Work and Energy Calculations in GCSE CIE Maths | GCSE CIE 数学：功与能量考点精讲

📚 Work and Energy Calculations in GCSE CIE Maths | GCSE CIE 数学：功与能量考点精讲

In GCSE CIE Mathematics, questions on work and energy appear not as pure physics, but as real‑world applications of algebra, proportional reasoning and unit conversions. You are expected to use standard formulae such as W = F × d, KE = ½mv² and GPE = mgh, substitute values correctly, and solve for an unknown. This article covers the key mathematical techniques required to handle work, energy and power problems confidently in your exam.

在 GCSE CIE 数学考试中，功和能量的题目并非纯粹物理题，而是代数、比例推理和单位换算的实际应用。你需要熟记并使用标准公式，如 W = F × d、KE = ½mv² 和 GPE = mgh，正确代入数值并求解未知量。本文将涵盖解决功、能量和功率问题时必需的关键数学技巧，帮助你在考试中得心应手。

1. What Are Work and Energy in the Maths Context? | 数学背景下的功与能量是什么？

In a CIE Maths paper, ‘work’ refers to the product of a force and the distance moved in the direction of that force. Energy appears as kinetic energy and gravitational potential energy. The questions often ask you to find a missing force, mass, speed or height, using the link between work and the change in energy. The mathematics is based on substituting numbers into a formula and rearranging equations.

在 CIE 数学试卷中，“功”指的是力与该力方向上移动距离的乘积。能量通常表现为动能和重力势能。题目常要求你利用功与能量变化之间的联系，求出缺失的力、质量、速度或高度。其数学本质是将数值代入公式并进行方程变形。

2. The Work Formula W = F × d | 做功公式 W = F × d

Work done (W) is measured in joules (J). The formula is W = F × d, where F is the constant force in newtons (N) and d is the distance moved in metres (m) along the line of the force. If the force is not completely in the direction of motion, you may be asked to use the component Fcosθ, but at GCSE level problems are usually straightforward, with the force parallel to the displacement.

做功 (W) 以焦耳 (J) 为单位。公式为 W = F × d，其中 F 为恒力（牛顿，N），d 为该力方向上移动的距离（米，m）。若力并非完全沿着运动方向，你可能需要使用分量 Fcosθ，但 GCSE 层次的题目通常很简单，力与位移方向平行。

W = F × d

Example: A horizontal force of 40 N pushes a box 5 m across a floor. Work done = 40 × 5 = 200 J.

示例：一个 40 N 的水平力将箱子推过 5 m 的地面。做功 = 40 × 5 = 200 J。

3. Units and Conversions You Must Know | 必须掌握的单位及换算

Questions can mix units, forcing you to convert before calculating. Typically, speed might be given in km/h, but the kinetic energy formula requires m/s. Mass may appear in grams, but must be in kg for energy calculations. Time is often in minutes or hours for power, but the standard unit is the second.

题目常常混合单位，要求你先换算再计算。通常，速度可能以 km/h 给出，但动能公式需要 m/s。质量可能以克为单位，但能量计算必须用 kg。时间在功率计算中常以分钟或小时出现，但标准单位是秒。

1 km = 1000 m, 1 hour = 3600 seconds.

1 km = 1000 m，1 小时 = 3600 秒。
To convert km/h to m/s: divide by 3.6.

将 km/h 转换为 m/s：除以 3.6。
1 gram = 0.001 kg.

1 克 = 0.001 千克。
1 joule = 1 N·m = 1 kg·m²/s².

1 焦耳 = 1 N·m = 1 kg·m²/s²。

Always check that the final answer is in joules (J) for work and energy, and in watts (W) for power.

始终检查最终答案：功和能量的单位是焦耳 (J)，功率单位为瓦特 (W)。

4. Kinetic Energy KE = ½ m v² | 动能 KE = ½ m v²

Kinetic energy is the energy of a moving object. The formula is KE = ½ m v², where m is the mass in kg and v is the speed in m/s. Squaring the speed means the formula is quadratic; you may need to rearrange it to find v by taking the square root.

动能是运动物体所具有的能量。公式为 KE = ½ m v²，其中 m 为质量（kg），v 为速度（m/s）。速度的平方使得该公式是二次的；你可能需要对其进行变形，通过开平方根求出 v。

KE = ½ m v²

If KE = 450 J and m = 10 kg, then 450 = ½ × 10 × v² → 450 = 5 v² → v² = 90 → v = √90 ≈ 9.49 m/s.

若 KE = 450 J 且 m = 10 kg，则 450 = ½ × 10 × v² → 450 = 5 v² → v² = 90 → v = √90 ≈ 9.49 m/s。

5. Gravitational Potential Energy GPE = mgh | 重力势能 GPE = mgh

Gravitational potential energy depends on mass (m), gravitational field strength (g) and height (h) above a reference level. On Earth, g is usually taken as 10 m/s² or 9.8 m/s² if specified. The formula GPE = mgh gives the energy stored by an object because of its position. Typical questions ask you to find the height reached by an object, or the mass given the energy gained.

重力势能取决于质量 (m)、重力场强度 (g) 和相对于参考平面的高度 (h)。在地球上，g 通常取 10 m/s²，如果题目指定则用 9.8 m/s²。公式 GPE = mgh 表示物体因位置而储存的能量。典型题目会要求你求出物体上升的高度，或根据获得的能量求出质量。

GPE = m g h

Example: A 2 kg book is lifted 1.5 m. GPE gained = 2 × 10 × 1.5 = 30 J.

