📚 A-Level Physics: Work and Energy Key Points | A-Level 物理：功与能量 考点精讲
Work and energy are fundamental concepts in A-Level Physics, linking forces, motion, and the capacity to do work. This guide covers the essential definitions, formulas, and problem-solving techniques that appear regularly in exams. By mastering work, kinetic energy, potential energy, conservation of energy, power, and efficiency, you can tackle a wide range of mechanics problems with confidence.
功和能量是A-Level物理中的基本概念，将力、运动和做功能力联系起来。本指南涵盖考试中经常出现的重要定义、公式和解题技巧。掌握功、动能、势能、能量守恒、功率和效率，你就可以自信地解决各种力学问题。
1. Definition of Work | 功的定义
d the displacement (s) in the direction of the force: W = F s cos θ, where θ is the angle between the force and the displacement.
当一个力使物体沿力的方向发生位移时，就做了功。功的科学定义是力与在力方向上位移的乘积：W = F s cos θ，其中θ是力与位移之间的夹角。
he direction of the force (1 J = 1 N m). Work is a scalar quantity; it has magnitude only.
功的国际单位是焦耳(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(1 J = 1 N m)。功是标量，只有大小。
you carry a bag horizontally at constant velocity, the upward force from your hand does no work on the bag as the displacement is horizontal.
如果力与位移垂直(θ = 90°)，则不做功，因为cos 90° = 0。例如，当你提着包水平匀速运动时，手对包向上的力不做功，因为位移是水平的。
2. Work Done by a Constant Force | 恒力做功
the force and displacement are in the same direction (θ = 0°), this simplifies to W = F s.
当一个恒定的力F作用在物体上，物体沿直线移动距离s，所做的功为W = F s cos θ。如果力和位移方向相同(θ = 0°)，则简化为W = F s。
Always check the angle. If a force is applied at an angle to the horizontal, only the horizontal component (F cos θ) does work when the object moves horizontally. The vertical component does no work if there is no vertical displacement.
始终要检查角度。如果力与水平方向有一个角度，当物体水平运动时，只有水平分力(F cos θ)做功。如果没有竖直位移，竖直分力不做功。
Work can be positive, negative, or zero. Positive work is done when the force has a component in the direction of displacement (speeding up the object). Negative work is done when the force opposes the displacement (slowing it down), such as friction. Zero work occurs when there is no displacement or when the force is perpendicular to the displacement.
功可以是正功、负功或零功。正功是在力有沿位移方向的分量时做的功（使物体加速）；负功是力与位移方向相反时做的功（使物体减速），比如摩擦力；当没有位移或力垂直于位移时，功为零。
3. Work Done by a Variable Force (Force-Displacement Graph) | 变力做功（力-位移图）
When the force is not constant, the work done is found by calculating the area under a force–displacement graph. The area between the graph and the displacement axis represents the work done. This is because work equals the integral of force with respect to displacement.
当力不是恒力时，可以通过计算力-位移图下的面积来求得所做的功。图线与位移轴之间的面积代表所做的功。这是因为功等于力对位移的积分。
ph is a straight line, the area can be a triangle or trapezium. For a spring obeying Hooke's law (F = kx), the force– k done in stretching the spring is the area of a triangle: ½ × force × extension = ½ k x².
如果图线是直线，面积可以是三角形或梯形。对于遵从胡克定律(F = kx)的弹簧，力-伸长量图是一条过原点的直线，因此拉伸弹簧所做的功是三角形的面积：½ × 力 × 伸长量 = ½ k x²。
In exam problems, you might be asked to estimate the work done by counting squares under a curve or using the formula for the area. Always check the axes labels: force (N) on y-axis, displacement (m) on x-axis.
在考试问题中，你可能会被要求通过数曲线下的方格或使用面积公式来估算所做的功。一定要检查坐标轴标注：y轴是力(N)，x轴是位移(m)。
4. Kinetic Energy and Work-Energy Theorem | 动能与动能定理
the mass and v is the speed. Kinetic energy is always positive and is measured in joules.
动能(Eₖ)是物体由于运动而具有的能量。公式为Eₖ = ½ m v²，其中m是质量，v是速率。动能总是正值，单位为焦耳。
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: Wnet = ΔEₖ = ½ m vfinal² – s is a powerful tool for solving problems where forces cause acceleration or deceleration.
动能定理表明，物体所受合外力做的功等于其动能的变化量：Wnet = ΔEₖ = ½ m vfinal² – ½ m v_initial²。这是解决力引起加速或减速问题的有力工具。
20 m/s, the work done by the engine (net work, ignoring friction) is ΔEₖ = ½ × 1000 × (20² – J. Note: the change in speed squared is not the same as the square of the change in speed.
例如，一辆质量为1000 kg的汽车从10 m/s加速到20 m/s，发动机做的净功（忽略摩擦）为ΔEₖ = ½ × 1000 × (20² – 10²) = 150 000 J。注意：速度平方的变化量不等于速度变化量的平方。
5. Gravitational Potential Energy | 重力势能
he energy stored in an object due to its position in a gravitational field. Near the Earth' ight Δh is ΔEₚ = m g Δh, where g is the gravitational field strength (9.81 N/kg on Earth).
重力势能(Eₚ或GPE)是物体由于在引力场中的位置而储存的能量。在地球表面附近，质量为m的物体升高Δh时，重力势能的变化量为ΔEₚ = m g Δh，其中g是重力场强度（地球表面约9.81 N/kg）。
0, often the ground or the lowest point. The work done against gravity to lift an object is equal to the gain in GPE (if no other forces act).
重力势能是一个相对量；我们通常定义一个参考面，令其GPE = 0，常取地面或最低点。
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