📚 Work and Energy in IGCSE CIE Mathematics | IGCSE CIE 数学：功和能量 考点精讲
In IGCSE CIE Mathematics, questions on work and energy often appear as contextual applications of algebraic formulas, proportional reasoning, and graphical interpretation. You will not be expected to recall complex physics derivations, but you must be able to manipulate equations, convert units, and solve multi‑step word problems accurately. Understanding the basic definitions of work, energy, and power allows you to translate physical scenarios into straightforward mathematical calculations, boosting your confidence in the exam.
在 IGCSE CIE 数学考试中，功和能量题目通常以代数公式应用、比例推理和图像分析的形式出现。你不需要记忆复杂的物理推导，但必须能熟练变换公式、换算单位，并准确求解多步骤文字题。理解功、能量和功率的基本定义，能帮助你把物理情景转化为清晰的数学运算，从而在考试中更加得心应手。
1. What is Work? | 什么是功？
f how large the force is. This concept is almost always linked to the formula W = F × d, where F is the constant force applied and d is the distance moved in the direction of the force.
在数学题中，当一个力使物体沿力的方向移动时，就说力做了功。如果物体没有移动，无论力有多大，都没有做功。这一概念几乎总是与公式 W = F × d 相联系，其中 F 是施加的恒力，d 是沿力方向移动的距离。
2. Calculating Work: W = F × d | 计算功：W = F × d
. When the force and distance are perpendicular, no work is done in that direction. You must check that the distance used is strictly the component in the direction of the force. Many exam questions provide a force parallel to the displacement, making the calculation direct substitution.
你要使用的核心公式是 W = F d。当力与位移垂直时，该力不做功。你必须确保所用的距离严格沿力的方向。很多考题给出的力与位移平行，此时只需直接代入数值即可。
W = F × d
3. Units of Work: Joules | 功的单位：焦耳
Work is measured in joules (J). One joule is the work done when a force of 1 newton moves an object 1 metre in the direction of the force. In CIE Mathematics, you may need to convert units such as cm to m, g to kg, or minutes to seconds before substituting into the formula. Always check that force is in newtons (N) and distance in metres (m).
功的单位是焦耳（J）。1 焦耳等于 1 牛顿的力使物体沿力的方向移动 1 米所做的功。在 CIE 数学中，你可能需要先将单位转换为标准单位，例如将厘米转换为米、克转换为千克、分钟转换为秒，再代入公式。务必确保力以牛顿（N）为单位，距离以米（m）为单位。
- 1 kJ = 1000 J
- 1 g = 0.001 kg → weight = mg (N)
4. Work Done at an Angle | 角度力做功
d × cos θ. Questions often give the angle, and you simply multiply force, distance, and cosine of the angle. A table of exact trigonometric values may be provided or required from memory for common angles like 30°, 45°, and 60°.
如果力与水平方向的夹角为 θ，只有水平分量沿运动方向做功。此时公式变为 W = F × d × cos θ。题目通常会给出角度，你只需将力、距离和该角度的余弦值相乘。对于 30°、45° 和 60° 等常见角度，你可能需要牢记或会使用精确三角值表。
W = F d cos θ
5. Energy and Work Relationship | 能量与功的关系
Energy is the capacity to do work. In mathematical terms, the work done on an object equals the energy transferred. You will often be asked to equate work done by a force to the change in kinetic or potential energy, leading to equations that need solving. This connection is fundamental for energy conservation problems.
能量是做功的本领。从数学角度看，对一个物体做的功等于能量的转移量。题目经常让你将力做的功与动能或势能的变化量建立等量关系，从而得到一个需要求解的方程。这一联系是能量守恒问题的基础。
6. Kinetic Energy Formula | 动能公式
ss (kg) and v is speed (m/s). In CIE Mathematics, you must be able to substitute values, rearrange to find mass or velocity, and compare kinetic energies before and after a change. Remember that squaring the velocity can produce large numbers, so careful arithmetic is essential.
