📚 Mastering Work and Energy in IB & WJEC Mathematics | IB & WJEC 数学功与能量考点精讲
Work and energy are central to mechanics in both the IB Mathematics: Applications and Interpretation (AI) HL course and the WJEC Mathematics Mechanics modules. These topics link forces, motion, and calculus, allowing you to solve problems without directly handling time-dependent equations. This article covers core definitions, the work–energy principle, conservation of energy, power, and common exam pitfalls, with methods tailored to IB and WJEC syllabuses.
功和能量是 IB 数学：应用与解释 (AI) HL 课程和 WJEC 数学力学模块中的核心内容。这些主题将力、运动和微积分联系起来，让你无需直接处理含时间的方程就能解决问题。本文涵盖基本定义、功-能原理、能量守恒、功率以及常见考试陷阱，并结合 IB 与 WJEC 大纲要求的方法进行讲解。
1. Definition of Work | 功的定义
In mechanics, work is done when a force causes a displacement. The work W done by a constant force F acting on an object that moves through a displacement d in a straight line is given by W = F d cos θ, where θ is the angle between the force and the displacement vector.
在力学中，当力使物体发生位移时，力就对物体做了功。恒力 F 作用在物体上，物体沿直线发生位移 d，力所做的功为 W = F d cos θ，其中 θ 是力与位移矢量之间的夹角。
W = F d cos θ
If the force and displacement are in the same direction (θ = 0°), work is simply W = F d. If the force is perpendicular to the displacement (θ = 90°), no work is done.
如果力与位移方向相同 (θ = 0°)，功简化为 W = F d。若力垂直于位移 (θ = 90°)，则不做功。
Work is a scalar quantity measured in joules (J). 1 J = 1 N m. In mathematical modelling, work can be positive (energy transferred to the object) or negative (energy taken from the object).
功是标量，单位为焦耳 (J)。1 J = 1 N m。在数学建模中，功可以为正（能量传递给物体）或负（能量从物体移出）。
2. Work Done by a Constant Force | 恒力做功
For a single constant force, calculate work directly using W = F d cos θ. Always identify the component of force in the direction of motion. In inclined plane problems, for instance, the work done by gravity along the slope is mg d sin α, where α is the angle of inclination.
对于单个恒力，可直接使用 W = F d cos θ 计算功。务必找出沿运动方向的分力。例如在斜面问题中，重力沿斜面做的功为 mg d sin α，其中 α 为斜面倾角。
When several constant forces act, the total work is the sum of the work done by each force, or equivalently the work done by the resultant force in the direction of displacement.
多个恒力作用时，总功等于各力做功的代数和，也等于合力沿位移方向所做的功。
Work can also be expressed using the scalar (dot) product: W = F · d, which is especially useful in vector geometry questions found in both IB and WJEC papers.
功还可以用标量积（点积）表示：W = F · d，这在 IB 和 WJEC 试卷中的矢量几何题里格外有用。
3. Work Done by a Variable Force | 变力做功
When force varies with displacement, as with springs or resistive forces, the work done is found by integration. If a force F(x) acts along the x-axis from x = a to x = b, work done is:
当力随位移变化时（如弹簧或阻力），做功可以通过积分求得。若力 F(x) 沿 x 轴作用，从 x = a 到 x = b，做功为：
W = ∫ab F(x) dx
This is a key skill in IB AI HL, where students must interpret a force–displacement graph or integrate a given force function. Area under the force–distance curve represents work done.
这是 IB AI HL 的关键技能，学生需要解读力-位移图或对给定的力函数进行积分。力-距离曲线下的面积代表所做的功。
For a spring obeying Hooke’s law, F = kx, the work done in stretching from extension 0 to e is W = ½ k e².
对于遵循胡克定律的弹簧，F = kx，从伸长量 0 拉伸到 e 所做的功为 W = ½ k e²。
In WJEC Mechanics, variable forces may appear in the context of motion with resistance proportional to velocity, requiring integration as well.
在 WJEC 力学中，变力可能出现在阻力与速度成正比的情景中，同样需要积分处理。
4. Kinetic Energy (KE) | 动能
Kinetic energy is the energy an object possesses due to its motion. For a body of mass m moving with speed v, kinetic energy is:
动能是物体因运动而具有的能量。质量为 m、速度为 v 的物体，其动能为：
KE = ½ m v²
The unit is the joule. Note that KE is always non-negative and depends on v², so direction does not matter. In exams, you may be asked to find the change in KE when speed changes from u to v: ΔKE = ½ m (v² – u²).
单位是焦耳。注意动能始终非负，并且取决于 v²，因此方向无关。考试中可能要求计算速度从 u 变为 v 时的动能变化量：ΔKE = ½ m (v² – u²)。
This expression is directly linked to the work–energy principle and is used to find velocities without explicit equations of motion.
该表达式与功-能原理直接相关，可用于求速度而无需求解显式运动方程。
5. Gravitational Potential Energy (GPE) | 重力势能
Gravitational potential energy is the energy stored in an object due to its height above a reference level. Near Earth’s surface, for an object of mass m raised by a vertical height h:
重力势能是物体由于位于某参考面之上的高度而储存的能量。在地表附近，质量为 m 的物体升高竖直高度 h：
GPE = mgh
where g is the acceleration due to gravity (usually 9.8 m s⁻²). Choose a zero level consistently; only changes in GPE are physically meaningful.
其中 g 为重力加速度（通常取 9.8 m s⁻²）。需一致地选取零势能面；只有重力势能的变化才具有物理意义。
On an inclined plane, the vertical height is h = d sin α. The work done against gravity is mgh, independent of the path taken.
