Edexcel Maths: Work and Energy – Key Points Explained | Edexcel 数学：功和能量 考点精讲

📚 Edexcel Maths: Work and Energy – Key Points Explained | Edexcel 数学：功和能量 考点精讲

Work and energy form a core part of Edexcel A Level Mechanics. Understanding these concepts not only helps you solve problems involving moving objects and forces, but also builds a foundation for more complex topics like momentum and circular motion. The work‑energy principle provides a powerful alternative to Newton’s laws, especially when dealing with varying forces or motion over a distance. This article walks you through the essential points tested in the Edexcel specification, from defining work and kinetic energy to tackling inclined planes and power calculations.

功和能量是 Edexcel A Level 力学部分的核心内容。掌握这些概念不仅能帮助你解决物体运动和受力的问题，也为动量、圆周运动等更复杂的主题打下基础。功‑能原理提供了一种替代牛顿定律的强有力方法，尤其是在处理变力或沿路径的运动时。本文带你逐一梳理 Edexcel 考试大纲中的关键考点，从功和动能的定义到斜面问题与功率计算，助你精准备考。

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

When a constant force moves an object through a displacement, the work done by the force is defined as the product of the force magnitude, the displacement magnitude and the cosine of the angle between them. In symbols: W = F s cos θ, where F is the force, s is the displacement and θ is the angle between the force and the direction of motion. Work is measured in joules (J).

当恒力使物体发生位移时，力所做的功定义为力的大小、位移的大小以及力与位移夹角余弦的乘积。用公式表示：W = F s cos θ，其中 F 是力，s 是位移，θ 是力与运动方向之间的夹角。功的单位是焦耳 (J)。

W = F s cos θ

If the force acts in the same direction as the displacement, θ = 0° and W = F s. If the force is perpendicular to the displacement, θ = 90° and no work is done. For example, the normal reaction or the weight component perpendicular to an inclined plane does zero work.

如果力与位移同向，θ = 0°，W = F s；如果力垂直于位移，θ = 90°，做功为零。例如，法向反力或垂直于斜面的重力分量不做功。

2. Work Done Against Gravity and Friction | 克服重力和摩擦力做功

Work done against gravity when lifting an object vertically upwards is stored as gravitational potential energy: W = m g h, where m is mass, g is acceleration due to gravity (9.8 m s⁻²) and h is the vertical height gained. This work is independent of the path taken – only the change in vertical height matters.

竖直提升物体时克服重力所做的功储存为重力势能：W = m g h，其中 m 是质量，g 是重力加速度 (9.8 m s⁻²)，h 是上升的竖直高度。此功与路径无关——只取决于竖直高度的变化量。

Work done against a constant frictional or resistive force R over a distance d is W = R d, and this energy is dissipated as heat or sound.

克服恒定摩擦力或阻力 R 移动距离 d 所做的功为 W = R d，这些能量以热或声音的形式耗散。

• Gravity: a conservative force; work done against gravity is recoverable.
• 重力：保守力；克服重力做的功可以回收。
• Friction: a non‑conservative force; work done against friction is not recoverable.
• 摩擦力：非保守力；克服摩擦做的功不可回收。

3. Kinetic Energy | 动能

The kinetic energy (KE) of an object of mass m moving with speed v is KE = ½ m v². Kinetic energy is a scalar quantity and is always non‑negative. A change in kinetic energy occurs when a net force does work on the object.

质量为 m、速度为 v 的物体所具有的动能（KE）为 KE = ½ m v²。动能是标量，且恒为非负。当合外力对物体做功时，物体的动能发生变化。

KE = ½ m v²

It may also be expressed using initial speed u and final speed v: ΔKE = ½ m v² – ½ m u².

也可用初速度 u 和末速度 v 表示：ΔKE = ½ m v² – ½ m u²。

Remember that speed must be in m s⁻¹ and mass in kg to give KE in joules.

