Work and Energy Key Points | 功和能量考点精讲

📚 Work and Energy Key Points | 功和能量考点精讲

Work and energy provide powerful tools for solving mechanics problems in A-Level CCEA Mathematics. These concepts link forces, displacements, velocities and masses, allowing you to avoid complicated kinematics when forces are not constant. In this revision guide we cover the definitions, the work–energy principle, conservation of energy, power and variable forces, all aligned with the CCEA M2/M3 specification.

功和能量为解决A-Level CCEA数学中的力学问题提供了有力工具。这些概念把力、位移、速度和质量联系起来，让你在力不恒定时可以避开复杂的运动学计算。本篇复习指南涵盖功的定义、功–能原理、能量守恒、功率以及变力做功，完全贴合CCEA M2/M3考纲。

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1. Definition of Work Done | 做功的定义

Work is done when a force causes a displacement in the direction of the force. For a constant force F acting on an object moving through a displacement s, the work done is given by the scalar product of force and displacement.

当一个力使物体沿力的方向发生位移时，该力就对物体做了功。对于作用在物体上的恒力F，物体移动位移s时，所做的功等于力与位移的点积。

W = F s cos θ

displacement (θ = 90°), no work is done. If the force has a component opposing motion (θ > 90°), work is negative, which represents energy being taken from the object.

这里θ是力矢量与位移矢量之间的夹角。功的单位是焦耳(J)；1 J = 1 N m。如果力与位移垂直（θ = 90°），则不做功。如果力的分量与运动方向相反（θ > 90°），则做负功，表示能量从物体中移走。

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2. Work Done Against Gravity | 克服重力做功

When an object is lifted at constant speed, the lifting force must overcome the object's weight. The work done against gravity is equal to the gain in gravitational potential energy, assuming no energy losses. For a vertical lift of height h, the work done by the lifting force is:

当物体匀速上升时，提升力必须克服物体的重量。假设没有能量损失，克服重力所做的功等于物体增加的重力势能。对于竖直上升高度h，提升力所做的功为：

W = m g h

where m is mass and g is the acceleration due to gravity (typically 9.8 m s⁻²). This expression holds even if the lifted path is not vertical, provided h is the vertical height gained. CCEA questions often involve dragging an object up a slope; then h = d sin α, where d is the distance along the slope and α the angle of inclination.

其中m为质量，g为重力加速度（通常取9.8 m s⁻²）。即使提升的路径不是竖直的，只要h是增加的竖直高度，该式依然成立。CCEA试题常涉及沿斜面拖动物体，此时h = d sin α，其中d为沿斜面的距离，α为斜面的倾角。

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3. Gravitational Potential Energy | 重力势能

n in a gravitational field. If we choose a reference level where GPE = 0, then at a height h above that level:

重力势能(GPE)是物体因在重力场中的位置而具有的能量。如果我们选定一个GPE为零的参考面，则在高于该参考面h的高度处：

GPE = m g h

nds to positive work done by gravity, which often converts into kinetic energy. In problems, always define your zero level clearly to avoid sign errors.

将物体举高时克服重力所做的功等于GPE的改变量：ΔGPE = mg Δh。GPE的减少对应重力做正功，这常转化为动能。解题时务必明确零势能参考面，以避免符号错误。

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4. Kinetic Energy | 动能

Kinetic energy (KE) is the energy a body possesses by virtue of its motion. For a body of mass m moving with speed v:

动能(KE)是物体因运动而具有的能量。对于质量为m、速度为v的物体：

KE = ½ m v²

Kinetic energy is a scalar quantity, always non-negative, and is measured in joules. The change in kinetic energy is central to the work–energy principle. Note that direction of velocity does not affect KE – only the magnitude of velocity matters.

动能是标量，始终非负，单位为焦耳。动能的改变是功–能原理的核心。注意速度的方向不影响动能——只有速度的大小才起作用。

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5. The Work–Energy Principle | 功–能原理

The work–energy principle states that the total work done by all forces acting on a particle equals the change in its kinetic energy. This includes work done by driving forces, resistances, components of weight, and tension. It does not require forces to be constant.

功–能原理指出：作用在质点上的所有力所做的总功等于该质点动能的变化量。这包括驱动力、阻力、重力分量和张力所做的功。它不要求力是恒定的。

W_total = ΔKE = ½ m v² – ½ m u²

Here u is initial speed and v is final speed. If the total work is positive, kinetic energy increases; if negative, it decreases. This principle is especially useful in finding speed when acceleration is not uniform. CCEA examiners often expect you to apply this principle rather than using suvat equations when a force varies.

这里u为初速度，v为末速度。若总功为正，动能增加；若为负，动能减少。该原理在加速度非均匀时特别适合求速度。CCEA阅卷老师常期望你应用功–能原理，而非在力变化时使用匀加速运动公式。

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6. Conservation of Mechanical Energy | 机械能守恒

When the only forces doing work are conservative – such as gravity – mechanical energy is conserved. This means the sum of kinetic energy and gravitational potential energy remains constant throughout the motion, provided resistive forces are negligible.

