions. A common error is to treat 1 cm² as 0.01 m², but it is actually 0.0001 m².

要特别注意平方和立方换算。常见错误是将 1 cm² 视为 0.01 m²，但实际应为 0.0001 m²。

10. Checking Dimensional Consistency | 检查量纲一致性

Before you accept any derived formula, check that both sides have identical dimensions. If they do not, the equation is not valid.

在接受任何推导公式之前，检查两边是否具有相同量纲。若不一致，该方程无效。

For example, consider the equation s = ut + ½at². Each term must have the dimension of length L:

例如，考虑方程 s = ut + ½at²。每一项必须具有长度 L 的量纲：

s has dimension L.

s 的量纲为 L。
ut has dimensions (LT⁻¹)(T) = L.

ut 的量纲为 (LT⁻¹)(T) = L。
½at² has dimensions (LT⁻²)(T²) = L.

½at² 的量纲为 (LT⁻²)(T²) = L。

Since all terms have dimension L, the equation is dimensionally consistent. This method allows you to verify unfamiliar formulas quickly.

由于所有项的量纲均为 L，该方程在量纲上是一致的。此方法可帮助你快速验证不熟悉的公式。

11. Units in Vector Resolution | 矢量分解中的单位

When resolving vectors into perpendicular components, each component retains the same unit as the original vector. For example, a force of 10 N at 30° above the horizontal has horizontal component 10 cos 30° N and vertical component 10 sin 30° N.

当将矢量分解为垂直分量时，每个分量保持与原矢量相同的单位。例如，一个 10 N 的力与水平方向成 30°，其水平分量为 10 cos 30° N，垂直分量为 10 sin 30° N。

Angles are dimensionless in the sense that they do not affect the units of a physical quantity. The trigonometric functions cos θ and sin θ are dimensionless ratios.

角度在某种意义上是无量纲的，它们不影响物理量的单位。三角函数 cos θ 和 sin θ 是无量纲的比值。

When writing final answers, include the correct unit. A numerical value without a unit is considered incomplete in an examination.

写出最终答案时，必须包含正确的单位。没有单位的数值在考试中被视为不完整。

12. Common Mistakes and Exam Tips | 常见错误与考试提示

Many students lose marks in mechanics not because of weak mathematics, but because of unit errors and sign errors. Here are the most common pitfalls to avoid.

许多学生在力学中丢分并非因为数学能力弱，而是因为单位错误和符号错误。以下是需要避免的最常见陷阱。

Always convert minutes to seconds, kilometres to metres, and grams to kilograms before substitution.

代入前务必把分钟转换为秒、公里转换为米、克转换为千克。
Do not confuse mass with weight. Weight is a force measured in newtons; mass is measured in kilograms.

不要混淆质量与重量。重量是力，以牛顿为单位；质量以千克为单位。
Remember that suvat equations apply only to constant acceleration, not to variable acceleration.

请记住，suvat 方程只适用于匀加速运动，不适用于变加速运动。
When using g = 9.8 ms⁻², check whether the question asks for the value to be taken as 10 ms⁻².

使用 g = 9.8 ms⁻² 时，注意题目是否要求取 10 ms⁻²。
Write down the unit at every stage of your working, not only in the final answer.

在计算过程的每一步都写下单位，而不仅在最终答案中。
Pay attention to significant figures. A-Level mark schemes often require answers to 3 significant figures unless stated otherwise.

注意有效数字。A-Level 评分标准通常要求答案保留 3 位有效数字，除非另有说明。

Finally, practise converting compound units such as kmh⁻¹ to ms⁻¹ by dividing by 3.6. This single conversion appears repeatedly in kinematics questions.

最后，练习复合单位换算，例如将 kmh⁻¹ 转换为 ms⁻¹ 时除以 3.6。这一换算在运动学问题中反复出现。

13. Summary of Key Units | 关键单位总结

The table below summarises the most important mechanical quantities, their symbols, SI units, and vector or scalar nature.

下表总结了最重要的力学量、符号、SI 单位以及矢量或标量性质。

Quantity	Symbol	SI Unit	Scalar / Vector
Mass	m	kg	Scalar
Displacement	s	m	Vector
Velocity	v	ms⁻¹	Vector
Acceleration	a	ms⁻²	Vector
Force	F	N	Vector
Weight	W	N	Vector
Momentum	p	kg ms⁻¹	Vector
Impulse	J or I	Ns	Vector
Work	W	J	Scalar
Energy	E	J	Scalar
Power	P	W	Scalar

Mastering units is not just about memorising symbols. It is about understanding the relationships between physical quantities, so that equations make sense and results can be trusted.

掌握单位不仅是记忆符号，更是理解物理量之间的关系，从而使方程有意义、结果可信。

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Physical Quantities and Units in Mechanics | 力学中的量与单位