Work and Energy in GCSE Edexcel Maths | 功和能量考点精讲

📚 Work and Energy in GCSE Edexcel Maths | 功和能量考点精讲

In GCSE Edexcel Mathematics, the topic of work and energy appears as part of compound measures and formula-based problem solving. You are expected to use given physical formulas to calculate work, kinetic energy, gravitational potential energy, power and efficiency, and to rearrange them to find unknown quantities. This revision guide consolidates every key concept, formula, unit and strategy you need for exam success.

在 GCSE Edexcel 数学中，功和能量作为复合单位与公式应用的一部分出现。你需要运用给定的物理公式来计算功、动能、重力势能、功率和效率，并会通过公式变形求解未知量。本篇复习指南将梳理所有核心概念、公式、单位换算及解题策略，帮助你轻松应对考试。

1. The Work Done Formula | 功的定义与公式

Work done is the energy transferred when a force moves an object in the direction of the force. The fundamental formula is:

功表示力使物体沿力的方向移动时所传递的能量。基本公式为：

W = F × d

where W is work done in joules (J), F is the force in newtons (N) and d is the distance moved in metres (m).

其中 W 为功，单位焦耳（J）；F 为力，单位牛顿（N）；d 为物体在力的方向上移动的距离，单位米（m）。

Remember: the force and the distance must be in the same direction. If a force is applied at an angle, only the component in the direction of motion does work (though simplified GCSE questions rarely require trigonometry).

注意：力与距离必须处于同一方向。若力与运动方向有夹角，只有力在运动方向上的分力做功（不过 GCSE 数学题目通常不涉及三角函数的复杂情况）。

2. Units of Work and Conversions | 功的单位与换算

One joule is defined as the work done when a force of 1 newton moves an object 1 metre in the direction of the force: 1 J = 1 N m.

1 焦耳定义为 1 牛顿的力使物体沿力的方向移动 1 米所做的功：1 J = 1 N m。

In exam problems, you may need to convert compound units. Common conversions include:

在考试中，你可能需要换算复合单位。常见的换算有：

Kilojoules to joules: 1 kJ = 1000 J
千焦与焦耳：1 kJ = 1000 J
Centimetres to metres: 1 m = 100 cm, so to convert cm to m divide by 100.
厘米与米：1 m = 100 cm，将厘米换算成米需除以 100。
Time conversions: Power calculations often involve seconds; ensure minutes or hours are converted to seconds (1 min = 60 s, 1 h = 3600 s).
时间换算：功率计算常涉及秒；务必把分钟或小时换算成秒（1 分 = 60 秒，1 小时 = 3600 秒）。

Always check that the quantities you substitute into formulas have consistent units (SI units). Otherwise your answer will be incorrect.

一定要确保代入公式的物理量单位一致（均使用国际单位制），否则答案就会出错。

3. Calculating Work – Worked Examples | 功的计算例题

Let’s look at a straightforward example:

请看一个简单的例子：

Example 1: A gardener pushes a wheelbarrow with a constant force of 120 N over a distance of 15 m. How much work is done?

示例 1：一位园丁用 120 N 的恒力推动手推车前行 15 m。做了多少功？

Step 1: Write the formula: W = F × d.

步骤 1：写出公式：W = F × d。

Step 2: Substitute the values: W = 120 N × 15 m.

步骤 2：代入数值：W = 120 N × 15 m。

Step 3: Calculate: W = 1800 J.

步骤 3：计算：W = 1800 J。

Example 2: A box is pulled with a force of 50 N. The work done is 2000 J. Find the distance the box moved.

示例 2：一个箱子被 50 N 的力拉动，做的功为 2000 J。求箱子移动的距离。

Rearrange the formula: d = W / F.

公式变形：d = W / F。

Substitution: d = 2000 J / 50 N = 40 m.

代入：d = 2000 J / 50 N = 40 m。

These examples show that you must be comfortable substituting numbers and rearranging simple equations.

