📚 Work and Energy for IB CCEA Mathematics: Key Points | IB CCEA 数学：功和能量考点精讲
Understanding work and energy is crucial for solving mechanics problems in IB CCEA Mathematics. This article provides a comprehensive revision of key concepts, formulas, and typical exam questions involving work, kinetic energy, potential energy, the work-energy theorem, conservation of energy, and power.
理解功和能量对于解决 IB CCEA 数学中的力学问题至关重要。本文全面复习了关键概念、公式和典型考题，包括功、动能、势能、动能定理、能量守恒和功率。
1. Definition of Work | 功的定义
In physics, work is done when a force moves an object through a displacement in the direction of the force. Mathematically, for a constant force F and displacement d, work W is defined as W = F d cos θ, where θ is the angle between the force vector and the displacement vector. The SI unit of work is the joule (J).
在物理学中，当力使物体沿力的方向发生位移时，力就做了功。对于恒力 F 和位移 d，功 W 的定义是 W = F d cos θ，其中 θ 是力矢量与位移矢量之间的夹角。功的国际单位是焦耳（J）。
Work is a scalar quantity; it can be positive, negative, or zero. Positive work adds energy to the system, negative work removes energy. For instance, lifting a book increases its gravitational potential energy, so you do positive work against gravity; lowering the book involves negative work.
功是标量；可为正、负或零。正功为系统增加能量，负功从系统移除能量。例如，举起一本书增加了它的重力势能，因此你对重力做正功；放下书则做负功。
2. Work Done by a Constant Force | 恒力做功
When a constant force acts on a particle moving in a straight line, the work done is W = F s cos θ. If the force is parallel to the displacement (θ = 0°), then W = F s. If perpendicular (θ = 90°), no work is done. In exam problems, you often resolve forces into components and calculate the work done by each component separately.
当恒力作用于沿直线运动的质点时，做功为 W = F s cos θ。若力与位移平行（θ = 0°），则 W = F s。若垂直（θ = 90°），则不做功。在考题中，常需将力分解为分量，分别计算每个分量所做的功。
Example: A force of 10 N acts at 60° to the horizontal, moving a box 5 m along the floor. The work done is 10 × 5 × cos60° = 25 J. Notice that only the horizontal component of the force contributes to the work.
示例：一个10 N的力与水平方向成60°角，推动箱子沿地面移动5 m。做功为 10 × 5 × cos60° = 25 J。注意，只有力的水平分量做了功。
On an inclined plane, the work done by gravity when a block slides down a distance d along the slope is given by W_gravity = m g sin α × d, where α is the angle of inclination. This expression will be combined with friction and other forces in full problems.
在斜面上，当滑块沿斜面下滑距离 d 时，重力做功为 W_重力 = m g sin α × d，其中 α 为倾角。该表达式将在综合问题中与摩擦力及其他力结合使用。
3. Work Done by a Variable Force | 变力做功
If the force is not constant, work is calculated as the area under the force-displacement graph or by integration: W = ∫ₐᵇ F(x) dx, where F(x) is the force as a function of position x. In IB CCEA Mathematics, you may need to integrate functions like F(x) = kx (spring force) or polynomial expressions.
若力不恒定，功可通过力-位移图下的面积计算，或通过积分：W = ∫ₐᵇ F(x) dx，其中 F(x) 是力关于位置 x 的函数。在 IB CCEA 数学中，可能需要积分如 F(x) = kx（弹簧力）或多项式表达式。
For a spring obeying Hooke’s law, the work done in stretching it from extension a to extension b is W = ∫ₐᵇ kx dx = ½k(b² – a²). If the spring starts at its natural length (a = 0), the work simplifies to ½k b², which is the elastic potential energy stored.
对于符合胡克定律的弹簧，将其从伸长量 a 拉伸至 b 所做的功为 W = ∫ₐᵇ kx dx = ½k(b² – a²)。若弹簧从原长开始（a = 0），功简化为 ½k b²，即储存的弹性势能。
In exam contexts, you might be given a force-distance graph where the area can be found by counting squares or using trapezium rules. Always consider the sign: areas above the distance axis correspond to positive work, while areas below correspond to negative work.
在考试中，可能给出力-距离图，通过数方格或梯形法则求面积。务必考虑正负：距离轴上方的面积对应正功，下方的面积对应负功。
4. Kinetic Energy and the Work-Energy Theorem | 动能与动能定理
Kinetic energy (KE) is the energy possessed by an object due to its motion, given by KE = ½ m v², where m is mass and v is speed. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ∆KE = ½ m v² – ½ m u², where u is initial speed and v is final speed.
