Work and Energy Key Points for CIE A-Level Mathematics | A-Level CIE 数学：功和能量考点精讲

📚 Work and Energy Key Points for CIE A-Level Mathematics | A-Level CIE 数学：功和能量考点精讲

Mastering work and energy is essential for solving mechanics problems in CIE A-Level Mathematics. This guide covers the core concepts, formulas, and solving techniques you need to confidently tackle exam questions on work done by forces, kinetic and potential energy, the work-energy principle, conservation of mechanical energy, and power. We break down each topic with clear explanations and practical examples.

掌握功和能量对于解决 CIE A-Level 数学中的力学问题至关重要。本指南涵盖了核心概念、公式和解题技巧，帮助你自信应对关于力做功、动能与势能、功能原理、机械能守恒以及功率的考试题目。我们对每个主题进行了清晰的解释并结合实例进行说明。

1. What is Work? | 什么是功？

In mechanics, work is done when a force moves its point of application in the direction of the force. For a constant force F applied at an angle θ to the direction of displacement s, the work done is W = F s cos θ. When the force is parallel to the displacement (θ = 0), this simplifies to W = F s. Work is a scalar quantity measured in joules (J).

在力学中，当一个力使其作用点在力的方向上移动时，该力就做了功。对于恒力 F 与位移 s 方向夹角为 θ 的情况，所做的功为 W = F s cos θ。当力与位移平行（θ = 0）时，可简化为 W = F s。功是标量，单位为焦耳 (J)。

If the force is not constant along the path, work is calculated by integration: W = ∫ F · ds. In A-Level problems, you will mostly encounter constant forces, weight, friction, tension, and occasionally variable forces given as functions of position, requiring basic integration.

如果沿路径力不是恒定的，功通过积分计算：W = ∫ F · ds。在 A-Level 题目中，你大多会遇到恒力、重力、摩擦力、拉力，偶尔也会遇到作为位置函数给出的变力，需要简单积分。

Remember that work can be positive, negative, or zero. Positive work means the force component is in the direction of motion (e.g., pulling a sledge forward). Negative work occurs when the force opposes motion (e.g., friction). Zero work happens when the force is perpendicular to displacement (e.g., normal reaction).

记住，功可以为正、负或零。正功表示力的分量与运动方向相同（例如，向前拉雪橇）。当力阻碍运动时做负功（例如，摩擦力）。当力垂直于位移时做功为零（例如，法向反作用力）。

2. Kinetic Energy (KE) | 动能 (KE)

Kinetic energy is the energy an object has due to its motion. For a particle of mass m moving with speed v, kinetic energy is given by KE = ½ m v². Kinetic energy is also measured in joules and is always non-negative.

动能是物体因运动而具有的能量。对于质量为 m、以速度 v 运动的粒子，动能为 KE = ½ m v²。动能也以焦耳为单位，且始终为非负值。

The change in kinetic energy is a central part of the work-energy principle. When calculating KE changes, remember that the speed is squared: a small increase in speed can produce a large increase in kinetic energy. Also, KE depends only on instantaneous speed, not on direction of motion.

动能的变化是功能原理的核心部分。在计算动能变化时，记住速度是平方项：速度的小幅增加会导致动能大幅增加。此外，动能仅取决于瞬时速率，与运动方向无关。

In many exam problems, you will be asked to find the final speed of an object after several forces have acted on it. Fairly straightforward, but be careful with units: mass in kg and speed in m/s.

在许多考试题目中，你将被要求求出一个物体在受多个力作用后的最终速率。这比较直接，但要注意单位：质量用 kg，速率用 m/s。

3. Gravitational Potential Energy (GPE) | 重力势能 (GPE)

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. Near the Earth’s surface, the change in GPE when an object of mass m is raised through a vertical height h is ΔGPE = m g h. The zero level of GPE can be chosen arbitrarily – only changes in GPE matter.

重力势能是物体因在重力场中的位置而具有的能量。在地球表面附近，当质量为 m 的物体被提升垂直高度 h 时，重力势能的变化量为 ΔGPE = m g h。GPE 的零势能面可以任意选择——只有 GPE 的变化才是重要的。

In A-Level CIE Mechanics, it is common to consider downward motion losing GPE and upward motion gaining GPE. Take care with signs: when an object falls, GPE decreases, and this loss often converts to kinetic energy.