注意速度单位必须为 m s⁻¹，质量单位为 kg，才能得到以焦耳为单位的动能。

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Near the Earth’s surface, the change in GPE when an object moves through a vertical height Δh is ΔGPE = m g Δh. For most problems you can set a reference level where h = 0.

重力势能（GPE）是物体因在重力场中的位置而拥有的能量。在地球表面附近，物体竖直移动 Δh 时重力势能的变化量为 ΔGPE = m g Δh。在多数问题中，可设定一个 h = 0 的参考面。

If an object moves downwards, GPE decreases and kinetic energy usually increases. When moving upwards, GPE increases, requiring work to be done against gravity.

物体向下运动时，重力势能减小，通常动能增加；向上运动时，重力势能增大，需要克服重力做功。

5. The Work‑Energy Principle | 功‑能原理

The work‑energy principle states that the net work done on a particle by all the forces acting on it equals the change in its kinetic energy. This can be written as: W_net = ΔKE = ½ m v² – ½ m u². It is a very useful tool for solving problems where you know the initial and final speeds rather than time or acceleration.

功‑能原理指出：作用在质点上的所有力做的净功等于该质点的动能变化量。可写为：Wₙₑₜ = ΔKE = ½ m v² – ½ m u²。当已知初末速度而不关注时间或加速度时，功‑能原理是解决问题的一大利器。

Wₙₑₜ = ½ m v² – ½ m u²

Net work includes work done by driving forces, resisting forces, components of weight, etc. Be careful to subtract work done against friction or resistance, as these reduce the net work available to increase KE.

净功包括驱动力、阻力、重力分量等所做的功的总和。注意要减去克服摩擦力或阻力所做的功，因为这些会减少用于增加动能的净功。

Force type	Work contribution to W_net
Driving force parallel to motion	+ F s
Friction or resistance opposite to motion	– R s
Component of weight down slope	+ m g sin θ × s (if aids motion)
Tension or thrust	± T s depending on direction

6. Conservation of Mechanical Energy | 机械能守恒

If only conservative forces (like gravity) do work, the total mechanical energy (KE + GPE) of a system remains constant. This is the principle of conservation of mechanical energy. For a projectile or a particle sliding on a smooth slope, you can often equate initial and final total energy: ½ m u² + m g h₁ = ½ m v² + m g h₂.

如果只有保守力（如重力）做功，系统的总机械能（动能 + 重力势能）保持不变。这就是机械能守恒定律。对于抛体或在光滑斜面上滑动的质点，通常可令初态和末态的总能量相等：½ m u² + m g h₁ = ½ m v² + m g h₂。

½ m v² + m g h = constant

When non‑conservative forces (friction, air resistance) act, mechanical energy is not conserved. The work done by these forces must be accounted for as a change in total energy: W_nc = (KE₂ + GPE₂) – (KE₁ + GPE₁).

当存在非保守力（摩擦力、空气阻力）时，机械能不守恒。这些力所做的功必须计入总能量的变化：Wₙc = (KE₂ + GPE₂) – (KE₁ + GPE₁)。

7. Power and Its Calculation | 功率及其计算

Power is the rate of doing work or transferring energy. The average power P_avg = W / t or ΔE / t, where t is the time taken. The instantaneous power exerted by a constant force F moving its point of application with velocity v is P = F v, provided F and v are in the same direction.

功率是做功或能量转化的速率。平均功率 Pₐᵥ = W / t 或 ΔE / t，其中 t 为所用时间。恒力 F 在其作用点以速度 v 移动时产生的瞬时功率为 P = F v，前提是 F 与 v 同向。

P = F v

In problems involving vehicles, P = F v is often used to relate the driving force, the constant power of an engine and the speed. Remember to convert speeds to m s⁻¹.

在涉及车辆的问题中，P = F v 常用于关联驱动力、发动机恒定功率与速度。注意将速度换算为 m s⁻¹。

The unit of power is the watt (W): 1 W = 1 J s⁻¹. A common exam trick is to find the driving force when moving at a constant speed against resistance; then net force is zero, so driving force = resistance, and power = resistance × speed.