动能（KE）是物体因运动而具有的能量，公式为 KE = ½ m v²，其中 m 为质量，v 为速率。动能定理指出，作用于物体的净功等于其动能的变化量：W_net = ΔKE = ½ m v² – ½ m u²，其中 u 为初速率，v 为末速率。
This theorem is a powerful tool because it applies regardless of the nature of the forces (conservative or non-conservative) and bypasses the need for acceleration and time calculations. It is especially convenient when forces vary with position.
该定理是一个强大工具，因为它与力的性质（保守力或非保守力）无关，且无需计算加速度和时间。当力随位置变化时，它特别方便。
When using this theorem, carefully identify all forces doing work and sum their contributions with appropriate signs. The normal reaction force and the centripetal force usually do zero work because they act perpendicular to the displacement.
使用该定理时，仔细找出所有做功的力，并用恰当的正负号求和。法向反作用力和向心力通常不做功，因为它们与位移垂直。
5. Potential Energy: Gravitational and Elastic | 势能：重力势能和弹性势能
Potential energy is stored energy due to an object’s position or configuration. In IB CCEA problems, two types are common: gravitational potential energy (GPE) and elastic potential energy (EPE).
势能是由物体的位置或形状决定的储存能量。在 IB CCEA 问题中，常见两种类型：重力势能（GPE）和弹性势能（EPE）。
Gravitational potential energy change near Earth’s surface is ∆GPE = m g h, where h is the vertical height change. The zero level can be selected arbitrarily; only differences matter. For a more general approach, GPE = –G M m / r, but this is rarely needed in this course.
地表附近的重力势能变化为 ΔGPE = m g h，h 为垂直高度变化。零势能面可任意选择；只有差值才有意义。更一般的形式为 GPE = –G M m / r，但本课程很少用到。
Elastic potential energy for a spring obeying Hooke’s law is EPE = ½ k x², where k is the spring constant and x is the extension or compression from the natural length. Note that both extension and compression store positive potential energy.
遵循胡克定律的弹簧的弹性势能为 EPE = ½ k x²，其中 k 为弹簧劲度系数，x 为从原长计的伸长或压缩量。注意，伸长和压缩均储存正势能。
6. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces (such as gravity and ideal spring forces) do work, the total mechanical energy (KE + PE) remains constant. This principle provides a direct link between speed and position:
当只有保守力（如重力和理想弹力）做功时，总机械能（动能 + 势能）保持不变。该原理直接关联速度与位置：
½ m v₁² + m g h₁ + ½ k x₁² = ½ m v₂² + m g h₂ + ½ k x₂²
Common scenarios include pendulums, free fall, roller coasters, and spring-mass systems. For a simple pendulum, as the bob swings down, GPE converts to KE, and the speed at the lowest point can be found without analysing tension.
常见情境包括摆、自由落体、过山车和弹簧-质量系统。对于单摆，摆锤下摆时 GPE 转化为 KE，最低点速度无需分析张力即可求得。
A critical check: before applying conservation of mechanical energy, confirm that no non-conservative forces (friction, air resistance, applied external forces) are doing work. If they are present, the total mechanical energy changes, and you must use the work-energy theorem instead.
关键检查：在应用机械能守恒前，确认没有非保守力（摩擦、空气阻力、外加力）做功。若存在，总机械能会改变，必须改用动能定理。
7. Power | 功率
Power is the rate of doing work or transferring energy. Average power P_avg = W/t, where W is work done in time t. Instantaneous power for a constant force acting on a moving object is P = F v cos θ. The SI unit is the watt (W), equivalent to J/s.
功率是做功或能量传递的速率。平均功率 P_avg = W/t，其中 W 为时间 t 内做的功。作用于运动物体的恒力的瞬时功率为 P = F v cos θ。国际单位是瓦特（W），相当于 J/s。
If a car engine delivers a constant power P, and the vehicle moves at speed v, the driving force is given by F = P/v. As the car accelerates and v increases, the available driving force decreases, explaining why acceleration falls off at high speeds.
若汽车引擎输出恒定功率 P，车辆以速度 v 行驶，则驱动力为 F = P/v。随着汽车加速、v 增大，可用的驱动力减小，这解释了为何高速时加速度下降。
In problems with slopes, the power required to maintain a constant speed v up an incline can be found by multiplying the total resistive force (including component of weight and friction) by the velocity: P = (m g sin θ + f) v.
在斜坡问题中，保持匀速 v 上坡所需的功率，可由总阻力
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