📚 Work and Energy | 功与能量 考点精讲
In the study of mechanics, the concepts of work, energy, and power are fundamental to explaining how forces cause motion and how systems evolve. Whether you are analysing a falling apple, a stretched spring, or an accelerating car, the principles of energy conservation and work-energy transfer provide a powerful and universal framework. This revision guide unpacks every essential idea needed for CCEA Physics, blending precise definitions with worked examples and exam-focused insights.
在力学学习中，功、能量和功率是解释力如何引起运动以及系统如何演化的基础概念。无论是分析下落的苹果、被拉伸的弹簧还是加速的汽车，能量守恒和功能转换原理都提供了一个强大而普适的框架。这份复习指南将为你拆解 CCEA 物理考试所需的每一个核心概念，结合精准的定义、例题和应试要点。
1. Definition of Work | 功的定义
In physics, work is done when a force moves its point of application in the direction of the force. The symbol for work is W and it is measured in joules (J). One joule is the work done when a force of one newton moves through a distance of one metre in the direction of the force. Importantly, if the force is perpendicular to the displacement, no work is done. For example, the tension in the string of a conical pendulum does no work on the bob because the force is always at right angles to the motion.
在物理学中，当力使其作用点沿力的方向移动时，就说力做了功。功用符号 W 表示，单位为焦耳 (J)。1 焦耳等于 1 牛的力使物体沿力的方向移动 1 米所做的功。很重要的一点是，如果力的方向与位移方向垂直，则不做功。例如，圆锥摆中绳子的拉力对摆锤不做功，因为拉力始终与运动方向垂直。
Work is a scalar quantity even though it depends on two vector quantities – force and displacement. It can be positive, negative, or zero. Positive work is done when the force assists the motion (e.g. lifting a book); negative work is done when the force opposes the motion (e.g. friction slowing a block).
功是标量，尽管它依赖于两个矢量——力和位移。功可以为正、为负或为零。当力促进运动时做正功（如举起一本书）；当力阻碍运动时做负功（如摩擦力使物块减速）。
2. Work Done by a Constant Force | 恒力做功
When a constant force F acts on a body and causes a displacement s, the work done is given by:
W = F s cosθ
the displacement, cosθ = 1 and W = F s. If the force is perpendicular, cosθ = 0 and W = 0.
当恒力 F 作用在物体上并产生位移 s 时，做功为：
W = F s cosθ
其中 θ 是力矢量与位移矢量之间的夹角。若力与位移平行，cosθ = 1，W = F s。若力与位移垂直，cosθ = 0，W = 0。
In many CCEA problems, you will resolve forces to find the component acting in the direction of motion. For example, a person pulling a sledge with a rope at 30° to the horizontal does work only through the horizontal component of the tension. You must be careful to use the correct angle and to recognise that the displacement is always measured along the path actually moved.
在许多 CCEA 试题中，你需要对力进行分解，找出沿运动方向的分量。例如，一个人用与水平方向成 30° 的绳子拉雪橇，做功的只有拉力的水平分量。你必须注意使用正确的角度，并明确位移总是沿实际移动路径测量。
3. Work Done by a Varying Force | 变力做功
ed force is not constant, the work done can be found by calculating the area under a force– rce × distance and for a varying force the total work is the sum of many tiny increments F ∆ curve and the displacement axis. A common case is the extension of a spring obeying Hooke' ly from zero to kx, so the work done in stretching the spring by an extension e is ½ k e².
当外力不恒定时，可以通过计算力–位移图下的面积来求功。这是因为功 = 力 × 位移，而对于变力，总功是许多微小增量 F ∆x 的总和。从图形上看，这个总和等于力曲线与位移轴之间的面积。一个常见情景是遵循胡克定律的弹簧伸长，力从零线性增大到 kx，因此将弹簧拉伸 e 长度所做的功为 ½ k e²。
You may be required to estimate the area by counting squares or using geometric methods for a given graph. This concept directly connects to the work done in compressing a gas or deforming a material before the elastic limit. Always check the units: area under a force (N) vs. displacement (m) graph yields energy in joules.
你可能会被要求通过数格或几何方法估算给定图线下的面积。这一概念直接与压缩气体或材料弹性极限前变形所做的功相联系。务必检查单位：力 (N)–位移 (m) 图下的面积得出的能量单位为焦耳。
4. Kinetic Energy | 动能
Kinetic energy (KE or Ek) is the energy a body possesses due to its motion. For a body of mass m moving with speed v, the kinetic energy is:
Ek = ½ m v²
This is a scalar quantity measured in joules. The derivation comes from the work– mass from rest to speed v over a distance s, then F s = ½ m v² (assuming no other forces).