在 A-Level CIE 力学中，通常考虑向下运动失去 GPE，向上运动获得 GPE。注意符号：物体下落时，GPE 减少，这部分损失通常转化为动能。

For objects moving on a slope, the change in vertical height is used, not the distance along the slope. If an object slides down a distance d along a slope inclined at angle α, the height lost is d sin α. The change in GPE is then m g d sin α.

对于在斜面上运动的物体，使用的是垂直高度的变化，而不是沿斜面的距离。如果物体沿倾角为 α 的斜面滑下距离 d，则损失的高度为 d sin α。此时 GPE 的变化为 m g d sin α。

4. The Work-Energy Principle | 功-能原理

The work-energy principle states that the total work done by all forces acting on a particle (both conservative and non-conservative) equals the change in the particle’s kinetic energy: W_total = ΔKE = ½ m v² – ½ m u². This principle is often the quickest way to solve problems involving changes in speed without considering acceleration and time.

功-能原理指出，作用在粒子上所有力（保守力和非保守力）做的总功等于粒子动能的变化：W_total = ΔKE = ½ m v² – ½ m u²。这一原理通常是解决涉及速率变化但无需考虑加速度和时间的问题的最快方法。

You must include work done by all forces: driving forces, resistance, friction, weight component along the displacement, tension, etc. Remember that the normal reaction does no work if there is no motion perpendicular to the surface. When a force is applied via a rope or an engine, the work done is often given indirectly by power problems.

你必须包括所有力做的功：驱动力、阻力、摩擦力、沿位移方向的重力分量、张力等。记住，如果没有垂直于表面的运动，法向反作用力不做功。当力通过绳子或发动机施加时，功通常通过功率问题间接给出。

A typical exam question: “A car of mass 800 kg accelerates from 10 m/s to 20 m/s on a straight horizontal road. The driving force is constant at 2000 N and the resistance to motion is 400 N. Find the distance travelled.” You would set the net work (driving force minus resistance) × distance = change in KE.

一个典型的考试题目：“一辆质量为 800 kg 的汽车在笔直的水平公路上从 10 m/s 加速到 20 m/s。驱动力恒为 2000 N，运动阻力为 400 N。求行驶的距离。”你需要将净值功（驱动力减去阻力）乘以距离 = 动能的变化。

5. Conservative Forces and Mechanical Energy Conservation | 保守力与机械能守恒

A conservative force is one for which the work done is independent of the path taken, depending only on the initial and final positions. Weight and elastic spring force are conservative; friction and air resistance are non-conservative. For conservative forces, we can define a potential energy (gravitational or elastic).

保守力是指做功与路径无关、仅取决于初末位置的力。重力和弹簧弹力是保守力；摩擦力和空气阻力是非保守力。对于保守力，我们可以定义势能（重力势能或弹性势能）。

When only conservative forces do work, the total mechanical energy (KE + PE) remains constant: KE_initial + PE_initial = KE_final + PE_final. This is the principle of conservation of mechanical energy. It is extremely useful for problems like pendulums, projectiles (ignoring air resistance), and roller coasters.

当只有保守力做功时，总机械能（KE + PE）保持不变：KE_初 + PE_初 = KE_末 + PE_末。这就是机械能守恒定律。它对于摆、抛体（忽略空气阻力）和过山车等问题极为有用。

If non-conservative forces are present, the work done by these forces equals the change in mechanical energy: W_nc = Δ(KE + PE). This is essentially a restatement of the work-energy principle separating work done by conservative forces into potential energy changes.

如果存在非保守力，这些力做的功等于机械能的变化：W_nc = Δ(KE + PE)。这本质上是功能原理的另一种表述，将保守力做的功归入势能变化。

6. Power: The Rate of Doing Work | 功率：做功的速率

Power is the rate at which work is done. Average power = work done / time taken, or W / t. Instantaneous power for a constant force acting on an object moving with velocity v is given by P = F v, provided the force and velocity are in the same direction. The unit of power is the watt (W), where 1 W = 1 J/s.

