📚 Kinetic Energy | 动能
Kinetic energy is one of the most fundamental concepts in classical mechanics, linking motion to a measurable scalar quantity that appears in energy conservation and work–energy calculations. A thorough understanding of kinetic energy is essential for tackling topics from collisions and explosions to the behaviour of gas molecules in A-Level Physics.
动能是经典力学中最基本的概念之一，它将运动与一个可测量的标量联系起来，出现在能量守恒和功能计算中。透彻理解动能对于掌握 A-Level 物理中从碰撞、爆炸到气体分子行为等课题至关重要。
1. What is Kinetic Energy? | 什么是动能？
Kinetic energy is the energy that a body possesses by virtue of being in motion. Any object that has mass and moves with a certain speed relative to a given reference frame has kinetic energy. The faster the object moves or the greater its mass, the larger its kinetic energy. Kinetic energy is a scalar quantity, meaning it has magnitude but no direction, and is always non-negative.
动能是物体由于运动而具有的能量。任何具有质量并相对于给定参考系以一定速度运动的物体都具有动能。物体运动越快或质量越大，其动能就越大。动能是一个标量，只有大小没有方向，并且总是非负的。
2. Mathematical Definition and Units | 数学定义和单位
For an object of mass m moving with translational speed v, the kinetic energy Eₖ is given by the expression:
对于质量为 m、平动速度为 v 的物体，动能 Eₖ 的表达式为：
Eₖ = ½ m v²
joule (J), where 1 J = 1 kg m² s⁻². Because speed is squared, the kinetic energy depends strongly on the magnitude of velocity; doubling the speed quadruples the kinetic energy.
在国际单位制（SI）中，质量以千克（kg）为单位，速度以米每秒（m s⁻¹）为单位。能量的导出单位是焦耳（J），其中 1 J = 1 kg m² s⁻²。由于速度被平方，动能对速度的大小非常敏感，速度翻倍会使动能变为原来的四倍。
3. Derivation from a Constant Net Force | 从恒力做功推导
Consider a constant net force F acting on an object of mass m over a displacement s in the direction of the force. The work done by the net force is W = F s. Using Newton's second law, F = m a, and the kinematic equation v² = u² + 2 a s for uniform acceleration, we can eliminate acceleration a. Rearranging gives a s = (v² − u²)/2. Substituting into the work expression yields W = m × (v² − u²)/2 = ½ m v² − ½ m u². This shows that the net work done equals the change in the quantity ½ m v², which we define as kinetic energy.
设想一个恒定的净力 F 沿力的方向作用在质量为 m 的物体上，产生的位移为 s。净力做的功为 W = F s。利用牛顿第二定律 F = m a 以及匀加速运动的运动学方程 v² = u² + 2 a s，我们可以消去加速度 a。整理得 a s = (v² − u²)/2。代入功的表达式得到 W = m × (v² − u²)/2 = ½ m v² − ½ m u²。这表明净功等于 ½ m v² 这个量的变化量，我们将其定义为动能。
4. Work–Energy Theorem | 功能定理
The work–energy theorem states that the net work done on an object by all the forces acting on it equals the change in its kinetic energy. In equation form:
功能定理指出，作用在物体上所有力的净功等于其动能的变化量。方程形式为：
Wnet = ΔEₖ = ½ m vf² − ½ m vi²
This theorem provides a powerful problem-solving tool: instead of analysing forces and accelerations in detail, we can relate the total work to the change in speed. It applies regardless of whether the net force is constant or variable, provided we can compute the total work done along the path. If the net work is positive, the kinetic energy increases; if negative, it decreases.
该定理是一个强大的解题工具：我们无需详细分析力和加速度，便可将总功与速度变化联系起来。无论净力是恒力还是变力，只要能计算出沿路径所做的总功，该定理就适用。如果净功为正，动能增加；如果为负，动能减少。
5. Relationship Between Kinetic Energy and Momentum | 动能与动量的关系
Momentum p = m v and kinetic energy are linked through the object's mass. By substituting v = p/m into the kinetic energy formula, we obtain a useful alternative form:
动量 p = m v 和动能通过物体的质量相联系。将 v = p/m 代入动能公式，我们得到一个有用的替代形式：
Eₖ = p² / (2m)
This relation reveals that for objects with the same momentum, the one with the smaller mass has a larger kinetic energy. Conversely, for a given kinetic energy, momentum scales as the square root of mass. The relationship is particularly helpful when dealing with collisions where momentum is conserved but kinetic energy may not be.
这个关系表明，对于动量相同的物体，质量较小的物体具有更大的动能。反之，对于给定的动能，动量与质量的平方根成正比。当处理动量守恒但动能可能不守恒的碰撞问题时，这个关系特别有帮助。
6. Kinetic Energy in Elastic Collisions | 弹性碰撞中的动能
In an elastic collision, both momentum and total kinetic energy are conserved. This idealised interaction is approached by collisions between hard steel balls or gas molecules at low temperatures. For two colliding bodies of masses m₁ and m₂, the conservation laws give:
在弹性碰撞中，动量和总动能都守恒。硬质钢球或低温下气体分子之间的碰撞接近这种理想化的相互作用。对于质量分别为 m₁ 和 m₂ 的两个碰撞体，守恒定律给出：
½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²
where u and v represent speeds before and after the collision. The elastic condition imposes a specific relationship between relative speeds: the relative speed of approach equals the relative speed of separation. Kinetic energy after the collision is simply rearranged but not lost to heat, sound or deformation.