在斜面上，竖直高度为 h = d sin α。克服重力做功为 mgh，与路径无关。
6. Elastic Potential Energy (EPE) | 弹性势能
Elastic potential energy is stored in deformed elastic objects such as springs. For a spring with stiffness k extended or compressed by an amount x from its natural length:
弹性势能储存在形变的弹性物体（如弹簧）中。对于劲度系数为 k 的弹簧，当它被拉伸或压缩 x 时：
EPE = ½ k x²
This expression is derived by integrating Hooke’s law force over distance and is valid as long as the elastic limit is not exceeded.
该公式通过对胡克定律位移积分得出，且仅当未超出弹性限度时才成立。
In problems involving springs, EPE is often combined with KE and GPE in energy conservation equations, a common theme in both IB and WJEC mechanics.
在涉及弹簧的问题中，弹性势能常与动能、重力势能一起出现在能量守恒方程中，这是 IB 和 WJEC 力学中常见的主题。
7. The Work-Energy Principle | 功-能原理
The net work done on a particle by all external forces equals the change in its kinetic energy:
所有外力对质点所做的净功等于其动能的变化：
Wnet = ΔKE = ½ m v² – ½ m u²
This principle allows you to connect forces and displacements directly to speed, bypassing acceleration and time. It is especially powerful when forces are not constant.
该原理让你能够将力和位移与速度直接联系起来，而跳过加速度和时间。当力不是恒力时，这一原理尤其有效。
In WJEC M1, you often apply this to objects on slopes, rough surfaces, or connected particles. Include work done by friction and tension.
在 WJEC M1 中，该原理经常用于处理斜面上的物体、粗糙表面上的物体或连接的质点。需计入摩擦力和张力所做的功。
For IB AI HL, questions may require you to set up an integral for net work and equate it to ΔKE, reinforcing the use of calculus in mechanics.
对于 IB AI HL，题目可能要求建立净功的积分式并令其等于 ΔKE，强化微积分在力学中的应用。
8. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces (gravity, elastic) do work, the total mechanical energy E = KE + PE remains constant. This yields equations like:
当只有保守力（重力、弹力）做功时，总机械能 E = KE + PE 保持不变。由此可得方程：
½ m v₁² + mgh₁ + ½ k x₁² = ½ m v₂² + mgh₂ + ½ k x₂²
Energy lost to friction or air resistance means mechanical energy is not conserved, but the work–energy principle still holds if you include the work done by non-conservative forces.
摩擦力或空气阻力造成的能量损失意味着机械能不守恒，但如果计入非保守力所做的功，功-能原理仍适用。
Typical exam applications include pendulums, roller coasters, and springs. Identify the initial and final states and equate total energies, taking care to define the zero of potential energy consistently.
典型的考试应用包括单摆、过山车和弹簧。确定初末状态并使总能量相等，同时注意始终一致地定义势能零点。
9. Power and Efficiency | 功率与效率
Power is the rate of doing work or transferring energy. Average power Pavg = W / t, and instantaneous power P = F v, where v is the velocity in the direction of the force.
功率是做功或能量传递的速率。平均功率 Pavg = W / t，瞬时功率 P = F v，其中 v 是沿力方向的速度。
P = F v
This formula is used when a vehicle’s engine provides a driving force at a certain speed. Units are watts (W), where 1 W = 1 J s⁻¹.
该公式用于车辆发动机以一定速度提供驱动力的情况。单位是瓦特 (W)，1 W = 1 J s⁻¹。
Efficiency η = (useful work out / total energy in) × 100%. For inclined conveyors or pumps, efficiency may be given, and you need to relate power input to the rate of gain of GPE.
效率 η = (有用输出功 / 总输入能) × 100%。对于倾斜传送带或水泵，可能给出效率，你需要将输入功率与重力势能增加的速率联系起来。
Both IB and WJEC exams include power questions, often requiring conversion of units and careful interpretation of “useful power”.
IB 和 WJEC 考试都含有功率题，通常要求进行单位换算并仔细解读“有用功率”。
10. Exam Tips and Common Pitfalls | 考试技巧与常见错误
Many students lose marks by confusing mass and weight, or by omitting the cos θ factor when force and displacement are not parallel. Always draw a clear force diagram and check the angle used.
许多学生因为混淆质量和重量，或在力和位移不平行时遗漏 cos θ 因子而丢分。务必画出清晰的受力图并检查所用的角度。
In variable force integrations, remember to use the correct limits; the lower limit often corresponds to the initial position. Sign errors are frequent when work is done against a force.
在处理变力积分时，记住使用正确的积分限；下限通常对应初始位置。当力做负功时，符号错误很常见。
When applying conservation of energy, do not double-count work – if you have already included a force’s potential energy, do not also add the work done by that force.
在应用能量守恒时，不要重复计算功——如果已经考虑了某个力的势能，就不要再加上该力所做的功。
For WJEC Mechanics, show all steps of substituting into equations; marks are awarded for correct substitution even if the final answer is wrong. In IB, correct notation and clear reasoning are essential for the higher marks.
对于 WJEC 力学，要展示代入方程的所有步骤；即使最终答案错误，正确代入也能得分。在 IB 中，正确的符号和清晰的推理对获取高分至关重要。
Practice converting units: speed in m s⁻¹, distance in m, mass in kg. Using km h⁻¹ without conversion will lead to incorrect energy values.
练习单位换算：速度用 m s⁻¹，距离用 m，质量用 kg。使用 km h⁻¹ 而不换算将导致能量值错误。
Finally, remember that the work–energy principle and conservation laws are often faster than kinematics equations, especially when time is not asked for. Use them strategically.
最后，记住功-能原理和守恒定律通常比运动学方程更快，特别是在不涉及时间时。要有策略地使用它们。
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