Work and Energy | 功和能量考点精讲

📚 Work and Energy | 功和能量考点精讲

In IGCSE Edexcel Mathematics, the topic of work and energy extends beyond pure arithmetic and algebra into applied mechanics. It requires students to use given formulas to calculate work done, kinetic energy, gravitational potential energy and power. These concepts frequently appear in multi‑step word problems, testing both algebraic manipulation and unit conversion. A solid grasp of these relationships will help you tackle questions where physical scenarios model real‑world situations, and you must interpret the results in context.

在 IGCSE 爱德思数学中，功和能量不仅涉及纯数与代数，还延伸到应用力学。学生需要利用给定的公式计算做功、动能、重力势能和功率。这些概念常出现在多步骤应用题中，既考查代数运算，也考查单位换算。扎实掌握这些关系，有助于你在将物理情境建模为数学问题后，正确地解释计算结果。

1. Work Done in Mathematics | 数学中的功

Work done is defined as the product of the force applied and the distance moved in the direction of the force. The formula provided on the formula sheet is W = F × d, where W is work in joules (J), F is the constant force in newtons (N), and d is the distance in metres (m). In exam questions, the force may be given directly, or you may need to resolve a force into the direction of motion when the angle is involved, though angle resolution is more common in physics; in IGCSE Maths, most questions will assume the force acts along the line of movement.

功定义为施加的力与在力的方向上移动的距离的乘积。公式表上给出的公式是 W = F × d，其中 W 是功（单位焦耳 J），F 是恒力（牛顿 N），d 是距离（米 m）。在考试题中，力可能是直接给出的，或者当涉及角度时，你可能需要将力分解到运动方向上——不过角度分解在物理中更常见，在 IGCSE 数学中，大部分题目假设力沿着运动方向作用。

W = F × d

When using this formula, always check that distance is measured in metres. If a question gives centimetres or kilometres, convert to metres before substituting. Failure to convert units is a very common error, so underline the unit conversions in your working.

使用该公式时，务必检查距离的单位是否为米。如果题目给出厘米或千米，需先换算成米再代入。不进行单位换算是非常常见的错误，因此在解题过程中一定要在单位换算处做标记。

2. Units and Conversions | 单位与换算

Energy, work and power all rely on standard SI units. Work and energy are both measured in joules (J). Force is in newtons (N) and distance in metres (m). For kinetic energy, mass must be in kilograms (kg) and speed in metres per second (m/s). Gravitational potential energy requires mass in kg, gravitational field strength g in N/kg (on Earth, usually taken as 9.8 or 10), and height in metres. Power is measured in watts (W), where 1 W = 1 J/s. Time is always in seconds (s) when calculating power.

能量、功和功率都依赖标准国际单位。功和能量均以焦耳（J）为单位。力的单位是牛顿（N），距离是米（m）。动能计算中质量必须用千克（kg），速度用米/秒（m/s）。重力势能要求质量单位为 kg，重力场强度 g 为 N/kg（地球上通常取 9.8 或 10），高度单位为米。功率的单位是瓦特（W），1 W = 1 J/s。计算功率时时间总是以秒（s）为单位。

Some questions deliberately mix units, for example giving a mass in grams (g) and speed in km/h. In such cases, you must convert grams to kilograms by dividing by 1000, and km/h to m/s by dividing by 3.6. Show these conversion steps clearly to gain method marks.

有些题目会故意混合单位，例如质量以克（g）给出，速度以千米/小时（km/h）给出。此时，需将克除以 1000 换成千克，将 km/h 除以 3.6 换为 m/s。清楚地写出这些换算步骤，才能得到方法分。

3. Kinetic Energy Formula | 动能公式

Kinetic energy (KE) is the energy possessed by a moving object. The formula is KE = ½ m v², where m is mass and v is velocity (or speed, since direction does not affect KE). This is the exact formula provided, so you can directly substitute. Note that the v is squared; doubling the speed quadruples the kinetic energy. Make sure you square the speed before multiplying by mass and 0.5.

动能是运动物体具有的能量。公式为 KE = ½ m v²，其中 m 是质量，v 是速度（方向不影响动能大小，所以也可以用速率）。这是给出的精确公式，可以直接代入。注意速度要平方；速度加倍，动能变为四倍。务必先将速度平方，再乘以质量和 0.5。

KE = ½ m v²

When a question asks you to find the speed given the kinetic energy and mass, you will need to rearrange the formula. Multiply both sides by 2, divide by m, then take the square root: v = √(2KE/m). Exam questions often test this rearrangement, so practise it until it becomes automatic.

