📚 Conservation of Energy | 能量守恒定律
Energy is one of the most fundamental concepts in physics. The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or transformed from one form to another. This powerful idea underpins everything from the motion of a simple pendulum to the fusion reactions inside stars. In IB Physics, understanding how to track energy through different processes is essential for solving problems in mechanics, thermodynamics, and beyond.
能量是物理学中最基本的概念之一。能量守恒定律指出，能量既不能凭空产生，也不能凭空消失；它只能从一种形式转化为另一种形式，或从一个物体转移到另一个物体。从单摆的运动到恒星内部的核聚变反应，这一深刻思想都奠定了理论基础。在 IB 物理中，学会追踪不同过程中的能量流动，是解决力学、热学乃至更广泛物理问题的关键。
1. Work and Energy | 功与能量
In physics, work (W) is done when a force moves an object through a displacement in the direction of the force. The work done by a constant force is given by W = F s cos θ defined as the capacity to do work, and both work and energy share the SI unit joule (J).
在物理学中，当力使物体沿力的方向发生位移时，力就对物体做了功。恒力所做的功由公式 W = F s cos θ 给出，其中 F 是力的大小，s 是位移大小，θ 是力与位移之间的夹角。如果力与位移方向相同，θ = 0°，cos θ = 1，则 W = F s。能量被定义为做功的本领，功和能量的国际单位都是焦耳 (J)。
When work is done on an object, energy is transferred to that object. For instance, lifting a book increases its gravitational potential energy, and a moving bat doing work on a ball increases the ball's kinetic energy. The work–energy theorem, which states that the net work done on an object equals its change in kinetic energy, is a direct consequence of the conservation principle: Wnet = ΔEk.
当对物体做功时，能量就传递给该物体。例如，举高书本会增大它的重力势能；运动的球棒对球做功，会增大球的动能。功能原理指出，合力对物体所做的净功等于物体动能的变化量：Wnet = ΔEk，这是能量守恒原理的直接体现。
2. Kinetic Energy | 动能
Kinetic energy is the energy an object possesses due to its motion. For an object of mass m moving at speed v, the translational kinetic energy is given by the expression:
动能是物体由于运动而具有的能量。对于质量为 m、速度为 v 的物体，它的平动动能由下式给出：
Ek = ½ m v²
This equation shows that kinetic energy is proportional to the mass and to the square of the speed. Doubling the mass doubles the kinetic energy, but doubling the speed quadruples it. This quadratic dependence explains why high-speed collisions involve much larger amounts of energy.
该式表明，动能与质量成正比，与速度的平方成正比。质量加倍，动能加倍；但速度加倍，动能则会变为原来的四倍。这种平方关系解释了为什么高速碰撞涉及的能量要大得多。
It is important to remember that kinetic energy is a scalar quantity and is always positive (or zero when the object is at rest). In IB calculations, energy is often expressed in joules, and speeds must be in metres per second (m s⁻¹).
需要牢记的是，动能是标量，并且总为正值（物体静止时为零）。在 IB 计算中，能量通常用焦耳表示，速度则必须使用米每秒 (m s⁻¹)。
3. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object due to its position in a gravitational field. Near the Earth's surface, where the gravitational field strength g is approximately uniform (9.81 N kg⁻¹), the change in gravitational potential energy when an object of mass m is raised through a vertical height Δh is:
重力势能是物体由于其在引力场中的位置而储存的能量。在地球表面附近，重力场强度 g 近似均匀（9.81 N kg⁻¹），当质量为 m 的物体竖直升高 Δh 时，重力势能的变化量为：
ΔEp = m g Δh
We often define a reference level where GPE is taken to be zero, and then the absolute potential energy can be written as Ep gh. Only changes in GPE have physical meaning; the choice of reference point is arbitrary.
我们通常定义一个参考面，令该处的重力势能为零，于是绝对势能可以写作 Ep = mgh。具有物理意义的只是重力势能的变化；参考面的选择是任意的。
When an object falls freely, its gravitational potential energy is converted into kinetic energy. In the absence of air resistance, the decrease in GPE exactly equals the increase in kinetic energy, beautifully illustrating energy conservation.
当物体自由下落时，重力势能转化为动能。在无空气阻力的条件下，重力势能的减少恰好等于动能的增加，这正是能量守恒的精彩体现。
4. Elastic Potential Energy | 弹性势能
gy stored in deformed elastic objects, such as a stretched spring or a compressed bungee cord. For a spring that obeys Hooke' g constant and x is the extension or compression), the stored elastic potential energy is:
弹性势能是储存在发生弹性形变的物体（例如被拉伸的弹簧或被压缩的蹦极绳）中的能量。对于遵从胡克定律（F = k x，k 为劲度系数，x 为伸长量或压缩量）的弹簧，其储存的弹性势能为：
Ee = ½ k x²
Notice the similar mathematical form to kinetic energy – a half times a constant times the square of the variable. The energy stored depends on the spring constant and the square of the displacement. A stiffer spring (larger k) stores more energy for the same deformation.