运动物体的动能（KE）由 KE = ½ m v² 给出，其中 m 是质量（kg），v 是速率（m/s）。在 CIE 数学中，你必须能代入数值、通过移项求出质量或速度，并能比较变化前后的动能。注意速度平方可能产生很大的数值，因此算术计算需格外仔细。
KE = ½ m v²
7. Potential Energy: Gravitational | 重力势能
Gravitational potential energy (GPE) near the Earth' in the question. The height h must be vertical height in metres. Problems may involve lifting objects, sliding down slopes, or converting GPE into KE. The equation is linear in both mass and height, making direct proportion easy to spot.
近地表面的重力势能（GPE）计算公式为 GPE = m g h，其中 g = 9.8 m/s² 或 10 m/s²，视题目要求而定。高度 h 必须是竖直高度，单位用米。题目可能涉及举高物体、沿斜面下滑，或将重力势能转化为动能。该公式对质量和高度都是线性关系，很容易识别正比问题。
GPE = m g h
8. Conservation of Energy Problems | 能量守恒问题
nknown. Often, a mass cancels out, simplifying the algebra. This principle links together height, speed, and mass in a clear mathematical way suitable for CIE Mathematics exams.
在没有摩擦的情况下，总机械能（动能 + 重力势能）保持不变。你将建立诸如 ½ m v² + m g h = 常数 的方程，并求解未知量。质量往往会被消去，从而简化代数运算。这条原理以清晰的数学方式将高度、速度和质量联系在一起，非常适合 CIE 数学考试的要求。
½ m v₁² + m g h₁ = ½ m v₂² + m g h₂
9. Power: Rate of Doing Work | 功率：做功的快慢
nt speed v under a constant force F. In CIE Mathematics, power problems test your ability to rearrange formulas and convert time units (hours to seconds, etc.).
功率（P）是单位时间内所做的功。标准公式为 P = W / t，其中 t 是以秒为单位的时间。对于在恒力 F 作用下以恒定速度 v 运动的物体，你也可能看到 P = F × v 的形式。在 CIE 数学中，功率问题考察你变换公式和换算时间单位（如小时换算成秒）的能力。
P = W / t or P = F × v
10. Interpreting Force-Distance Graphs | 解读力-位移图像
A graph of force (y-axis) against distance (x-axis) represents the work done as the area under the curve. If the force is constant, the area is a rectangle. In more complex cases, you may need to calculate the area of a trapezium or triangle. CIE Mathematics questions expect you to find work by counting squares, applying area formulas, or splitting composite shapes.
力（y 轴）随距离（x 轴）变化的图像，其下方的面积代表所做的功。如果力是恒定的，该面积是一个矩形。在更复杂的情形中，你可能需要计算梯形或三角形的面积。CIE 数学题要求你能通过数方格、应用面积公式或分割组合图形的方法求出功。
- Constant force → area of rectangle = F × d
- Linearly changing force → area of triangle or trapezium
11. Solving Multi-Step Problems | 多步骤问题求解
Work and energy problems in CIE Mathematics are rarely single-step. A typical question may ask you to first calculate GPE, then equate it to KE, solve for speed, and finally find power. Write down each step clearly, substituting only at the end to minimise rounding errors. Using algebra before arithmetic is a valuable technique to improve accuracy.
CIE 数学中的功和能量问题极少是单步计算。典型的题目可能先让你计算重力势能，再将其与动能建立等量关系，求出速度，最后计算功率。请清晰写出每一步的推导过程，尽量在最后才代入数值，以减少舍入误差。先做代数运算再做数值计算是提高准确性的好方法。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always read the question to identify which form of energy is involved. Check that all quantities are in SI units: mass in kg, distance in m, time in s, force in N. When a problem involves an angle, be certain whether cos θ is needed. If g is given as 10, use that value; otherwise, stick to 9.8 or the instruction given. Show your full working to gain marks even if the final answer is incorrect.
务必仔细读题，明确涉及哪种形式的能量。检查所有量是否已化为国际单位制：质量用 kg，距离用 m，时间用 s，力用 N。当问题涉及角度时，一定要确认是否需要乘 cos θ。如果题目规定 g 取 10，就采用该值；否则使用 9.8 或题中指定的数值。写出完整的解题步骤，这样即使最终答案有误，也能得到过程分。
- Mistake: Forgetting to square velocity in KE formula
- Mistake: Using horizontal distance instead of vertical height for GPE
- Tip: Cancel mass early where possible to simplify equations
- Tip: Draw a force-distance diagram if you need to find work from a graph
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