动能 (KE 或 Ek) 是物体由于运动而具有的能量。对于质量为 m、运动速度为 v 的物体，动能为：
Ek = ½ m v²
这是一个标量，单位为焦耳。其推导来自于功能原理：物体所受合外力做的功等于其动能的变化量。如果一恒力将质量为 m 的物体从静止加速到速度 v，移动了距离 s，则有 F s = ½ m v²（假设无其他力）。
Kinetic energy depends on the square of speed, so doubling the speed quadruples the kinetic energy. This is crucial in road safety: a car travelling at 100 km/h has four times the kinetic energy of the same car at 50 km/h, leading to much larger stopping distances. Also, note that the mass must be in kilograms and speed in m/s to obtain energy in joules.
动能与速度的平方成正比，因此速度加倍会使动能变为原来的四倍。这在道路安全中至关重要：一辆以 100 km/h 行驶的汽车具有的动能是同一辆车以 50 km/h 行驶时的四倍，导致制动距离大大增加。此外，注意质量必须用千克、速度用米/秒，才能得到焦耳为单位的能量。
5. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE or Ep) is the energy stored in an object due to its position in a gravitational field. Near the Earth's surface, where the gravitational field strength g is approximately 9.81 N/kg, the change in GPE when an object of mass m is raised through a vertical height ∆h is:
∆Ep = m g ∆h
The zero of GPE is arbitrary – you can choose any convenient reference level. Only changes in GPE are physically meaningful. For an object lifted at constant speed, the lifting force does work equal to the gain in GPE; if it is allowed to fall, the lost GPE is converted to kinetic energy.
重力势能 (GPE 或 Ep) 是物体由于在引力场中的位置而储存的能量。在地球表面附近，重力场强度 g 约为 9.81 N/kg，当质量为 m 的物体提升竖直高度 ∆h 时，重力势能的变化量为：
∆Ep = m g ∆h
重力势能的零点可以任意选取——你可以选择任何方便的参考面。只有重力势能的变化量才具有物理意义。对于以恒定速度提升的物体，提升力做的功等于增加的重力势能；如果让物体自由下落，减少的重力势能转化为动能。
In exam questions, you often use the conservation of mechanical energy to relate speeds and heights for objects sliding down frictionless slopes or swinging on pendulums. Remember that g can be taken as 9.8 m/s² or 9.81 m/s² as specified in the data booklet, and the vertical height must be measured perpendicular to the ground, not the slant distance.
在考试题目中，你经常会应用机械能守恒将无摩擦斜面滑下或摆动的物体的速度与高度联系起来。记住重力加速度 g 可根据数据手册取 9.8 m/s² 或 9.81 m/s²，并且竖直高度必须垂直于地面测量，而不是斜面长度。
6. Elastic Potential Energy | 弹性势能
obeys Hooke’s law, the force F is proportional to the extension x: F = k x, where k is the spring constant (N/m). The work done to stretch the spring from 0 to extension e is equal to the area under the F–x graph, which is a triangle. Therefore, the elastic potential energy stored is:
Ee = ½ k e²
This formula assumes the spring is not strained beyond its elastic limit. If a spring is compressed rather than extended, the same expression applies, with e being the compression distance. The energy stored can be released to do work, for example in a catapult or a mechanical clock.
弹性势能是储存在发生弹性形变的物体（如被拉伸的弹簧或压缩的橡皮筋）中的能量。对于服从胡克定律的弹簧，力 F 与伸长量 x 成正比：F = k x，其中 k 为劲度系数（N/m）。将弹簧从原长拉伸到伸长量 e 所做的功等于 F–x 图下的面积，这是一个三角形。因此，储存的弹性势能为：
Ee = ½ k e²
该公式假设弹簧未超过弹性极限。如果是压缩弹簧，同样适用，此时 e 为压缩距离。储存的能量可以释放用来做功，例如在弹弓或机械钟表中。
Elastic potential energy is a specific type of potential energy and is included in the total mechanical energy when dealing with oscillating spring-mass systems.
你必须能够根据 W = 平均力 × 距离推导出该结果，其中平均力为 (0 + F)/2 = ½ k e。弹性势能是一种特殊的势能，在研究弹簧振子系统时，会包含在总机械能中。
7. Work-Energy Theorem | 功能原理
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:
Wnet = ∆Ek = ½ m v² − ½ m u²
This is one of the most powerful tools in mechanics because it links forces and motion without requiring detailed knowledge of acceleration or time. The net work is the algebraic sum of the work done by all forces acting on the object, including friction, applied forces, and weight. If the net work is positive, the kinetic energy increases; if negative, it decreases.