请注意它和动能的数学形式相似——二分之一乘以常数再乘以变量的平方。储存的能量取决于劲度系数和形变量的平方。对于相同的形变量，较硬的弹簧（k 值更大）能储存更多能量。
When a compressed spring is released, the elastic potential energy can be converted into kinetic energy of a mass, or into gravitational potential energy, while obeying the conservation law.
当被压缩的弹簧释放时，弹性势能可以转化为物体的动能或重力势能，整个过程始终遵循能量守恒定律。
5. The Principle of Conservation of Energy | 能量守恒原理
The full principle of conservation of energy states that the total energy of an isolated system remains constant over time. Energy can change its form – from kinetic to thermal, from chemical to electrical, from nuclear to radiant – but the total quantity of energy never changes. This principle is one of the most important and well-tested laws in all of science.
完整的能量守恒原理指出：孤立系统的总能量不随时间改变。能量可以改变形式——从动能变为热能、从化学能变为电能、从核能变为辐射能——但能量的总量永不改变。这一原理是全部科学中最重要、经受了最充分检验的定律之一。
In any real process, it may appear that energy is 'lost', for example when a moving object comes to rest due to friction. However, careful measurement shows that the kinetic energy is not destroyed; it is converted into thermal energy, increasing the internal energy of the surfaces and the surroundings. The sum of all forms of energy remains unchanged.
在任何实际过程中，能量看似会“消失”，比如一个运动的物体因摩擦而停下。然而，细致的测量表明，动能并没有被消灭，而是转化为热能，增大了接触面和环境的内部能量。所有形式的能量之和仍然保持不变。
In IB Physics, you are expected to apply this principle qualitatively and quantitatively: identifying the energy stores involved, describing the energy transfers, and calculating quantities using the appropriate equations.
在 IB 物理中，你需要既能定性、也能定量地应用这一原理：识别相关的能量储存，描述能量的转移，并使用恰当的方程进行定量计算。
6. Mechanical Energy Conservation | 机械能守恒
Mechanical energy is the sum of kinetic energy and all forms of potential energy in a system (usually gravitational and elastic). When only conservative forces (such as gravity and ideal spring forces) do work, the total mechanical energy of the system remains constant. This is called the principle of conservation of mechanical energy.
机械能是系统中动能与所有形式势能（通常是重力势能和弹性势能）的总和。当只有保守力（如重力和理想弹力）做功时，系统的总机械能保持不变。这被称为机械能守恒定律。
Mathematically, we write:
数学上，我们写作：
Ek1 + Ep1 = Ek2 + Ep2
uations allow us to solve for unknown speeds or positions without needing to use equations of motion.
对自由落体而言，就是 ½ m v₁² + mgh₁ = ½ m v₂² + mgh₂。对水平弹簧上的物体，则是 ½ m v₁² + ½ k x₁² = ½ m v₂² + ½ k x₂²。利用这些方程，我们不必使用运动学方程就能求解未知的速度或位置。
It is essential to choose a consistent reference level for height and to clearly define the system. The mechanical energy conservation approach is particularly powerful in analysing pendulums, roller coasters, and projectile motions where air resistance is negligible.
必须为高度选定一致的参考面，并清晰定义系统。在分析单摆、过山车和忽略空气阻力的抛体运动时，机械能守恒的方法尤为有力。
7. Non-conservative Forces and Energy Dissipation | 非保守力与能量耗散
In many real-world situations, non-conservative forces such as friction, air resistance, and applied forces do work and cause mechanical energy to be converted into other forms, primarily thermal energy. In these cases, total mechanical energy is not conserved, but total energy still is.
在许多真实情境中，非保守力（如摩擦力、空气阻力、外加拉力）会做功，并使机械能转化为其他形式的能量，主要是热能。在这些情况下，总机械能不守恒，但总能量依然守恒。
The work done by non-conservative forces, Wnc, equals the change in mechanical energy of the system:
非保守力所做的功 Wnc 等于系统机械能的变化量：
Wnc = ΔEmech = (Ekf + Epf) − (Eki + Epi)
If friction opposes motion, Wnc is negative, and mechanical energy decreases; the 'lost' mechanical energy appears as an increase in internal energy, often observed as a temperature rise. This equation is a powerful tool for analysing inclined planes with friction, skidding vehicles, and many other dissipative systems.