这些例子说明，你必须熟练代入数据并对简单方程进行变形。

4. Work Done Against Gravity (Gravitational Potential Energy) | 克服重力做功（重力势能）

Lifting an object vertically requires a force equal to its weight. Weight is the product of mass (m) and gravitational field strength (g). On Earth, g is usually taken as 10 N/kg or 9.8 N/kg (the question will specify). The work done to lift the object, and the gain in gravitational potential energy (GPE), is:

竖直提升物体需要用等于其重量的力。重量等于质量（m）与重力场强度（g）的乘积。在地球上，g 通常取 10 N/kg 或 9.8 N/kg（题目会指明）。提升物体所做的功，即增加的重力势能（GPE）为：

W = m × g × h

where m is mass in kg, g is gravitational field strength in N/kg and h is vertical height in metres.

其中 m 为质量（kg），g 为重力场强度（N/kg），h 为竖直高度（m）。

For example, lifting a 5 kg bag through 2 m (using g = 10): W = 5 × 10 × 2 = 100 J. The bag gains 100 J of gravitational potential energy.

例如，将 5 kg 的包提升 2 m（取 g = 10）：W = 5 × 10 × 2 = 100 J。该包获得 100 J 的重力势能。

In problems where an object slides down a slope, you may need to use the vertical height, not the slope length, to calculate GPE change.

在物体沿斜面下滑的问题中，计算重力势能变化时要用竖直高度，而非斜面长度。

5. Kinetic Energy | 动能

Any moving object possesses kinetic energy (KE), given by the formula:

任何运动的物体都具有动能（KE），其公式为：

KE = ½ m v²

where m is mass in kg and v is speed in m/s. Notice that speed is squared, so doubling the speed quadruples the kinetic energy.

其中 m 为质量（kg），v 为速度（m/s）。注意速度是平方项，因此速度加倍会使动能变为原来的四倍。

Example: A cyclist of mass 70 kg is travelling at 8 m/s. Calculate the kinetic energy. (Take ½ = 0.5)

示例：一位质量为 70 kg 的自行车手以 8 m/s 的速度骑行，求动能。

KE = ½ × 70 × (8)² = 0.5 × 70 × 64 = 2240 J.

You must be able to rearrange the KE formula to find mass or speed. For instance, v = √(2 × KE / m). The square root means you must be careful with calculator work.

你必须能对动能公式变形，求出质量或速度。例如 v = √(2 × KE / m)。使用平方根时一定要用计算器仔细处理。

6. Energy Conservation and Efficiency | 能量守恒与效率

In a closed system, energy cannot be created or destroyed, only transferred. This means total energy before = total energy after, ignoring resistive forces like friction and air resistance. A typical GCSE maths problem might ask you to equate loss in GPE to gain in KE, or calculate efficiency.

在封闭系统中，能量既不能创造也不能消失，只能转化。这意味着初始总能量等于最终总能量（忽略摩擦和空气阻力等耗散力）。典型的 GCSE 数学问题可能会让你根据重力势能的减少与动能的增加来列方程，或者计算效率。

Efficiency is the ratio of useful energy output to total energy input:

效率是指有用能量输出与总能量输入的比值：

Efficiency = (useful output energy / total input energy) × 100%

You may also see it expressed using power: Efficiency = (useful power output / total power input) × 100%.

也可能用功率表示：效率 = (有用功率输出 / 总功率输入) × 100%。

If a machine receives 200 J of energy and only 150 J does useful work, efficiency = (150/200) × 100% = 75%. The remaining 50 J is wasted, often as heat.

若机器接收 200 J 能量，仅 150 J 做了有用功，效率 = (150/200) × 100% = 75%。剩余 50 J 被浪费，通常转化为热能。

7. Power | 功率

Power is the rate at which work is done or energy is transferred. The straightforward formula is:

功率表示做功或能量转换的速率。其简洁的公式为：

P = W / t

where P is power in watts (W), W is work done in joules (J) and t is time in seconds (s). 1 watt equals 1 joule per second.