功能原理指出，作用在物体上的合外力所做的功等于物体动能的变化量：
Wnet = ∆Ek = ½ m v² − ½ m u²
这是力学中最强大的工具之一，因为它将力与运动联系起来，而无需详细了解加速度或时间。合外力所做功是所有作用在物体上的力（包括摩擦力、外加力、重力）做功的代数和。如果合外力做正功，动能增加；做负功，则动能减少。
In many CCEA problems, you apply this theorem by first drawing a free-body diagram, calculating the work done by each force over a given displacement, summing them, and then equating to the change in kinetic energy. This is especially useful on rough slopes where gravitational work, frictional work, and possibly applied work all contribute.
在许多 CCEA 题目中，你应用这一定理时首先要画受力分析图，计算每个力在给定位移上所做的功，求和，然后使其等于动能的变化量。这在存在粗糙斜面的情境中尤其有用，此时重力做功、摩擦力做功、可能还有外加力做功共同起作用。
8. Conservation of Mechanical Energy | 机械能守恒
In a closed system where only conservative forces (such as gravity or elastic spring forces) do work, the total mechanical energy (sum of kinetic and potential energies) remains constant:
Ek + Ep = constant
the surroundings by resistive forces like friction or air resistance. For a falling object, ½ m v² + m g h = constant; for a simple pendulum, at the highest point energy is all GPE, at the lowest point it is all KE.
在一个只有保守力（如重力或弹簧弹力）做功的封闭系统中，总机械能（动能与势能之和）保持不变：
Ek + Ep = 常数
这意味着能量可以在动能和势能之间相互转化，但总和不变，前提是没有因摩擦力或空气阻力等耗散力将能量传递到周围环境。对于自由下落的物体，½ m v² + m g h = 常数；对于单摆，在最高点能量全部为重力势能，在最低点全部为动能。
When non-conservative forces like friction are present, mechanical energy is not conserved; instead, some mechanical energy is converted into thermal energy (heat). In such cases you must account for the work done against friction, which equals the decrease in mechanical energy. Many problems ask for the frictional force or the energy dissipated as heat.
当存在摩擦力等非保守力时，机械能不守恒；一部分机械能会转化为热能。这类情况下，你必须考虑克服摩擦力所做的功，它等于机械能的减少量。许多题目会要求计算摩擦力或耗散为热能的能量。
9. Power | 功率
Power is the rate at which work is done or energy is transferred. It is a scalar quantity measured in watts (W), where one watt equals one joule per second. The average power is given by:
P = W / t or P = ∆E / t
For a constant force moving an object at constant speed v, the power can also be expressed as:
P = F v
ulating the power output of engines moving vehicles at maximum speed or the power needed to lift a load at a steady rate.
功率是做功或能量转化的快慢程度。它是一个标量，单位为瓦特 (W)，1 瓦特等于每秒 1 焦耳。平均功率由下式给出：
P = W / t 或 P = ∆E / t
对于以恒定速度 v 移动物体的恒力，功率也可表示为：
P = F v
该关系可由 P = Fs/t = Fv 导出。它对于计算发动机在最大速度下驱动车辆时的输出功率，或以恒定速率提升重物所需的功率特别有用。
When the force is not parallel to the velocity, you must use the component of force in the direction of motion. Also, for a car moving at constant speed, the engine power equals the work done per second against resistive forces. Remember to convert speeds to m/s and forces to newtons to yield power in watts.
当力与速度不平行时，必须使用力在运动方向上的分量。此外，对于匀速行驶的汽车，发动机功率等于每秒克服阻力所做的功。记住将速度换算为 m/s，力换算为牛顿，才能得出以瓦特为单位的功率。
10. Efficiency | 效率
Efficiency measures how much of the input energy or power is converted into useful output. It is defined as:
Efficiency = (useful output energy / total input energy) × 100%
Or, using power: Efficiency = (useful power output / total power input) × 100%. Efficiency is always less than 100% due to energy losses (mainly as heat) from friction, sound, vibration, etc. In mechanics, a common question involves calculating the efficiency of an electrical motor lifting a weight: the input is the electrical energy or power, the useful output is the gain in gravitational potential energy per second.
效率衡量输入的能量或功率中有多少被转化为有用的输出。其定义为：
效率 = (有用的输出能量 / 总输入能量) × 100%
或者使用功率：效率 = (有用的输出功率 / 总输入功率) × 100%。由于摩擦、声音、振动等原因，总会有能量损失（主要为热能），因此效率始终小于 100%。在力学中，常见题目涉及计算电动机提升重物时的效率：输入为电能或电功率，有用输出为每秒增加的重力势能。
When solving efficiency problems, clearly identify the energy or power forms involved and ensure all values are in the same units (joules or watts). The difference between input and useful output represents the energy wasted, often dissipated as heat in the surroundings.
