📚 GCSE OCR Maths: Work and Energy Key Points | GCSE OCR 数学：功和能量 考点精讲
Work and energy are fundamental concepts in physics, but they also appear frequently in OCR GCSE Mathematics as contextual application problems. Understanding the key formulas, their units, and how to rearrange equations allows you to solve real‑world problems involving forces, motion, and energy transfers. This revision guide covers everything you need to know, from calculating work done to interpreting force–distance graphs.
功和能量是物理学的基本概念，但也经常作为应用题出现在 OCR GCSE 数学考试中。掌握关键公式、单位以及如何变换方程，能帮助你解决涉及力、运动和能量转移的实际问题。本复习指南涵盖了你需要了解的全部内容，从计算功到解读力–距离图像。
1. Work Done: Force × Distance | 功的定义：力 × 距离
Work is done when a force moves an object through a distance. In mathematics, you calculate work using the simple equation: work done = force × distance moved in the direction of the force. This relationship only applies when the force is constant and acts along the line of motion.
当力使物体移动一段距离时就做了功。在数学中，你使用简单方程计算功：功 = 力 × 沿力方向移动的距离。这种关系仅适用于力恒定且与运动方向一致的情况。
W = F × d
Here W is work done in joules (J), F is force in newtons (N), and d is distance in metres (m). If the force is applied at an angle, you must use the component of force in the direction of travel, but GCSE questions typically keep things parallel.
这里 W 表示功，单位焦耳 (J)；F 是力，单位牛顿 (N)；d 是距离，单位米 (m)。如果力是斜着施加的，你需要使用沿运动方向的分力，但 GCSE 题目通常只涉及平行情况。
2. Units of Work and Energy | 功和能量的单位
Both work and energy are measured in joules (J). One joule is defined as the work done when a force of one newton moves an object one metre in the direction of the force. In calculations, you must ensure all quantities are expressed in base SI units: newtons for force, metres for distance, kilograms for mass, and metres per second for speed. Using centimetres or grams without conversion will lead to incorrect answers.
功和能量都以焦耳 (J) 为单位。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。在计算中，必须确保所有量都用国际单位制基本单位表示：力用牛顿，距离用米，质量用千克，速度用米/秒。未经换算直接使用厘米或克会导致错误答案。
Common unit conversions you might need in work–energy problems include: 1 kJ = 1000 J, 1 km = 1000 m, 1 hour = 3600 s. Always convert before plugging numbers into formulas.
功–能量问题中可能用到的常见单位换算包括：1 kJ = 1000 J，1 km = 1000 m，1 小时 = 3600 s。务必先换算再代入公式。
3. Kinetic Energy Formula | 动能公式
Kinetic energy (KE) is the energy an object possesses due to its motion. The formula is: kinetic energy = ½ × mass × (speed)². In algebraic form, this gives a quadratic relationship: if speed doubles, kinetic energy quadruples.
动能 (KE) 是物体由于运动而具有的能量。公式为：动能 = ½ × 质量 × 速度²。写成代数式后呈现平方关系：速度加倍，动能变为原来的 4 倍。
KE = ½ m v²
Here m is mass in kilograms (kg) and v is speed in metres per second (m/s). Rearranging this equation allows you to find mass or speed when the other quantities are known. For example, to find speed: v = √(2 × KE ÷ m).
这里 m 是质量（kg），v 是速度（m/s）。对该方程进行变换，可以在已知其他量时求出质量或速度。例如求速度：v = √(2 × KE ÷ m)。
4. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object because of its height above ground. The formula is: GPE = mass × gravitational field strength × height. On Earth, the gravitational field strength g is approximately 9.8 N/kg, but most GCSE questions use 10 N/kg for simplicity.
重力势能 (GPE) 是物体因离地高度而储存的能量。公式为：GPE = 质量 × 重力场强度 × 高度。在地球上，重力场强度 g 约为 9.8 N/kg，但大多数 GCSE 题目为简便取 10 N/kg。
GPE = m g h
Mass m is in kg, g in N/kg, and height h in metres. Note that height must be measured vertically from a chosen reference point. GPE is directly proportional to both mass and height.
质量 m 以 kg 为单位，g 以 N/kg 为单位，高度 h 以米为单位。注意高度必须从所选参考点垂直测量。重力势能与质量和高度均成正比。
5. Conservation of Energy | 能量守恒
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In a closed system, total energy remains constant. This allows you to equate different types of energy; for example, a falling object loses GPE and gains KE, so mgh = ½mv² (ignoring air resistance). Cancelling m from both sides gives gh = ½v², often used to find the speed of an object dropped from a height.
能量守恒定律指出，能量既不会凭空产生也不会消失，只能从一种形式转移或转化为另一种。在封闭系统中，总能量保持不变。这让你可以将不同类型的能量等量代换；例如，一个下落的物体会损失 GPE 并增加 KE，因此 mgh = ½mv²（忽略空气阻力）。两边消去 m 得到 gh = ½v²，常用于求从高处下落物体的速度。
Energy conservation problems often ask you to calculate the maximum height reached by a projectile or the speed just before impact. Set initial energy equal to final energy and solve for the unknown.