示例：将一本 2 kg 的书举高 1.5 m。获得的重力势能 = 2 × 10 × 1.5 = 30 J。

6. Conservation of Energy and the Work‑Energy Principle | 能量守恒与功能原理

Energy cannot be created or destroyed, only transferred. In many maths problems, the work done by a force equals the change in kinetic energy, or the loss in gravitational potential energy equals the gain in kinetic energy (if air resistance is negligible). The work‑energy principle states: W = ΔKE = KE_final − KE_initial. This link allows you to set up an equation connecting force, distance, mass and speed.

能量不能被创造或消灭，只能被转移。在许多数学问题中，力所做的功等于动能的变化量，或者重力势能的减少量等于动能的增加量（若空气阻力可忽略）。功能原理表述为：W = ΔKE = KE_末 − KE_初。这一联系使你能够建立连接力、距离、质量和速度的方程。

W = ΔKE = ½ m v_f² − ½ m v_i²

For a falling object, mgh = ½ m v² can be used to find the speed v = √(2gh), provided the initial velocity is zero.

对于下落的物体，可以用 mgh = ½ m v² 得出速度 v = √(2gh)，前提是初速度为零。

7. Power Calculations P = W / t | 功率计算 P = W / t

Power is the rate of doing work or transferring energy. The formula is P = W / t, where P is power in watts (W), W is work done in joules (J) and t is time in seconds (s). Sometimes you are given the power and time and asked to calculate the total work done: W = P × t. Combined with other energy formulas, this can link force, speed and power.

功率是做功或转移能量的速率。公式为 P = W / t，其中 P 为功率（瓦特，W），W 为做功（焦耳，J），t 为时间（秒，s）。有时题目会给出功率和时间，要求计算总做功：W = P × t。与其他能量公式结合，就能将力、速度和功率联系起来。

P = W / t

In a car moving at constant speed, the engine’s power overcomes resistive forces, and you may also use P = F × v for a constant driving force F.

在匀速行驶的汽车中，发动机的功率用于克服阻力，你也可以使用 P = F × v，其中 F 为恒定的驱动力。

8. Solving Equations Involving Energy | 解含能量的方程

Many exam questions require you to substitute values, rearrange the equation and then solve for an unknown. This might involve squaring or square roots, dividing both sides by a variable, or linking two expressions through energy conservation. You must be comfortable with algebraic manipulation, including working with fractions like ½, and using correct BIDMAS order.

许多考题要求你代入数值、对方程进行变形，然后解出未知量。这可能涉及平方或开平方根、两边同时除以一个变量，或通过能量守恒关联两个表达式。你必须熟练掌握代数操作，包括处理 ½ 这样的分数，并遵循正确的 BIDMAS 运算顺序。

Worked example: A cyclist of mass 80 kg is travelling at 6 m/s. She coasts to a stop over a distance of 24 m against a constant resistive force. Find the force. Initial KE = ½ × 80 × 6² = 1440 J. Work done against force F: W = F × 24. By work‑energy: 1440 = F × 24, so F = 60 N.

例题：一名质量为 80 kg 的自行车手以 6 m/s 的速度行驶。她滑行 24 m 后停止，受到恒定的阻力。求阻力大小。初动能 = ½ × 80 × 6² = 1440 J。克服阻力 F 做功：W = F × 24。根据功能原理：1440 = F × 24，得 F = 60 N。

9. Graphical Interpretation and Real‑World Problems | 图形解释与现实问题

Work can also be found from a force‑distance graph: the area under the graph represents the work done. In a force‑extension graph for a spring, the area under the line gives the elastic potential energy (½ × force × extension). While not always a core GCSE CIE topic, it may appear as an extension question. You may also be asked to interpret speed‑time or distance‑time graphs to extract data needed for energy calculations.

功也可以通过力‑距离图求得：图下的面积代表所做的功。在弹簧的力‑伸长量图中，线下的面积表示弹性势能（½ × 力 × 伸长量）。虽然这并非 GCSE CIE 的核心考点，但可能作为拔高题出现。你还可能被要求解读速度‑时间图或距离‑时间图，从中提取能量计算所需的数据。

Example: A constant force of 15 N acts over a distance of 4 m. The area of the rectangle (15 × 4) = 60 J is the work done.

示例：15 N 的恒力作用了 4 m。矩形面积 (15 × 4) = 60 J 即为所做的功。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Watch out for these typical errors: forgetting to convert grams to kilograms or km/h to m/s; mixing up kinetic and potential energy formulas; forgetting the ½ in KE; using the wrong value of g; and not squaring the speed correctly. Always list the given values with their units and decide which formula links them before substituting.

要当心这些常见错误：忘记将克转换为千克，或将 km/h 转换为 m/s；混淆动能和势能公式；忘了 KE 中的 ½；使用了错误的 g 值；以及没有正确对速度进行平方。代入数值之前，始终列出已知量及其单位，并确定哪个公式能够关联它们。

Write the formula first, then substitute numbers.

先写公式，再代入数字。
Check that the units are consistent (SI units).

检查单位是否一致（使用 SI 单位）。
Show all your steps, especially when rearranging.

展示所有步骤，尤其是变形过程。
If an answer looks unrealistic (e.g. speed of 500 m/s for a bicycle), re‑check your conversions.

如果答案看起来不真实（例如，自行车速度 500 m/s），请重新检查单位换算。

Practising past paper questions will help you become familiar with the style of word problems involving work and energy.

练习历年真题将帮助你熟悉涉及功和能量的文字题风格。

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Work and Energy Calculations in GCSE CIE Maths | GCSE CIE 数学：功与能量考点精讲