其中 P 为功率，单位瓦特（W）；W 为功，单位焦耳（J）；t 为时间，单位秒（s）。1 瓦特等于 1 焦耳每秒。

You can also use energy instead of work: P = E / t.

你也可以用能量替代功：P = E / t。

Example: A motor lifts a weight, doing 2400 J of work in 2 minutes. Calculate the power output.

示例：一台电动机提升重物，在 2 分钟内做了 2400 J 的功，求输出功率。

Convert time to seconds: 2 min = 120 s. P = 2400 J / 120 s = 20 W.

将时间换算成秒：2 分 = 120 s。P = 2400 J / 120 s = 20 W。

Rearranging: W = P × t and t = W / P are equally common in multi-step problems.

公式变形：W = P × t 与 t = W / P 在稍复杂的题目中也经常用到。

8. Force–Distance Graphs | 力-距离图

The work done by a force can also be found from a force–distance graph. Because work = force × distance, the area under the graph represents the work done.

力所做的功也可以从力-距离图中求得。因为功 = 力 × 距离，所以图线下的面积就代表功。

For a constant force, the graph is a horizontal line. The area is a rectangle:

对于恒力，图像是一条水平线。其面积为矩形：

Work done = force × distance = area of rectangle

For a force that increases uniformly from zero, the graph is a triangle. The work done is the area of that triangle:

对于从零开始均匀增大的力，图像为三角形。功等于该三角形的面积：

Work done = ½ × base × height = ½ × distance × maximum force

You may need to interpret such graphs or draw them in applied contexts. Remember that distance must be on the horizontal axis, force on the vertical axis.

你可能需要解读这类图像，或者在应用题中绘制图像。请记住距离在横轴，力在纵轴。

9. Problem-Solving Strategies – Rearranging and Substituting | 解题策略 – 公式变形与代入

GCSE Edexcel Maths questions will often give you a formula and ask you to find a missing quantity. Follow these steps:

GCSE Edexcel 数学题常给出公式，要求你求出未知量。遵照以下步骤：

Identify the known variables and their units, converting if needed.
确定已知量和单位，必要时进行换算。
Write down the relevant formula.
写出相关公式。
Rearrange the formula to make the unknown the subject (e.g., to find speed from KE: v = √(2KE / m)).
变形公式，使未知量成为主体（例如从动能求速度：v = √(2KE / m)）。
Substitute the numbers carefully, watching out for squares and square roots.
仔细代入数值，注意平方与平方根。
Calculate the final answer, including correct units.
计算出最终答案，包含正确单位。

Practice with multi-step problems, such as finding the height a ball will reach given its initial speed, using GPE = KE and assuming all energy is transferred.

多练习多步问题，比如已知初速度求小球上升的高度，利用 GPE = KE 并假设所有能量都得到转化。

10. Common Mistakes and Tips | 常见错误与考场提示

Avoid these pitfalls to secure full marks:

避免以下常见错误，确保拿下满分：

Mixing units: Never use cm or g without converting to m and kg first.
单位混用：切勿在未换算成 m 和 kg 的情况下直接使用 cm 或 g。
Forgetting to square the speed in the KE formula; 3 m/s gives v² = 9, not 3.
忘记在动能公式里将速度平方：3 m/s 时 v² = 9，而非 3。
Confusing mass and weight: Mass is in kg; weight is mass × g and is in newtons. The work done against gravity uses weight, not mass.
混淆质量与重量：质量单位是 kg；重量等于质量 × g，单位是牛顿。克服重力做功用的是重量，不是质量。
Using the slope distance instead of vertical height for GPE calculations.
用斜面距离替代竖直高度计算重力势能。
Neglecting energy loss in efficiency questions: total input is always larger than useful output for real machines.
在效率问题中忽略能量损失：对于实际机器，总输入能量总是大于有用输出能量。

On exam day, underline key values and write the formula before substituting. Show all steps so you can gain method marks, even if a calculation error occurs.

考试时，划出关键数值，先写公式再代入。展示所有步骤，这样即使计算有误也能得到分步分数。

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Work and Energy in GCSE Edexcel Maths | 功和能量考点精讲