当题目给出动能和质量，要求速度时，你需要对公式变形。两边乘 2，除以 m，再开平方根：v = √(2KE/m)。考试中经常会考查这种变形，务必熟练到能自动反应。

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy stored due to an object’s height above a reference level. The formula is PE = m g h, where m is mass, g is the acceleration due to gravity (or gravitational field strength), and h is the vertical height. In IGCSE Maths, g is usually taken as 10 m/s² for simplicity, but sometimes 9.8 is used. Read the question carefully to see which value is specified.

重力势能是由于物体相对于参考水平面的高度而储存的能量。公式为 PE = m g h，其中 m 是质量，g 是重力加速度（或引力场强度），h 是垂直高度。在 IGCSE 数学中，g 常取 10 m/s² 以简化计算，但有时也用 9.8。要仔细读题，看题目指定了哪个数值。

PE = m × g × h

The height h must be the vertical height, not the slant distance along a slope. If a question describes an object sliding down a slope of length L inclined at an angle θ, you may need to use h = L sin θ. However, more commonly in Maths, the vertical height is given directly, so you can simply plug it into the formula.

高度 h 必须是垂直高度，而不是沿斜面的距离。如果题目描述一个物体沿长度为 L、倾角为 θ 的斜面下滑，可能需要用 h = L sin θ 计算高度。不过在数学卷中，更常见的是直接给出垂直高度，此时只需代入公式即可。

5. Principle of Conservation of Energy | 能量守恒原理

In a closed system with no external work done, the total energy remains constant. Energy can transform from one form to another, but the sum of kinetic energy, gravitational potential energy and any work done against friction remains the same. The typical exam scenario is a falling object or a pendulum: at the top, energy is all GPE; at the bottom, it is all KE, assuming no air resistance. This gives the equation m g h = ½ m v², and the mass cancels out, allowing you to find v.

在一个没有外部做功的封闭系统中，总能量保持不变。能量可以从一种形式转化为另一种，但动能、重力势能和克服摩擦所做的功的总和不变。典型的考试情景是自由落体或单摆：在最高点，能量全为重力势能；在最低点，如果没有空气阻力，则全为动能。由此可得到 m g h = ½ m v²，质量会约掉，从而求出速度 v。

m g h = ½ m v²

g h = ½ v² → v = √(2 g h)

If friction or air resistance is present, some energy is converted into heat or sound, so the mechanical energy is not conserved. In such questions, you might be asked to calculate the work done against friction by comparing the total mechanical energy before and after motion. The work done is the difference in total energy.

如果存在摩擦或空气阻力，部分能量会转化为热能或声能，因此机械能不守恒。在这类问题中，可能需要通过比较运动前后的总机械能来计算克服摩擦所做的功，做功的大小即为总能量之差。

6. Power – Rate of Doing Work | 功率 – 做功的快慢

Power is defined as the rate at which work is done or energy is transferred. The two key formulas are P = W / t and P = E / t, where W is work in joules, E is energy transferred, and t is time in seconds. The unit of power is the watt (W). In many problems, you first calculate work or energy change, then divide by time to find power.

功率定义为做功或能量转移的速率。两个关键公式是 P = W / t 和 P = E / t，其中 W 是功（焦耳），E 是转移的能量，t 是时间（秒）。功率的单位是瓦特（W）。许多问题都是先计算功或能量变化，再除以时间得到功率。

P = W / t

Another useful form relates to constant speed motion: when an object moves at constant speed v against a constant resistive force F, the power required is P = F × v. This is derived from P = W / t = (F × d) / t = F × v. This relationship is often tested when a vehicle travels at a steady speed up a slope or against friction.

另一种有用的形式涉及匀速运动：当物体以恒定速度 v 克服恒定阻力 F 运动时，所需功率为 P = F × v。这由 P = W / t = (F × d) / t = F × v 推导而来。当车辆以稳定速度上坡或克服摩擦行驶时，经常会考查这一关系。

7. Work-Energy Theorem | 功能定理

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, W_net = ΔKE = ½ m v² − ½ m u², where u is the initial speed and v is the final speed. This theorem can be used to find the braking distance or the force needed to change an object’s speed without directly using equations of motion.

