AS Physics: Work and Energy Key Points Review | AS 物理：功与能量考点精讲

📚 AS Physics: Work and Energy Key Points Review | AS 物理：功与能量考点精讲

Welcome to this focused AS-level revision guide on work and energy. Understanding the relationship between force, displacement, kinetic energy, potential energy, and power is essential for solving mechanics problems and forms the foundation for more advanced physics. This article covers key definitions, formulas, graphical interpretations and common pitfalls, all presented in clear bilingual pairs to help you master the topic.

欢迎阅读这份针对 AS 物理功与能量的考点精讲。理解力、位移、动能、势能和功率之间的关系，对于解决力学问题至关重要，也是进一步学习物理的基础。本文涵盖关键定义、公式、图像解释和常见误区，全部以中英对照的清晰段落呈现，助你彻底掌握这一主题。

1. Work Done by a Constant Force | 恒力做的功

In physics, work is done when a force causes a displacement. For a constant force F acting at an angle θ to the direction of displacement s, the work done W is defined by the product of the force component along the displacement and the magnitude of the displacement.

在物理学中，力使物体发生位移时便做了功。对于大小为 F 的恒力，若其与位移 s 方向的夹角为 θ，则所做的功 W 定义为力在位移方向的分量与位移大小的乘积。

W = F s cos θ

Work is a scalar quantity measured in joules (J). One joule is equal to the work done when a force of one newton moves an object through one metre in the direction of the force.

功是一个标量，单位为焦耳 (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。

When the force is parallel to the displacement (θ = 0°), cosθ = 1 and W = F s. If the force is perpendicular (θ = 90°), cosθ = 0 and no work is done. If θ is between 90° and 270°, cosθ is negative and the work done is negative, meaning energy is taken away from the object.

当力与位移平行时 (θ = 0°)，cosθ = 1，W = F s。若力与位移垂直 (θ = 90°)，cosθ = 0，不做功。若 θ 介于 90° 到 270° 之间，cosθ 为负，做功为负，表示能量从物体中被抽走。

2. Work Done by a Varying Force | 变力做的功

When the applied force is not constant, the work done cannot be calculated simply by W = F s cosθ. Instead, we consider the area under a force–displacement (F–x) graph. For a spring obeying Hooke’s law, the force is directly proportional to extension: F = k x, where k is the spring constant.

当作用力不是恒力时，不能简单地用 W = F s cosθ 计算功。此时我们要考虑力–位移 (F–x) 图线下的面积。对于遵循胡克定律的弹簧，力与伸长量成正比：F = k x，其中 k 为劲度系数。

F = k x

The work done in stretching a spring from its natural length to an extension x is given by the area of the triangle under the F–x graph, which leads to the formula for elastic potential energy stored.

将弹簧从自然长度拉伸至伸长量 x 所做的功，等于 F–x 图下三角形的面积，由此得到储存的弹性势能公式。

W = ½ k x²

More generally, for any varying force, the work done equals the definite integral of force with respect to displacement, which at AS level is interpreted as the area under the curve.

更一般地，对于任意变力，做功等于力对位移的定积分，在 AS 阶段可理解为图线下的面积。

3. Kinetic Energy | 动能

Kinetic energy (KE) is the energy an object possesses due to its motion. For a body of mass m moving with speed v, kinetic energy is given by:

动能 (KE) 是物体因自身运动而具有的能量。对于质量为 m、速度为 v 的物体，动能由下式给出：

KE = ½ m v²

Kinetic energy is a scalar quantity and is measured in joules. It depends on the square of the speed, so doubling the speed results in four times the kinetic energy.

动能是标量，单位为焦耳。它与速度的平方成正比，因此速度加倍会使动能变为原来的四倍。

Kinetic energy is always positive or zero; it does not have direction. The concept of kinetic energy is central to understanding collisions and the work–energy theorem.

动能始终非负，没有方向。动能概念是理解碰撞和动能定理的核心。

4. Work–Energy Theorem | 动能定理

The work–energy theorem states that the net work done by all forces acting on an object is equal to the change in its kinetic energy.

