📚 Work and Energy | 功和能量考点精讲
In A-Level WJEC Mathematics, particularly within the Mechanics modules, the principles of work and energy provide powerful tools for analysing motion without relying solely on kinematic equations. Mastering these concepts allows you to solve problems involving forces, velocities, and displacements more efficiently. This revision guide covers every key definition, essential formula, and typical exam question type you will encounter.
在 A-Level WJEC 数学的力学模块中，功和能量的原理为我们提供了强有力的分析工具，使解题不再仅仅依赖运动学公式。掌握这些概念能让你更高效地处理涉及力、速度和位移的问题。本考点精讲涵盖了你将遇到的所有关键定义、核心公式和典型考试题型。
1. Definition of Work | 功的定义
In mechanics, work is done when a force moves an object through a displacement. The amount of work done depends on the magnitude of the force, the distance moved, and the angle between the force and the direction of motion. Work is a scalar quantity measured in joules (J).
在力学中，当力使物体移动一段位移时，力就对物体做了功。功的大小取决于力的大小、移动的距离以及力与运动方向之间的夹角。功是标量，单位为焦耳 (J)。
2. Work Done by a Constant Force | 恒力做的功
When a constant force F acts on an object that moves a distance s with an angle θ between the force and the displacement, the work done is given by:
当一个恒力 F 作用于物体，物体移动距离 s，且力与位移之间的夹角为 θ 时，做功的计算公式为：
W = F s cos θ
to W = F s. If the force is perpendicular (θ = 90°), no work is done because cos 90° = 0.
如果力与位移平行 (θ = 0)，公式简化为 W = F s。如果力与位移垂直 (θ = 90°)，则不做功，因为 cos 90° = 0。
The component of force in the direction of motion is the one that does work. Always resolve forces carefully when dealing with inclined planes or angled pulls.
只有沿运动方向的分力才会做功。在处理斜面或斜向拉力时，务必仔细分解力。
3. Kinetic Energy | 动能
Kinetic energy (K.E.) is the energy an object possesses due to its motion. For an object of mass m moving with speed v, the kinetic energy is:
动能 (K.E.) 是物体由于运动而具有的能量。对于质量为 m、速度为 v 的物体，其动能为：
K.E. = ½ m v²
Kinetic energy is always non-negative and is measured in joules. The change in kinetic energy is central to the work-energy principle.
动能总是非负的，单位为焦耳。动能的变化是功-能原理的核心内容。
4. Gravitational Potential Energy | 重力势能
Gravitational potential energy (G.P.E.) is the energy stored in an object due to its height above a reference level. Near the Earth's surface, it is calculated as:
重力势能 (G.P.E.) 是物体由于距离参考平面的高度而储存的能量。在地球表面附近，其计算公式为：
G.P.E. = m g h
where m is the mass, g is the acceleration due to gravity (9.8 m s⁻² unless specified otherwise), and h is the vertical height. A common error is using the slant distance along a slope instead of the vertical height.
其中 m 为质量，g 为重力加速度 (除非另有说明，一般取 9.8 m s⁻²)，h 为竖直高度。常见的错误是使用沿斜面的距离而不是竖直高度来计算。
5. The Work-Energy Principle | 功-能原理
The work-energy principle states that the total work done by all forces acting on an object equals the change in the object's kinetic energy:
功-能原理表明，作用在物体上的所有力所做的总功等于物体动能的变化量：
Total Work Done = Δ K.E. = ½ m v² - ½ m u²
This principle is extremely useful when forces vary or when it is difficult to use equations of motion. It accounts for both driving forces and resistive forces like friction.
当力发生变化或很难用运动学方程求解时，这一原理格外有用。它同时考虑了驱动力和摩擦力等阻力所做的功。
6. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces (such as gravity) do work, the total mechanical energy of the system remains constant. This means the sum of kinetic and potential energy is conserved:
当只有保守力 (如重力) 做功时，系统的总机械能保持不变。这意味着动能与势能之和守恒：
Initial K.E. + Initial G.P.E. = Final K.E. + Final G.P.E.
If non-conservative forces like friction are present, mechanical energy is not conserved, and the work done against friction must be included as a loss.
如果存在摩擦力等非保守力，机械能就不再守恒，这时需要将克服摩擦力所做的功作为能量损失计入。
7. Power | 功率
Power is the rate at which work is done or energy is transferred. The average power P is given by:
功率是做功或能量传递的速率。平均功率 P 的计算公式为：
P = W / t
where W is the work done and t is the time taken. Power is measured in watts (W). For an object moving at constant speed v under a driving force F, power can also be expressed as:
其中 W 是所做的功，t 是所用时间。功率的单位是瓦特 (W)。对于在驱动力 F 作用下以恒定速度 v 运动的物体，功率还可以表示为：
P = F v
This relationship is particularly common in questions about vehicles moving against resistances.