功率的单位是瓦特 (W)：1 W = 1 J s⁻¹。常见的考试套路是：当物体克服阻力匀速运动时，驱动力等于阻力，因此功率 = 阻力 × 速度。

8. Work‑Energy Problems on Inclined Planes | 斜面上的功能问题

On a rough inclined plane, the work‑energy approach is particularly efficient. Resolve the weight along and perpendicular to the plane. The component of weight down the slope is m g sin θ. The frictional force is F_friction = μ R, where the normal reaction R = m g cos θ unless other forces are acting.

在粗糙斜面上，用功‑能方法尤为高效。将重力沿斜面和垂直于斜面分解。沿斜面向下的重力分量为 m g sin θ。摩擦力为 F_friction = μ R，其中法向反力 R = m g cos θ（除非有其他力作用）。

Work done by the weight component over a distance s along the slope is (m g sin θ) s. Work done against friction is μ m g cos θ × s. Apply the work‑energy principle: net work = change in KE.

重力沿斜面分量移动距离 s 做功为 (m g sin θ) s。克服摩擦力做功为 μ m g cos θ × s。应用功‑能原理：净功 = 动能变化量。

Example: A block slides down a rough slope of length s from rest. Net work = m g sin θ × s – μ m g cos θ × s = ½ m v² – 0. Solve for v.

例：一物块从静止沿粗糙斜坡下滑长度 s。净功 = m g sin θ × s – μ m g cos θ × s = ½ m v² – 0，由此可求解 v。

9. Overcoming Variable Resistance & Connected Particles | 变阻力与连接体

Although the Edexcel specification mainly deals with constant forces, you may encounter a variable resistance that depends on displacement. The work done against such a force can be found by the area under a force–displacement graph. The work‑energy principle still applies: the area under a net force–displacement graph equals the change in KE.

尽管 Edexcel 大纲主要处理恒力，但你可能遇到随位移而变的变阻力。克服该力所做的功可由力‑位移图下的面积求得。功‑能原理仍然适用：净力‑位移图下的面积等于动能变化量。

For connected particles (e.g., one on a table, one hanging), you can treat the system as a whole using the work‑energy principle, or apply it separately to each particle, accounting for the work done by tension. Remember that the tension in an inextensible string does no net work on the system if both particles move the same distance, because the work done by tension on one particle cancels with the work done by tension on the other.

对于连接体（如一物在桌面上，一物悬挂），既可以整体运用功‑能原理，也可以分别对每个物体应用，此时需计入拉力做的功。要记住，不可伸长的绳中张力对整个系统不做净功，因为张力对其中一个物体做正功的同时对另一物体做等量负功（若两者位移大小相等）。

10. Common Errors and Exam Strategies | 常见错误与应试策略

Many marks are lost by mixing up units, forgetting to square the speed in KE, or misidentifying the angle in W = F s cos θ. Always draw a clear diagram and label all forces. Check whether forces do positive or negative work depending on their direction relative to displacement.

许多失分源于单位混淆、动能中遗忘速率平方、或弄错 W = F s cos θ 中的夹角。务必画出清晰的受力示意图并标明所有力。根据力与位移的方向关系，判断力做正功还是负功。

• Always convert g to 9.8 unless told otherwise. Use consistent SI units: m, kg, s, N, J, W.
• 除非另有说明，g 取 9.8。始终使用统一的国际单位：米、千克、秒、牛顿、焦耳、瓦特。
• Remember that work done by a force perpendicular to displacement is zero.
• 记住垂直于位移的力不做功。
• In work‑energy problems, pick initial and final states where speed or height is known conveniently.
• 在功能问题中，选择初末状态时应确保速度或高度已知。
• When power is constant and you need the maximum speed, set driving force = resistance and use P = F v.
• 当功率恒定时求最大速度，令驱动力等于阻力并用 P = F v。
• For conservation of energy questions, be explicit: decrease in GPE = increase in KE + work done against friction.
• 在能量守恒问题中，明确写出：重力势能减少量 = 动能增加量 + 克服摩擦力做的功。

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