其中 u 和 v 分别表示碰撞前后的速度。弹性条件对相对速度施加了特定关系：接近的相对速度等于分离的相对速度。碰撞后的动能仅仅是重新分配，但不会损失为热、声或形变。
7. Kinetic Energy in Inelastic Collisions | 非弹性碰撞中的动能
Real-world collisions are often inelastic: momentum is still conserved, but some kinetic energy is transformed into other forms such as internal thermal energy, sound or permanent deformation. In a perfectly inelastic collision, the two bodies stick together and move with a common velocity v after impact. The loss in kinetic energy is maximum for a given initial momentum and is given by:
现实世界中的碰撞通常是非弹性的：动量仍然守恒，但部分动能转化为其他形式，如内能、声能或永久形变。在完全非弹性碰撞中，两个物体粘在一起并以共同速度 v 运动。对于给定的初始动量，动能损失最大，可通过以下表达式计算损失：
ΔEₖ = − [m₁ m₂ / (2(m₁+m₂))] (u₁ − u₂)²
This loss represents the energy dissipated, and it is always negative, confirming that kinetic energy after the collision is less than that before. Even in partially inelastic collisions, a fraction of the initial kinetic energy is converted to other forms.
这个损失代表了耗散的能量，且始终为负，证实碰撞后的动能小于碰撞前。即使在部分非弹性碰撞中，也会有一部分初始动能转化为其他形式。
8. Conservative Forces and Mechanical Energy | 保守力与机械能
ne a potential energy so that the total mechanical energy E = Eₖ + Eₚ is conserved when only conservative forces act. The work done by a conservative force decreases the corresponding potential energy and increases kinetic energy by the same amount, and vice versa.
保守力（如重力或弹簧的弹性力）做功与路径无关。对于这类力，我们可以定义势能，使得只有保守力作用时总机械能 E = Eₖ + Eₚ 守恒。保守力做功会减少相应的势能，并以相同的数量增加动能，反之亦然。
9. Non-conservative Forces and Energy Dissipation | 非保守力与能量耗散
Non-conservative forces, such as friction, air resistance and tension in inelastic materials, cause mechanical energy to be dissipated as heat or sound. The work–energy theorem remains valid: the net work includes the work done by non-conservative forces, and this part appears as a loss in mechanical energy. The change in kinetic energy is then equal to the net work from both conservative and non-conservative contributions.
非保守力，例如摩擦力、空气阻力以及非弹性材料中的张力，会导致机械能转化为热和声能。功能定理依然成立：净功包含非保守力做的功，这部分表现为机械能的损失。动能变化量等于保守力贡献和非保守力贡献的总净功。
10. Average Kinetic Energy of Gas Molecules | 气体分子的平均动能
In the kinetic theory of ideal gases, the absolute temperature T of a gas is directly proportional to the average translational kinetic energy of its molecules. For a monatomic gas, each molecule has an average kinetic energy given by:
在理想气体的动理论中，气体的绝对温度 T 与其分子平均平动动能成正比。对于单原子气体，每个分子的平均动能由下式给出：
⟨Eₖ⟩ = (3/2) kB T
where kB ²³ J K⁻¹). This fundamental link explains why temperature is a measure of microscopic motion and why heating a gas increases the speed of its particles. It also leads to the expression for the root-mean-square speed crms = √3RT/M, which students frequently encounter in A-Level calculations.
其中 kB 是玻尔兹曼常量（1.38 × 10⁻²³ J K⁻¹）。这一基本联系说明了为什么温度是微观运动的一种量度，以及为何加热气体会增加其粒子的速度。它还导出了均方根速度 crms = √3RT/M 的表达式，学生在 A-Level 计算中会经常遇到。
11. Rotational Kinetic Energy (Brief Overview) | 转动动能（概述）
Objects that rotate also possess kinetic energy associated with their angular motion. The rotational kinetic energy of a rigid body about a fixed axis is expressed as:
旋转的物体还具有与角运动相关的动能。刚体绕定轴转动的转动动能表示为：
Eₖ,rot = ½ I ω²
Here I is the moment of inertia and ω is the angular speed. Although not always the main focus of all A-Level specifications, this form is directly analogous to the translational expression, with mass replaced by rotational inertia and linear speed by angular speed.
此处 I 为转动惯量，ω 为角速度。尽管并非所有 A-Level 考纲都将其作为重点，但这种形式与平动表达式完全类似，质量被转动惯量取代，线速度被角速度取代。
12. Problem-Solving Strategy and Common Pitfalls | 解题策略与常见误区
When solving kinetic energy problems, always begin by identifying the system and the forces doing work. Use the work–energy theorem wisely: calculate total net work or, if conservative forces dominate, apply mechanical energy conservation. Remember that kinetic energy is always relative to a chosen inertial frame and is a scalar. Ensure that speeds are in m s⁻¹ and masses in kg to obtain energy in joules. A frequent error is using the wrong sign for work or forgetting to square the speed in calculations.
在解决动能问题时，始终从确定系统和做功的力开始。巧妙地运用功能定理：计算总净功，或者当保守力占主导时，应用机械能守恒。请记住，动能始终相对于所选惯性参考系，并且是一个标量。确保速度单位为 m s⁻¹，质量单位为 kg，从而得到以焦耳为单位的能量。一个常见错误是功的符号使用不当或在计算中忘记对速度进行平方。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图