功率是做功的速率。平均功率 = 做功 / 所用时间，即 W / t。对于作用在以速度 v 运动的物体上的恒力，如果力与速度同向，则瞬时功率为 P = F v。功率的单位是瓦特 (W)，1 W = 1 J/s。

In many vehicle-motion problems, the engine produces a constant power P, while the driving force F varies with speed. As the vehicle speeds up, the driving force decreases because F = P / v. The car reaches maximum speed when the net force is zero, i.e., driving force equals total resistance.

在许多车辆运动问题中，发动机产生恒定功率 P，而驱动力 F 随速度变化。当车辆加速时，驱动力会减小，因为 F = P / v。当净力为零，即驱动力等于总阻力时，汽车达到最大速度。

Typical problem: “A cyclist maintains a constant power of 400 W. The total resistance is 40 N. Find the maximum speed.” At max speed, driving force = resistance, so 400 / v_max = 40 → v_max = 10 m/s. Simple yet powerful application.

典型问题：“一名自行车骑手保持恒定功率 400 W。总阻力为 40 N。求最大速度。”在最大速度时，驱动力等于阻力，所以 400 / v_max = 40 → v_max = 10 m/s。简单但实用。

7. Work Done by a Variable Force | 变力做的功

When the force is not constant but varies as a function of displacement x, the work done from x = a to x = b is given by the definite integral: W = ∫ₐᵇ F(x) dx. Graphically, this is the area under the force-displacement curve. For example, the force exerted by a spring obeys Hooke’s law: F = kx, so the work done in stretching the spring from extension e₁ to e₂ is W = ∫ₑ₁ᵉ² kx dx = ½ k (e₂² – e₁²).

当力不是恒力而是随位移 x 变化时，从 x = a 到 x = b 做的功由定积分给出：W = ∫ₐᵇ F(x) dx。从图形上看，这是力-位移曲线下的面积。例如，弹簧施加的力遵循胡克定律：F = kx，因此将弹簧从伸长量 e₁ 拉伸到 e₂ 所做的功为 W = ∫ₑ₁ᵉ² kx dx = ½ k (e₂² – e₁²)。

Elastic potential energy (EPE) stored in a stretched or compressed spring is EPE = ½ k x², where x is the extension or compression from the natural length. This is a conservative potential energy, so the work done by the spring force can be treated as a change in EPE.

储存在被拉伸或压缩的弹簧中的弹性势能 (EPE) 为 EPE = ½ k x²，其中 x 是相对于原长的伸长量或压缩量。这是一种保守势能，因此弹簧力做的功可以视为 EPE 的变化。

You might also encounter resistance forces proportional to v² or functions like F = 1000 – 200x. Integration skills are tested modestly; remember to include appropriate limits and keep track of units.

你还可以会遇到与 v² 成比例的阻力，或像 F = 1000 – 200x 这样的函数。对积分技能的要求适中；记住代入适当的上下限并注意单位。

8. Energy Methods on Inclined Planes | 斜面上的能量方法

Inclined plane problems combine GPE, work against friction, and kinetic energy changes. When an object slides down a rough incline, the work-energy principle becomes: (m g sin θ – μ m g cos θ) × distance along plane = ΔKE. Alternatively, energy conservation with friction: loss in GPE = gain in KE + work done against friction.

斜面问题结合了 GPE、克服摩擦力做功以及动能的变化。当物体沿粗糙斜面下滑时，功能原理变为：(m g sin θ – μ m g cos θ) × 沿斜面的距离 = ΔKE。或者使用考虑摩擦的能量守恒：GPE 的损失 = KE 的增加 + 克服摩擦力做的功。

When a particle is projected up a rough slope, the kinetic energy is converted to GPE and work against friction. To find the distance travelled up the plane, you can set: initial KE = m g h + μ m g cos θ × d, where h = d sin θ. This avoids using equations of constant acceleration.

当一个粒子被向上发射到一个粗糙斜面上时，动能转化为 GPE 和克服摩擦力做的功。为了求出沿斜面向上运动的距离，你可以设定：初始 KE = m g h + μ m g cos θ × d，其中 h = d sin θ。这避免了使用匀加速运动方程。

Remember that the normal reaction R on an incline is mg cos θ, so friction is μ mg cos θ. Common exam traps: forgetting to account for the work done by the component of weight when using the work-energy principle, or using the wrong angle in height calculation.