当做功的力只有保守力（如重力）时，机械能守恒。这意味着在运动过程中，动能和重力势能的总和保持不变，只需忽略小到可忽略的阻力。

KE₁ + GPE₁ = KE₂ + GPE₂

This equation can be applied directly to problems involving pendulums, projectiles (ignoring air resistance) or smooth slopes. However, if friction or air resistance does significant work, conservation does not hold – you must use the work–energy principle accounting for those non-conservative forces.

该方程可直接应用于摆、抛体（忽略空气阻力）或光滑斜面的问题。但若摩擦力或空气阻力做了不可忽略的功，机械能守恒便不再成立——你必须使用功–能原理，将这些非保守力考虑在内。

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7. Work Done by a Variable Force | 变力做功

k done cannot be found from W = F s cos θ alone. Instead, work is the area under the force–displacement graph. If the force F(x) is given as a function of position x, the work done from x = a to x = b is:

当力随位移变化时，不能只用W = F s cos θ求功。此时功等于力–位移图像下的面积。若力F(x)作为位置x的函数给出，从x = a到x = b所做的功为：

W = ∫ₐᵇ F(x) dx

h or using a given function. For example, the force needed to stretch a spring obeys Hooke' ½ k e². Integration is often tested, so be comfortable with basic polynomial integration.

在CCEA M2考试中，你可能需要根据一条直线图像或给定的函数计算功。例如，拉伸弹簧所需的力遵循胡克定律，F = kx，因此将弹簧拉伸距离e所做的功为W = ½ k e²。积分经常被考察，所以要对基本多项式积分感到熟练。

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8. Power | 功率

Power is the rate at which work is done, or energy is transferred. The average power is given by:

功率是做功或能量转移的速率。平均功率为：

P_avg = ΔW / Δt

Instantaneous power, especially useful when a vehicle is moving under a constant power engine, is the product of the driving force and velocity:

瞬时功率（尤其当车辆在恒定功率引擎下运动时很有用）是驱动力与速度的乘积：

P = F v

y: F = P / v. As the vehicle speeds up, the available force drops, eventually balancing resistance and reaching a maximum speed. This topic frequently appears in CCEA exam questions involving cars or trains.

功率的单位是瓦特(W)，1 W = 1 J s⁻¹。若恒定动力源驱动车辆，则驱动力与速度成反比：F = P / v。随着车辆加速，可用的力会下降，最终与阻力平衡并达到最大速度。这一课题经常出现在涉及汽车或火车的CCEA试题中。

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9. Work and Energy on an Inclined Plane | 斜面上的功和能量

up a slope of length d, inclined at an angle α to the horizontal. The component of weight parallel to the slope is mg sin α, so the work done against gravity is mg d sin α = mgh, where h = d sin α. If friction acts, with coefficient of friction μ, the frictional force is μR = μmg cos α, and work done against friction is μmg cos α × d.

斜面问题综合了克服重力做功、克服摩擦力做功以及动能的变化。设质量为m的物块被沿长度为d的斜面向上拉动，斜面与水平面的倾角为α。重力平行于斜面的分量为mg sin α，因此克服重力所做的功为mg d sin α = mgh，其中h = d sin α。若存在摩擦，摩擦系数为μ，则摩擦力为μR = μmg cos α，克服摩擦力做功为μmg cos α × d。

Applying the work– s = change in KE. The pulling force T does positive work T × d, gravity does negative work –mg d sin α, friction does negative work –μmg d cos α. The equation becomes T d – mg d sin α – μmg d cos α = ½ m v² – ½ m u². This unified approach handles many CCEA inclined plane questions efficiently.

对物块应用功–能原理：所有力的总功等于动能的变化量。拉力T做正功T × d，重力做负功–mg d sin α，摩擦力做负功–μmg d cos α。方程变为T d – mg d sin α – μmg d cos α = ½ m v² – ½ m u²。这种统一的方法可以高效地处理许多CCEA斜面问题。

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10. Common Pitfalls and Exam Strategy | 常见陷阱与考试策略

Sign errors: Always decide a positive direction and assign signs consistently to forces and displacements. Work done against a force is negative work by that force. Unit consistency: Stick to SI units – mass in kg, displacement in m, force in N, work/energy in J. Zero level for GPE: State your chosen zero level explicitly and use vertical height to compute changes. Variable force integration limits: Match limits to the displacement interval. Using suvat incorrectly: Do not use constant-acceleration equations unless acceleration is truly constant; prefer the work–energy principle when forces vary.

符号错误：始终选定一个正方向，并对力和位移统一使用正负号。克服某个力做功意味着该力做负功。单位一致性：坚持使用国际单位制——质量用kg、位移用m、力用N、功 / 能用J。重力势能的零势能面：明确规定你选定的零势能面，并用竖直高度计算变化量。变力积分的上下限：使积分上下限对应位移区间。错误使用匀加速运动公式：只有在加速度真正不变时才使用匀加速公式；当力变化时，优先采用功–能原理。

In CCEA exams, always list the forces acting and their work contributions before forming an equation. Check that all terms are included, including work done by an applied force, weight component, tension, friction, and any other resistance. A clear diagram helps prevent missing forces.

在CCEA考试中，建立方程之前，应始终列出所有作用力及其做功的贡献。检查所有项是否已包括，如外加力、重力分量、张力、摩擦力及其他阻力的功。一个清晰的图示有助于避免遗漏力。

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