如果摩擦力与运动方向相反，Wnc 为负值，机械能减少；“消失”的机械能表现为内部能量的增加，常表现为温度的升高。该方程是分析存在摩擦的斜面、打滑的车辆以及其他众多耗散系统的有力工具。
8. Power and Efficiency | 功率与效率
unit of power is the watt (W), where 1 W = 1 J s⁻¹. Average power can be calculated using:
功率 (P) 是做功或能量转移的速率。功率的国际单位是瓦特 (W)，1 W = 1 J s⁻¹。平均功率可以用下式计算：
P = ΔW / Δt or P = ΔE / Δt
When a constant force moves an object at constant speed v, power can also be expressed as P = F v. This relationship is important in understanding transport, engines, and machinery.
当恒力使物体以恒定速度 v 运动时，功率还可以表示为 P = F v。这一关系对于理解交通工具、发动机和机械至关重要。
Efficiency (η) describes how much of the supplied energy is converted into useful output. It is a dimensionless ratio, usually expressed as a percentage:
效率 (η) 描述输入能量有多少转化为有用输出。它是一个无量纲比值，通常以百分数表示：
η = (useful work output / total energy input) × 100%
In any real device, efficiency is always less than 100% because some energy is inevitably dissipated as thermal energy due to friction, electrical resistance, or other losses. Maximising efficiency is a key goal in engineering and sustainable design.
在任何实际装置中，效率总是小于 100%，因为部分能量不可避免地因摩擦、电阻或其他损耗而以热能形式耗散。提高效率是工程和可持续设计的一个关键目标。
9. Energy Transfer Diagrams (Sankey Diagrams) | 能量传递图（桑基图）
Sankey diagrams are visual representations of energy transfers, where the widths of arrows are proportional to the amounts of energy. They clearly show the energy input, useful output, and wasted energy. In IB Physics, you may be asked to interpret or sketch a Sankey diagram.
桑基图是一种能量传递的直观表示方法，图中箭头的宽度与能量大小成正比。它们清晰地展示了能量输入、有用输出和浪费的能量。在 IB 物理考试中，你可能会被要求解读或绘制桑基图。
A typical diagram for an electric lamp shows an input of electrical energy (wide arrow). The arrow then splits: a narrower arrow to the right represents useful light output, while a much wider arrow branching downward represents the wasted thermal energy. The diagram makes efficiency easy to appreciate at a glance.
一个典型的电灯桑基图会显示一束电能输入（很宽的箭头），随后箭头分叉：向右的一个较窄箭头代表有用的光输出，而向下分支的一个粗得多的箭头则表示浪费的热能。这样的图示让效率高低一目了然。
Using the principle of conservation of energy, the total width of the output arrows must equal the input arrow width. This visual constraint reinforces that energy is conserved even though not all of it is useful.
根据能量守恒原理，所有输出箭头的总宽度必须等于输入箭头的宽度。这一视觉约束再次强调了尽管并非所有能量都有用，但能量总量是守恒的。
10. Real-world Applications | 实际应用
Energy conservation principles are used everywhere. A roller coaster constantly interconverts gravitational potential energy and kinetic energy: at the top of a hill, the train has maximum GPE; at the bottom, it has maximum kinetic energy. Friction and air resistance gradually reduce total mechanical energy, requiring the lift hill to input work.
能量守恒原理无处不在。过山车就是在重力势能和动能之间不断转换：在山顶，列车的重力势能最大；在山谷，动能最大。摩擦和空气阻力会逐渐减小总机械能，因此过山车需要利用提升坡来输入功。
In a hydroelectric dam, water stored at height has enormous gravitational potential energy. As water flows down through turbines, this GPE is converted into kinetic energy and then into electrical energy. The efficiency of this conversion can exceed 90%, but some energy is always lost to friction and turbulence.
在水力发电大坝中，蓄在高处的水具有巨大的重力势能。水向下流过水轮机时，重力势能先转化为动能，再转化为电能。这种转换的效率可以超过 90%，但总会有部分能量因摩擦和湍流而损耗。
A bungee jump provides a vivid demonstration of multiple energy transformations: initially, gravitational potential energy is converted into kinetic energy during the free fall; as the cord stretches, kinetic energy and GPE are transferred into elastic potential energy. At the lowest point, the jumper's kinetic energy is zero, and the maximum energy is stored in the stretched cord. Energy then oscillates back and forth, always respecting the conservation law.
蹦极则生动地展示了多种能量转化过程：最初，自由下落时重力势能转化为动能；当绳索拉伸时，动能和重力势能又转化为弹性势能。在最低点，跳跃者的动能为零，最大能量储存在拉伸的绳索中。随后能量来回振荡，始终遵循守恒定律。
Understanding these transformations not only helps solve IB examination problems but also deepens our appreciation of the natural world and informs the design of renewable energy systems.
理解这些能量转化不仅有助于解决 IB 考试问题，还能加深我们对自然界的理解，并为可再生能源系统的设计提供依据。
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