动能定理指出：作用在物体上所有力的净功，等于物体动能的变化量。

W_net = ΔKE = KE_f − KE_i

This theorem can be derived from Newton’s second law and the equations of motion. For a constant net force producing acceleration a, we have v² = u² + 2 a s, and multiplying by ½ m gives ½ m v² – ½ m u² = m a s = F_net s.

该定理可由牛顿第二定律和运动学方程导出。对于产生加速度 a 的恒定净力，有 v² = u² + 2 a s，两边乘以 ½ m 得到 ½ m v² – ½ m u² = m a s = F_net s。

The work–energy theorem applies even when forces are not constant, making it a powerful tool for solving problems where direct calculation of acceleration is cumbersome.

即使力不是恒力，动能定理也成立，这使得它成为解决复杂动力学问题的有力工具。

5. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy stored in an object due to its height above a reference level. Near the Earth’s surface, it is approximated as:

重力势能 (GPE) 是物体因相对于参考平面的高度而储存的能量。在地表附近，可近似表示为：

E_p = m g h

where m is mass, g is the gravitational field strength (9.81 m s⁻² on Earth), and h is the vertical height. The choice of reference level (where h = 0) is arbitrary; only changes in GPE have physical significance.

其中 m 为质量，g 为重力场强（地球取 9.81 m s⁻²），h 为竖直高度。零势能面（h = 0）的选取是任意的，只有势能的变化才有物理意义。

When an object is raised against gravity, its GPE increases; the work done by gravity is negative: W_g = −ΔE_p. Conversely, when an object falls, gravity does positive work and GPE decreases.

当物体克服重力上升时，重力势能增加；重力做功为负：W_g = −ΔE_p。反之，当物体下落时，重力做正功，重力势能减少。

6. Elastic Potential Energy | 弹性势能

Elastic potential energy is stored in a deformed elastic object, such as a stretched spring. Provided the spring obeys Hooke’s law (F = k x), the energy stored when stretched by an amount x from its natural length is:

弹性势能储存在形变的弹性物体中，例如拉伸的弹簧。只要弹簧遵循胡克定律（F = k x），则从自然长度拉伸 x 所储存的能量为：

E_e = ½ k x²

This expression comes from the work done to stretch the spring, which is the area under the F–x graph. The spring constant k has units N m⁻¹ and determines the stiffness of the spring.

这一表达式源于拉伸弹簧所做的功，即 F–x 图下的面积。劲度系数 k 的单位为 N m⁻¹，它决定了弹簧的软硬程度。

Elastic potential energy can be converted into kinetic energy and vice versa, as in a mass–spring oscillator. If the spring is compressed, the same formula applies, with x representing the compression distance.

弹性势能和动能可以相互转化，如弹簧振子。若弹簧被压缩，公式同样适用，x 表示压缩距离。

7. Conservation of Mechanical Energy | 机械能守恒

The principle of conservation of mechanical energy states that if only conservative forces (such as gravity and spring forces) do work, the total mechanical energy E = KE + PE remains constant.

机械能守恒定律指出：如果只有保守力（如重力和弹簧力）做功，那么系统的总机械能 E = KE + PE 保持不变。

KE_i + PE_i = KE_f + PE_f

A classic example is a pendulum swinging in a vacuum: at the highest points, speed is zero (all energy is GPE); at the lowest point, height is minimum (all energy is KE). Similarly, a mass falling freely converts GPE into KE.

典型例子是真空中摆动的单摆：在最高点，速度为零（全部为重力势能）；在最低点，高度最小（全部为动能）。同样，自由下落的物体将重力势能转化为动能。

When non-conservative forces like friction or air resistance act, mechanical energy is not conserved; some energy is transferred to thermal energy and sound, so the total energy of the system plus surroundings is still conserved.

当存在摩擦力或空气阻力等非保守力时，机械能不守恒；部分能量转化为热能和声能，但系统加环境的总能量依然守恒。

8. Power | 功率

Power is defined as the rate of doing work or the rate of energy transfer. The average power P over a time interval Δt is:

功率定义为单位时间内做功的多少或能量转移的速率。在时间间隔 Δt 内的平均功率 P 为：

P = W / t or P = ΔE / Δt

The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹. For a constant force F moving an object at constant velocity v in the direction of the force, the instantaneous power can be written as:

功率的国际单位是瓦特 (W)，1 W = 1 J s⁻¹。对于以恒定速度 v 沿力的方向移动的恒力 F，瞬时功率可表示为：

P = F v

If the force is at an angle to velocity, P = F v cos θ. This relationship is extremely useful for problems involving vehicles, motors and engines operating at constant maximum power.

