AQA Maths: Work and Energy Revision Notes | AQA 数学：功和能量 考点精讲

📚 AQA Maths: Work and Energy Revision Notes | AQA 数学：功和能量 考点精讲

Work and energy are fundamental topics in AQA Mechanics, bridging forces, motion, and power. Mastering these concepts means you can analyse how energy is transferred and conserved, solve for speed, distance, and power without relying solely on equations of motion, and confidently tackle typical exam scenarios involving slopes, friction, and vehicles. These notes cover all the key principles you need, from basic work done by constant forces to the work-energy theorem and power formulas, with worked examples and common pitfalls.

功和能量是 AQA 力学中的基础主题，将力、运动与功率连接起来。掌握这些概念意味着你能够分析能量的传递和守恒，求解速度、位移和功率而无需完全依赖运动学方程，并能自信地处理涉及斜面、摩擦和车辆的典型考题。本文涵盖了所有你需要的关键原理，从恒力做功的基础知识，到功能定理和功率公式，并配有实例解析和常见错误。

1. Introduction to Work and Energy | 功和能量导论

In mechanics, work represents the transfer of energy when a force acts over a distance. Energy is a scalar quantity measured in joules (J) and can exist in many forms, but our focus is kinetic energy (due to motion) and gravitational potential energy (due to height). The central idea is the work-energy principle: the net work done on a particle is equal to the change in its kinetic energy. This principle, together with conservation of energy and power calculations, forms the backbone of many AQA exam questions.

在力学中，功表示力作用于一段距离所产生的能量传递。能量是标量，单位为焦耳 (J)，可以以多种形式存在，但我们重点关注动能 (因运动产生) 和重力势能 (因高度产生)。核心思想是功能原理：对质点所作的净功等于其动能的变化量。这一原理与能量守恒和功率计算一起，构成了许多 AQA 考题的骨架。

2. Work Done by a Constant Force | 恒力做功

Work done (W) by a constant force is defined as the product of the force magnitude and the displacement in the direction of the force. The general formula is W = F s cos θ, where F is the force (N), s is the displacement (m), and θ is the angle between the force and the direction of motion. When force and displacement are parallel (θ = 0), the formula simplifies to W = F s. If they are perpendicular (θ = 90°), cos 90° = 0, so no work is done by that force (e.g., the normal reaction on a horizontal surface). Always check that the force is truly constant; if it varies, you cannot use W = F s cos θ directly.

恒力所做的功 (W) 定义为力的大小与沿力方向的位移的乘积。一般公式为 W = F s cos θ，其中 F 是力的大小 (N)，s 是位移 (m)，θ 是力与运动方向之间的夹角。当力与位移平行时 (θ = 0)，公式简化为 W = F s。若两者垂直 (θ = 90°)，则 cos 90° = 0，因此该力不做功 (例如水平面上的法向反力)。请务必确认力是真正的恒力；若力变化，则不能直接使用 W = F s cos θ。

W = F s cos θ

3. Kinetic Energy | 动能

The kinetic energy (Ek) of an object is the energy it possesses due to its motion. For a mass m moving at speed v, the kinetic energy is given by Ek = ½ m v². Both mass and the square of speed matter; doubling the speed quadruples the kinetic energy. The unit is joules. When a resultant force does work, the kinetic energy changes according to the work-energy principle. In many problems, you will calculate the change in Ek to find work or vice versa.

动能 (Ek) 是物体因运动而具有的能量。对于质量为 m、以速度 v 运动的物体，动能由 Ek = ½ m v² 给出。质量和速度的平方都很重要；速度加倍会使动能变为原来的四倍。单位是焦耳。当合外力做功时，动能会根据功能原理发生改变。在许多题目中，你将通过计算动能变化来求功，或反之。

Ek = ½ m v²

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (Ep) is the energy stored due to an object’s height above a chosen reference level. The change in Ep is ΔEp = m g Δh, where m is mass, g is the acceleration due to gravity (usually 9.8 m s⁻² in AQA problems), and Δh is the vertical height change. The reference level is arbitrary; only differences in Ep have physical meaning. When an object is lifted, its Ep increases; when it falls, Ep decreases and is converted to kinetic energy (if no other forces do work).

重力势能 (Ep) 是因物体相对于所选参考水平面的高度而储存的能量。Ep 的变化为 ΔEp = m g Δh，其中 m 为质量，g 为重力加速度 (AQA 题目中通常取 9.8 m s⁻²)，Δh 为竖直高度变化量。参考水平面可任意选取，只有 Ep 的差值才有物理意义。物体被升高时，Ep 增加；下落时 Ep 减少，并转化为动能 (若无其他力做功)。

ΔEp = m g Δh

5. Conservation of Mechanical Energy | 机械能守恒

In a system where the only forces doing work are gravity and other conservative forces (no friction, air resistance, or external applied forces), the total mechanical energy remains constant: Ek₁ + Ep₁ = Ek₂ + Ep₂. This means a falling object loses Ep and gains Ek, while a rising object gains Ep at the expense of Ek. In exam problems, this principle lets you quickly link speed and height without dealing with forces directly. However, if non-conservative forces (like friction) are present, mechanical energy is not conserved — some energy is transferred to heat.

在一个只有重力和其他保守力做功 (无摩擦、空气阻力或外加力) 的系统中，总

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