在涉及车辆克服阻力行驶的题目中，这一关系式尤为常见。
8. Work Done by a Variable Force | 变力做功
If the force is not constant, the work done can be found by calculating the area under a force-displacement graph. In WJEC Mathematics Mechanics, you may also encounter integration: if the force F(x) varies with position, then
如果力不是恒定的，可以通过计算力—位移图像下的面积来求功。在 WJEC 数学力学部分，你还可能遇到积分方法：如果力 F(x) 随位置变化，那么
W = ∫ F(x) dx
from the initial to the final position. Mastery of this technique links Mechanics directly to your Pure Mathematics skills.
积分限从初始位置到最终位置。掌握这一技巧将力学与你的纯数知识直接联系起来。
9. Energy and Inclined Planes | 能量与斜面
Problems on inclined planes often combine gravitational potential energy, work against friction, and kinetic energy. A systematic approach is to write an energy equation:
斜面问题常常综合了重力势能、克服摩擦力做功以及动能。系统的解法是列出能量方程：
Change in K.E. + Change in G.P.E. = Work Done by External Forces – Work Done Against Friction
Remember that the change in G.P.E. depends only on the vertical rise or fall, not on the length of the slope.
请记住，重力势能的变化只取决于竖直高度的升降，而与斜面长度无关。
10. Common Exam Questions and Solutions | 常见考题与解答
Typical WJEC questions ask you to find the final speed of a block sliding down a rough incline, the power developed by a car engine, or the work done by a tension force. Always start by identifying known energies and the work done by each force.
典型的 WJEC 考题会要求你计算滑块沿粗糙斜面滑下的末速度、汽车发动机产生的功率，或者拉力所做的功。解题时始终从识别已知能量和每个力所做的功入手。
- Example 1: A particle of mass 2 kg is pulled 5 m along a horizontal floor with a force of 10 N at 30° to the horizontal. Friction is 2 N. Find the increase in kinetic energy.
Solution: done against friction = 2 × 5 = 10 J. Net work = 33.3 J. This equals the increase in K.E. - 例1：质量为 2 kg 的物体在水平地面上被 10 N 的力拉动 5 m，该力与水平面成 30° 角。摩擦力为 2 N。求动能的增加量。
解：拉力分力做的功 = 10 cos30° × 5 = 43.3 J。克服摩擦做的功 = 2 × 5 = 10 J。净功 = 33.3 J，即等于动能的增加量。
- Example 2: A car of mass 800 kg travels up a slope inclined at sin⁻¹(0.05) at a constant speed of 15 m s⁻¹. Resistive forces total 300 N. Find the engine power.
Solution: esistance = 800 × 9.8 × 0.05 + 300 = 392 + 300 = 692 N. Power = F v = 692 × 15 = 10 380 W. - 例2：一辆质量为 800 kg 的汽车以 15 m s⁻¹ 的恒定速度驶上倾斜角为 sin⁻¹(0.05) 的斜坡。总阻力为 300 N。求发动机功率。
解：驱动力 = 重力沿斜面的分力 + 阻力 = 800 × 9.8 × 0.05 + 300 = 392 + 300 = 692 N。功率 = F v = 692 × 15 = 10 380 W。
11. Tips and Pitfalls | 技巧与易错点
e h is measured vertically. Do not confuse energy with force: many students mistakenly add mgh directly to a force equation. Pay close attention to units; speed must be in m s⁻¹ for K.E. calculations. If a problem involves a sudden change in motion, consider energy methods rather than equations of motion which require constant acceleration.
务必检查做功的力是恒力还是变力。使用 G.P.E. = mgh 时，要确保 h 是竖直测量值。不要混淆能量和力：很多学生错误地将 mgh 直接加在力的方程里。要特别注意单位；计算动能时速度必须以 m s⁻¹ 为单位。如果问题涉及运动的突变，请考虑使用能量方法，而不是要求加速度恒定的运动学公式。
12. Summary of Key Formulas | 关键公式总结
The following table summarises the essential formulas you must memorise for the WJEC Mechanics exam on work and energy:
下表总结了 WJEC 力学考试中关于功和能量你必须牢记的核心公式：
| Concept / 概念 | Formula / 公式 | Notes / 备注 |
|---|---|---|
| Work (constant force) | W = F s cos θ | θ is angle between F and s |
| Kinetic energy | K.E. = ½ m v² | Always in joules |
| Gravitational P.E. | G.P.E. = m g h | h is vertical height |
| Work-energy principle | ΣW = Δ K.E. | Net work = change in K.E. |
| Conservation of energy | K.E.ᵢ + P.E.ᵢ = K.E.բ + P.E.բ + W_friction | Include work against friction |
| Power | P = W/t = F v | v is speed in m s⁻¹ |
| Work (variable force) | W = ∫ F(x) dx | Area under F-x graph |
Regularly testing yourself on these formulas and understanding when to apply each one is the key to exam success.
经常自测这些公式并理解各自的适用条件，是取得考试成功的关键。
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