若力与速度有夹角，则 P = F v cos θ。这一关系在分析车辆、电机和发动机的恒定最大功率问题时非常有用。

9. Efficiency | 效率

Efficiency η (eta) quantifies how well a system converts input energy into useful output energy. It is expressed as a percentage:

效率 η 用于衡量系统将输入能量转化为有用输出能量的程度，通常用百分比表示：

η = (useful energy output / total energy input) × 100%

Alternatively, in terms of power: η = (useful power output / total power input) × 100%. Because of energy dissipation due to friction, sound and heat, efficiency is always less than 100% for real machines.

也可用功率表示：η = (有用输出功率 / 总输入功率) × 100%。由于摩擦、声和热的耗散，实际机械的效率总是低于 100%。

Improving efficiency often involves reducing unwanted energy transfers, such as using lubrication to reduce friction or streamlining shapes to reduce air resistance.

提高效率常常意味着减少不必要的能量转移，例如使用润滑剂减少摩擦，或采用流线型设计减少空气阻力。

10. Graphical Interpretation of Work | 功的图像解释

A force–displacement graph is a powerful visual tool. The area between the force curve and the displacement axis represents the work done. For a constant force, the graph is a horizontal line, and the area is a rectangle: W = F s.

力–位移图是一个强大的可视化工具。力曲线与位移轴之间的面积表示所做的功。对于恒力，图线为水平线，面积为矩形：W = F s。

For a spring, the graph is a straight line through the origin with slope k, and the work done up to extension x is the triangular area: W = ½ F x = ½ (k x) x = ½ k x². In exam questions, you may be asked to estimate work from a graph by counting squares or using geometric formulas.

对于弹簧，图线为过原点斜率为 k 的直线，伸长到 x 所做的功为三角形面积：W = ½ F x = ½ (k x) x = ½ k x²。在考试题中，你可能需要数格子或用几何公式来估算功。

If the force changes direction, the area is taken as positive or negative accordingly, and the net work is the algebraic sum of the areas.

如果力的方向改变，面积相应地取正或负，净功为这些面积的代数和。

11. Common Pitfalls and Problem-Solving Tips | 常见误区与解题技巧

Angle confusion: Always identify the angle between the force vector and the displacement vector. Students often incorrectly use the angle of an incline or the angle of a cable relative to the horizontal instead of the true angle between the force direction and motion.

角度混淆：务必确认力矢量与位移矢量之间的夹角。学生经常误用斜面倾角或缆绳与水平面的夹角，而没有找准力方向与运动方向之间的真实角度。

Sign of work: Work done by a force is positive if it adds energy to the object (e.g., lifting force) and negative if it removes energy (e.g., friction). Use the sign carefully in the work–energy theorem.

功的正负：若力给物体增加能量，则该力做正功（如提升力）；若力带走能量，则做负功（如摩擦力）。在动能定理中要谨慎使用正负号。

Zero reference for potential energy: You can choose any convenient reference level for GPE as long as you use it consistently. The change in GPE is what matters, not the absolute value.

势能零点选取：重力势能的参考平面可任意选取，只要前后一致即可。重要的是势能的变化量，而非绝对值。

Units: Always convert mass to kg, distance to m, and time to s to obtain energy in joules. Velocity in m s⁻¹ is essential for kinetic energy calculati**.

单位：务必把质量转换为 kg，距离转换为 m，时间转换为 s，才能得到以焦耳为单位的能量。计算动能时速度必须以 m s⁻¹ 为单位。

System boundary: When applying conservation of energy, clearly define what is inside the system. External work done on the system changes its total mechanical energy, while internal conservative forces only exchange KE and PE.

系统边界：应用能量守恒时，要清晰界定系统内外的范围。外部对系统做功会改变其总机械能，而系统内部的保守力只是在动能和势能之间转换。

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AS Physics: Work and Energy Key Points Review | AS 物理：功与能量 考点精讲