能量守恒问题经常要求你计算抛射体达到的最大高度或撞击前的速度。让初始能量等于最终能量，然后解未知数。
6. Power: Work Done per Second | 功率：每秒完成的功
Power is the rate at which work is done or energy is transferred. It is calculated using the formula: power = work done ÷ time taken, or equivalently power = energy transferred ÷ time. The unit of power is the watt (W), where 1 W = 1 J/s.
功率是做功或能量转移的速率。计算公式为：功率 = 做的功 ÷ 所用时间，或等价地，功率 = 转移的能量 ÷ 时间。功率的单位是瓦特 (W)，1 W = 1 J/s。
P = W ÷ t or P = E ÷ t
You may also see a combined formula linking power, force, and velocity: when an object moves at constant speed v against a constant force F, the power needed is P = F × v. This is derived from W = Fd by dividing both sides by time t, giving W/t = F × (d/t).
你可能还会看到联系功率、力和速度的组合公式：当物体以恒定速度 v 克服恒力 F 运动时，所需功率为 P = F × v。这由 W = Fd 两边同除以时间 t 导出，得到 W/t = F × (d/t)。
7. Converting Units for Calculations | 计算中的单位换算
Before substituting values into any formula, check that all quantities are in the correct units. Common mistakes involve using grams instead of kilograms, cm instead of m, or km/h instead of m/s. For speed conversions: 1 m/s = 3.6 km/h. Use the conversion m/s = (km/h) ÷ 3.6.
在任何公式中代入数值之前，请检查所有量是否使用了正确单位。常见错误包括用克代替千克、厘米代替米、或千米/小时代替米/秒。速度换算：1 m/s = 3.6 km/h。使用 m/s = (km/h) ÷ 3.6 进行换算。
| Quantity | Correct SI unit | Common wrong unit |
|---|---|---|
| mass | kg | g |
| distance | m | cm, km |
| speed | m/s | km/h |
| force | N | kN (must ×1000) |
| energy/work | J | kJ (must ×1000) |
Always write the conversion step clearly in your working to avoid errors. For instance, if mass is given as 500 g, record it as m = 0.5 kg before you start.
务必在解题步骤中清晰写出换算过程，以免出错。例如，若质量给的是 500 g，应先在计算前记作 m = 0.5 kg。
8. Solving Problems with Work and Energy Equations | 用功和能量方程解决问题
OCR questions often combine work, energy, and power concepts. A typical problem might ask: ‘A car of mass 1200 kg accelerates from rest to 20 m/s. How much work is done by the engine?’ Here the work done equals the gain in kinetic energy: W = KE_final − KE_initial = ½ × 1200 × 20² − 0 = 240 000 J.
OCR 题目经常结合功、能量和功率的概念。一个典型问题可能会问：“一辆质量 1200 kg 的汽车从静止加速到 20 m/s，发动机做了多少功？”这时所做的功等于动能的增加量：W = KE_末 − KE_初 = ½ × 1200 × 20² − 0 = 240 000 J。
Another common format gives you the braking force and stopping distance, asking for the initial kinetic energy or speed. Since work done by brakes = braking force × braking distance, this work must equal the car’s initial kinetic energy (assuming all KE is dissipated). Use the equality F × d = ½mv² to find the unknown.
另一种常见形式是给出制动力和制动距离，要求初始动能或速度。由于制动器做的功 = 制动力 × 制动距离，这部分功必定等于汽车初始动能（假设所有动能都被耗散）。利用等式 F × d = ½mv² 求未知数。
9. Efficiency and Energy Transfers | 效率和能量转移
Efficiency describes how much of the input energy is converted into useful output. It is expressed either as a decimal or a percentage: Efficiency = useful output energy ÷ total input energy. Multiply by 100 to get a percentage. In mathematical problems, you often have to calculate wasted energy as the difference between total input and useful output.
效率描述输入能量中有多少转化为有用输出。它可以用小数或百分比表示：效率 = 有用输出能量 ÷ 总输入能量。乘以 100 得到百分比。在数学问题中，你经常需要计算浪费的能量，即总输入与有用输出之差。
Efficiency = (E_useful / E_total) × 100%
No device is 100% efficient, so the useful output is always less than the input because energy is transferred to thermal energy of the surroundings. Questions may ask you to calculate the amount of energy lost as heat or sound.
没有设备能达到 100% 效率，因此有用输出总是小于输入，因为部分能量会转化为周围环境的热能。题目可能要求你计算以热量或声音形式损失的能量数额。
10. Graphical Representations | 图形表示
A force–distance graph is a powerful visual tool. The area under the force–distance graph represents the work done. If the force is constant, the graph is a horizontal line, and the area is a rectangle: work = force × distance. If the force varies, the area can still be found by counting squares or by using formulas for triangles and trapeziums.
力–距离图像是一种强大的可视化工具。力–距离图像下方的面积代表做功的大小。如果力恒定，图像是一条水平线，面积是矩形：功 = 力 × 距离。如果力在变化，仍可通过数格点或使用三角形、梯形面积公式求出面积。
A graph of kinetic energy against speed is a parabola (y ∝ x²), while a graph of kinetic energy against mass is a straight line through the origin (direct proportion). Recognising these shapes helps you interpret data and solve related problems.
动能–速度图像是一条抛物线（y ∝ x²），而动能–质量图像是一条过原点的直线（正比例关系）。识别这些形状有助于分析数据并解决相关问题。
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