在解答效率问题时，要明确涉及的每一种能量或功率形式，并确保所有数值单位一致（焦耳或瓦特）。输入与有用输出的差值代表浪费的能量，通常以热的形式耗散到环境中。
11. Summary of Key Formulas | 核心公式汇总
Here is a concise table of the essential equations you must be able to recall and apply in a CCEA Physics examination. Being fluent with these formulas will save vital time and help you identify the correct principle to use in each problem.
下表简要列出了在 CCEA 物理考试中必须熟记并能应用的核心公式。熟练运用这些公式将为你节省宝贵的时间，并帮助你准确判断每道题适用的原理。
| Quantity | Formula | Notes |
|---|---|---|
| Work (constant force) | W = F s cosθ | θ between F and s |
| Kinetic energy | Ek = ½ m v² | Scalar, always positive |
| Change in gravitational PE | ∆Ep = m g ∆h | ∆h is vertical height |
| Elastic PE (spring) | Ee = ½ k e² | Hookean spring |
| Work–energy theorem | Wnet = ½ m v² − ½ m u² | Net work = ΔKE |
| Power | P = W/t = F v | v as constant speed |
| Efficiency | η = (useful output/total input)×100% | No unit, expressed as % |
Remember: always check that your working uses SI units – metres, kilograms, seconds, newtons – to ensure your final energy is in joules and power in watts. The data booklet may contain some of these, but you should be able to write them from memory and know exactly what each symbol represents.
记住：解题时要始终检查是否使用了国际单位制——米、千克、秒、牛顿——以确保最终能量单位为焦耳，功率单位为瓦特。公式手册可能包含其中一部分，但你应该能够凭记忆写出它们并清楚每个符号代表什么。
12. Exam Tips for Work and Energy | 功与能量应试技巧
Mastering the topic of work and energy is not only about memorising formulas; it is about recognising which principle applies to a given scenario and carrying out accurate calculations under time pressure. Here are some targeted strategies that consistently help students score higher marks:
- Identify the system and forces: Always begin by clarifying what objects are included and whether external forces do work. Draw a labelled diagram.
- Decide your approach: If the situation has no friction, use conservation of mechanical energy. If friction is present, apply the work–energy theorem or account for energy dissipated.
- Use the dot product property: For work calculation, check that you are using the component of force along the displacement. Many students lose marks by omitting cosθ.
- Convert units early: Speeds in km/h must be converted to m/s (÷3.6); masses in grams to kg (÷1000); heights in cm to m. Work in SI throughout.
- Check the zero of potential energy: When using conservation of energy, clearly state the reference level for GPE. Only differences matter, so a consistent reference removes confusion.
- Area under graphs: Practice counting squares and using trapezium areas for non-linear force–displacement graphs. Show your workings for area clearly.
- Sign convention: Work done by a force can be negative. Use it consistently – for instance, friction does negative work because it opposes motion.
- Power questions: icle is moving at maximum speed, the driving force equals total resistive force, and P = Fresistive × v.
掌握功与能量专题不仅需要记住公式，还要能在给定情境中判断适用哪种原理，并在时间压力下准确计算。以下是一些能够持续帮助考生提高成绩的针对性策略：
- 明确系统与力：始终首先明确包含哪些物体，以及是否有外力做功。画出带标注的示意图。
- 选择方法：如果情境中无摩擦，使用机械能守恒。如果有摩擦，则应用功能原理或考虑耗散的能量。
- 利用点积特性：计算功时，确保使用的是力沿位移方向的分量。许多学生因遗漏 cosθ 而失分。
- 尽早转换单位：速度 km/h 需换算为 m/s（除以 3.6）；质量 g 换算为 kg（除以 1000）；高度 cm 换算为 m。全程使用国际单位。
- 确认势能零点：使用能量守恒时，明确说明重力势能的参考水平面。只有差值才重要，因此一致的参考可以消除混淆。
- 图线下的面积：练习对非线性力–位移图线数格和梯形面积计算。清晰地展示面积计算过程。
- 符号约定：力做的功可以为负。请前后一致地使用——例如，摩擦力做负功，因为它阻碍运动。
- 功率问题：如果车辆以最大速度行驶，驱动力等于总阻力，且 P = F阻力 × v。
Finally, always read the question carefully to determine whether it asks for work done by a specific force or net work done on an object. Practising past CCEA papers will build your confidence in choosing the most efficient pathway through each problem.
最后，务必仔细审题，明确是求某个特定力做的功还是作用在物体上的合外力做功。通过反复练习 CCEA 历年真题，你将能够自信地选择解决每个问题的最有效路径。
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