功能定理指出，对物体做的净功等于其动能的变化量。数学表达式为 W_net = ΔKE = ½ m v² − ½ m u²，其中 u 是初速度，v 是末速度。该定理可用来求解刹车距离或改变物体速度所需的力，无需直接使用运动学方程。

For instance, to find the constant braking force needed to stop a car of mass m from speed u over a distance d, you can set W = F × d and ΔKE = − ½ m u² (since final KE = 0), so F × d = ½ m u². Solve for F or d. This approach is extremely useful in IGCSE Maths word problems involving vehicles and stopping distances.

例如，要使质量为 m 的汽车从速度 u 开始，在距离 d 内停下，可设 W = F × d，而 ΔKE = − ½ m u²（因末动能为 0），因此 F × d = ½ m u²，解出 F 或 d。这种方法在涉及车辆刹车距离的 IGCSE 数学应用题中非常有用。

8. Typical Exam Question: Calculating Work Done | 典型考题：计算做功

A common question gives a force applied to push a box along a horizontal surface. For example: A man pushes a crate with a constant force of 200 N over a distance of 15 m. Calculate the work done. Solution: W = F × d = 200 × 15 = 3000 J. This looks simple but be careful: if the force is applied at an angle to the horizontal, only the horizontal component does work. However, in IGCSE Maths, the force is usually parallel to the direction of motion, as explicitly stated.

常见题目给出一个人用一个力沿水平面推箱子。例如：某人用 200 N 的恒力推一个板条箱，移动了 15 m。计算所做的功。 解答：W = F × d = 200 × 15 = 3000 J。看起来简单但要注意：如果力的方向与水平面有夹角，只有水平分力做功。不过在 IGCSE 数学中，力一般明确说明与运动方向平行。

Some questions involve lifting an object vertically. For instance, a weightlifter lifts a 80 kg barbell through a height of 2 m. The force required equals the weight: F = m g = 80 × 10 = 800 N. Then W = 800 × 2 = 1600 J. Notice we multiply the weight by the vertical height to get work done, which also equals the gain in GPE.

有些题目涉及竖直举起物体。例如，一名举重运动员将 80 kg 的杠铃举起 2 米高。所需力等于重量：F = m g = 80 × 10 = 800 N。然后 W = 800 × 2 = 1600 J。注意，将重量乘以垂直高度即得到做功，这也等于增加的重力势能。

9. Typical Exam Question: Energy Calculations | 典型考题：能量计算

A falling object question might state: A stone of mass 2 kg is dropped from a cliff 45 m high. Using g = 10 m/s², find the speed just before hitting the ground, assuming no air resistance. Use conservation of energy: m g h = ½ m v². Cancel m: 10 × 45 = ½ v² → 450 = ½ v² → v² = 900 → v = 30 m/s. This method is faster than using suvat equations.

落体问题可能是：一个 2 kg 的石块从 45 米高的悬崖上落下。取 g = 10 m/s²，假设无空气阻力，求落地前的速度。 用能量守恒：m g h = ½ m v²。约去 m：10 × 45 = ½ v² → 450 = ½ v² → v² = 900 → v = 30 m/s。这种方法比使用运动学公式更快捷。

Another classic problem is a pendulum: A pendulum bob of mass 0.5 kg is raised to a height of 0.2 m above the lowest point and released. Find its maximum speed. Again, GPE converts to KE: m g h = ½ m v² → g h = ½ v². With g = 10, 10 × 0.2 = ½ v² → 2 = ½ v² → v² = 4 → v = 2 m/s. These problems highlight that mass does not affect the final speed, only g and h.

另一经典题型是单摆：一个质量 0.5 kg 的摆球被拉到最低点以上 0.2 m 的高度释放。求其最大速度。 同样，GPE 转化为 KE：m g h = ½ m v² → g h = ½ v²。g 取 10，10 × 0.2 = ½ v² → 2 = ½ v² → v² = 4 → v = 2 m/s。这类问题突出最终速度只与 g 和 h 有关，与质量无关。

10. Typical Exam Question: Power and Time | 典型考题：功率与时间

A typical power question: A motor lifts a 50 kg load through 12 m in 8 seconds. Calculate the power output of the motor. Take g = 10. First, work done = force × distance = weight × height = (50 × 10) × 12 = 6000 J. Then power = W / t = 6000 / 8 = 750 W. Always express power in watts, but if the answer is large, they may ask for kilowatts (kW), so 750 W = 0.75 kW.