记住，斜面上的法向反作用力 R 为 mg cos θ，因此摩擦力为 μ mg cos θ。常见的考试陷阱：在使用功-能原理时忘记考虑重力分量做的功，或在高度计算中使用错误的角度。

9. Connected Particles and Energy | 连接体与能量

In systems with two particles connected by a light inextensible string passing over a pulley, energy methods can simplify finding speed after one particle has descended a certain height. Assuming no friction at the pulley, the loss in GPE of the descending mass equals the gain in KE of both masses plus the gain in GPE of the ascending mass.

在通过轻质不可伸长绳子跨过滑轮连接的两个粒子系统中，能量方法可以简化在其中一个粒子下降一定高度后求速率的问题。假设滑轮处无摩擦，下降物体的 GPE 损失等于两个物体 KE 的增加加上上升物体 GPE 的增加。

Example: Two particles of masses 3 kg and 2 kg connected over a smooth pulley. Released from rest, the 3 kg mass falls 2 m. Find the speed. Loss in GPE = 3g×2, gain in KE = ½×3×v² + ½×2×v², gain in GPE of 2 kg mass = 2g×2. Equation: 3g×2 = (5/2)v² + 2g×2 → (5/2)v² = 2g → v = √(4g/5).

例如：质量分别为 3 kg 和 2 kg 的两个粒子通过光滑滑轮连接。静止释放，3 kg 物体下降 2 m。求速率。GPE 损失 = 3g×2，KE 增加 = ½×3×v² + ½×2×v²，2 kg 物体 GPE 增加 = 2g×2。方程：3g×2 = (5/2)v² + 2g×2 → (5/2)v² = 2g → v = √(4g/5)。

If the pulley is not smooth or the string has mass, the problem becomes more complex, but CIE A-Level normally assumes light strings and smooth pulleys, allowing mechanical energy conservation (if no external non-conservative forces other than tension, which does no net work).

如果滑轮不光滑或绳子有质量，问题会更复杂，但 CIE A-Level 通常假设轻绳和光滑滑轮，允许机械能守恒（如果没有除张力外的外部非保守力，而张力不做净功）。

10. Exam Tips for Work and Energy | 功和能量的考试技巧

First, draw a clear diagram and label all forces and displacements. Identify the initial and final states, choose a reference level for GPE, and decide whether to use work-energy principle or conservation of energy. If the problem asks for speed or distance, energy methods are often faster than equations of motion.

首先，画出清晰的图表并标注所有力和位移。确定初末状态，选择一个 GPE 参考水平面，并决定使用功能原理还是能量守恒。如果问题要求速度或距离，能量方法通常比运动学公式更快。

Check the signs of work done by each force. Work done by a force acting in the direction of motion is positive; work done against a force (e.g., pushing against resistance) is negative in the work-energy equation. When writing energy conservation with non-conservative forces, remember: initial total energy = final total energy + work done against friction (or – work done by friction).

检查各个力做功的符号。与运动方向一致的力做正功；克服一个力做的功（例如，克服阻力推动）在功能方程中为负值。在考虑非保守力的能量守恒时，记住：初始总能量 = 最终总能量 + 克服摩擦做的功（或 – 摩擦力做的功）。

When power is involved, often you need to find the driving force at a particular speed, then use F = ma or equilibrium to solve the problem. For vehicles moving up a hill, the driving force must overcome both resistance and the component of weight.

当涉及功率时，通常需要先求出某一速度下的驱动力，然后使用 F = ma 或平衡条件求解问题。对于上坡的车辆，驱动力必须同时克服阻力和重力的分量。

Finally, always watch units: mass in kg, distance in metres, speed in m/s, force in newtons, work/energy in joules, power in watts. Use g = 9.8 or 10 as specified in the question. Double-check your answers to see if the magnitude is reasonable: a car doing 10⁶ J of work over a few metres of acceleration is plausible; a particle’s speed exceeding 100 m/s might indicate an error.

最后，始终注意单位：质量用 kg，距离用米，速率用 m/s，力用牛顿，功/能用焦耳，功率用瓦特。使用题目指定的 g = 9.8 或 10。复查答案，看看数量级是否合理：一辆汽车在几米内加速执行 10⁶ J 的功是合理的；一个粒子的速度超过 100 m/s 可能表明有误。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Work and Energy Key Points for CIE A-Level Mathematics | A-Level CIE 数学：功和能量 考点精讲