典型功率题目：一个电动机在 8 秒内将 50 kg 的重物提升 12 m。计算电动机的输出功率。g 取 10。 首先，做功 = 力 × 距离 = 重量 × 高度 = (50 × 10) × 12 = 6000 J。然后功率 = W / t = 6000 / 8 = 750 W。功率总是以瓦特表示，若答案较大，可能会要求以千瓦（kW）表示，即 750 W = 0.75 kW。

When a vehicle moves at constant speed, use P = F × v. For example: A car travels at a steady 20 m/s against a total resistive force of 800 N. Find the power developed by the engine. P = 800 × 20 = 16000 W = 16 kW. Remember that the speed must be in m/s; if given in km/h, convert first.

当车辆匀速行驶时，使用 P = F × v。例如：一辆汽车以 20 m/s 的稳定速度克服 800 N 的总阻力行驶。求发动机发出的功率。 P = 800 × 20 = 16000 W = 16 kW。记住速度单位必须是 m/s；若给出 km/h，要先转换。

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One of the most frequent errors is forgetting to square the speed in the kinetic energy formula. Always write the step v² clearly. Another mistake is using the wrong value of g: if the question says use g = 9.8, do not use 10. Misreading the direction of force is also common; work done is zero if the force is perpendicular to the motion. In mathematical problems, ensure you identify the component of force that does work. Additionally, many students forget to convert grams to kilograms or km/h to m/s. Create a checklist: mass in kg, distance/height in m, speed in m/s, time in s.

最常见的错误之一是忘记在动能公式中将速度平方。务必清晰写下 v² 这一步。另一个错误是使用错误的 g 值：如果题目要求 g=9.8，不可用 10。错误判断力的方向也很常见；当力垂直于运动方向时做功为零。在数学问题中，要确保识别出做功的力分量。此外，很多学生忘记将克换算为千克或 km/h 换算为 m/s。可制作一张检查表：质量 kg，距离/高度 m，速度 m/s，时间 s。

Also watch out for “useful power output” vs “total input power”: you may need to apply an efficiency percentage. But in IGCSE Maths, questions often assume 100% efficiency unless stated otherwise. Finally, in energy conservation problems, do not forget to include work done against friction if the question mentions a rough surface or air resistance. Subtract this work from the total energy change.

还应注意“有用输出功率”与“总输入功率”的区别：可能需要考虑效率百分比。不过在 IGCSE 数学中，除非另有说明，题目一般假设效率为 100%。最后，在能量守恒问题中，如果题目提到粗糙表面或空气阻力，别忘了计入克服摩擦所做的功。应从总能量变化中减去这部分功。

12. Revision and Exam Tips | 复习与应试技巧

Memorise the three core formulas: W = F d, KE = ½ m v², PE = m g h. They will be on your formula sheet, but knowing them by heart speeds up problem‑solving. Practise rearranging each one for the different variables. Work through past paper questions that combine work, energy and power, especially those with multiple conversions. Always show your unit conversions clearly, as method marks are awarded for correct conversions even if the final answer is wrong.

熟记三个核心公式：W = F d、KE = ½ m v²、PE = m g h。虽然公式表上会有，但熟记于心能加快解题速度。要练习就每个公式针对不同变量进行变形。多做真题中涉及功、能量和功率的题目，尤其是那些需要多次换算的题。清晰的单位换算过程能为你赢得方法分，即使最终答案有误。

When reading a question, underline the physical quantities given and the one you need to find. Draw a simple diagram if necessary, labelling forces, distances and heights. For energy conservation, write down initial total energy = final total energy + work done against friction. This equation organises your working. In power questions involving speed, check if the speed is constant so you can use P = F v. Finally, manage your time: these questions often appear in the latter part of the paper but are very structured – following the steps earns many marks.

读题时，在给出的物理量和待求量下画线。如有必要，画一个简图，标出力、距离和高度。对于能量守恒，写出初始总能量 = 最终总能量 + 克服摩擦做功。这样的方程能使解题过程有条理。在涉及速度的功率题中，先判断速度是否恒定，以便使用 P = F v。最后，管理好时间：这类题目常出现在试卷后半部分，但结构性强——按步骤作答能获得较多分数。

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Work and Energy | 功和能量考点精讲