📚 Calculating Energy Changes | 能量变化的计算
Energy calculations are at the heart of A-Level Physics. In CIE examinations you are expected to link work, kinetic energy, potential energy, power and efficiency in both theoretical and practical contexts. This article reviews the key equations, shows how to apply them, and highlights the most common pitfalls.
能量计算是 A-Level 物理的核心。在 CIE 考试中，你需要在理论和实验情境中联系功、动能、势能、功率和效率。本文回顾关键公式，展示如何应用，并指出最常见的失分点。
1. Energy, Work and the Joule | 能量、功与焦耳
In physics, energy is defined as the capacity to do work. Work is done when a force moves its point of application in the direction of the force. Both energy and work are scalar quantities and are measured in joules (J), where 1 J = 1 N m.
在物理学中，能量被定义为做功的本领。当力使其作用点沿力的方向移动时，就做了功。能量和功都是标量，单位为焦耳 (J)，其中 1 J = 1 N m。
W = F d cos θ
For a constant force F acting at an angle θ to the displacement d, only the component of force parallel to the displacement does work. This is why the cosine of the angle appears in the equation.
对于与位移 d 成 θ 角的恒力 F，只有平行于位移的力分量做功。这就是公式中出现 cos θ 的原因。
2. Kinetic Energy | 动能
Kinetic energy is the energy a body possesses because of its motion. It depends on the mass of the body and the square of its speed. The kinetic energy of a body of mass m moving at speed v is given by the following equation.
动能是物体由于运动而具有的能量。它取决于物体的质量和速度的平方。质量为 m、速度为 v 的物体的动能由以下公式给出。
Eₖ = ½ m v²
Because speed is squared, doubling the speed increases the kinetic energy by a factor of four for the same mass. This non-linear relationship is frequently tested in CIE questions involving braking distances and collisions.
由于速度被平方，质量相同时速度加倍会使动能增加为原来的四倍。这种非线性关系在 CIE 涉及制动距离和碰撞的题目中经常考查。
The work-energy principle states that the resultant work done on a body is equal to its change in kinetic energy. If the resultant force F acts over a displacement d, then F d = ½ m v² − ½ m u², where u is the initial speed and v is the final speed.
功能原理指出，作用在物体上的合力所做的功等于其动能的变化。如果合力 F 作用一段位移 d，则 F d = ½ m v² − ½ m u²，其中 u 为初速度，v 为末速度。
3. Gravitational Potential Energy | 重力势能
Gravitational potential energy is the energy stored in a body because of its position in a gravitational field. Near the Earth’s surface, the change in gravitational potential energy when a body of mass m is raised through a vertical height h is given by the following equation.
重力势能是物体因在引力场中的位置而储存的能量。在地球表面附近，质量为 m 的物体升高竖直高度 h 时，重力势能的变化由以下公式给出。
ΔEₚ = m g h
Here g is the gravitational field strength, which on Earth is approximately 9.81 N kg⁻¹. It is important to use the vertical height h, not the distance travelled along a slope, when calculating gravitational potential energy changes.
这里 g 是引力场强度，在地球上约为 9.81 N kg⁻¹。计算重力势能变化时，必须使用竖直高度 h，而不是沿斜面运动的距离。
When an object falls freely, its gravitational potential energy decreases and its kinetic energy increases. If air resistance is negligible, the loss in gravitational potential energy is equal to the gain in kinetic energy: m g h = ½ m v².
当物体自由下落时，其重力势能减少，动能增加。如果空气阻力可忽略，重力势能的减少量等于动能的增加量：m g h = ½ m v²。
4. Elastic Potential Energy | 弹性势能
Elastic potential energy is stored in a stretched or compressed spring or any object that obeys Hooke’s law within its elastic limit. For a spring with spring constant k extended or compressed by a displacement x from its natural length, the stored energy is given by the following equation.
弹性势能储存在被拉伸或压缩的弹簧或任何在弹性限度内遵循胡克定律的物体中。对于劲度系数为 k、从自然长度被拉伸或压缩位移 x 的弹簧，储存的能量由以下公式给出。
Eₑ = ½ k x²
This result can be understood from a force-extension graph. For a spring obeying Hooke’s law, F = k x, and the graph is a straight line through the origin. The work done in stretching the spring is equal to the area under the graph, which is the area of a triangle: ½ × F × x = ½ k x².
这一结果可以从力-伸长量图像理解。对于遵循胡克定律的弹簧，F = k x，图像为过原点的直线。拉伸弹簧所做的功等于图像下的面积，即三角形面积：½ × F × x = ½ k x²。
In many CIE problems, elastic potential energy is converted into kinetic energy when a spring is released. Assuming no external losses, ½ k x² = ½ m v² allows the launch speed to be calculated.
在许多 CIE 题目中，弹簧释放时弹性势能转化为动能。假设没有外部损失，½ k x² = ½ m v² 可用于计算发射速度。
5. Work Done by a Constant Force | 恒力做功
Work is the product of the force component in the direction of displacement and the displacement itself. If the force is perpendicular to the displacement, no work is done because cos 90° = 0. If the force opposes motion, the work done is negative.
功是力在位移方向上的分量与位移的乘积。如果力与位移垂直，则不做功，因为 cos 90° = 0。如果力阻碍运动，则做功为负。
For example, when a person pulls a sledge with a rope at an angle θ above the horizontal, the horizontal component F cos θ does useful work against friction. The vertical component F sin θ is balanced by the ground reaction and does no work along the horizontal displacement.
例如，当一个人以与水平方向成 θ 角的绳子拉雪橇时，水平分量 F cos θ 对抵抗摩擦做有用功。垂直分量 F sin θ 与地面反作用力平衡，沿水平位移不做功。
Negative work is important when considering kinetic energy reduction. If a resultant force acts opposite to the direction of motion, the work done is negative and the kinetic energy decreases.
在考虑动能减少时，负功很重要。如果合力方向与运动方向相反，则所做的功为负，动能减少。
6. Work Done by a Varying Force | 变力做功
When the applied force is not constant, the work done cannot simply be calculated as force times displacement. Instead, the work done is equal to the area under a force-distance graph. This is a common graphical skill assessed in CIE Physics.
当施加的力不是恒力时，不能简单地用力乘以位移来计算功。此时，功等于力-距离图像下方的面积。这是 CIE 物理中常见的图形分析技能。
For a spring being stretched, the force increases linearly from zero to F, so the average force is ½ F. Multiplying the average force by the extension x gives the work done ½ F x, which is consistent with the area of a triangle under the force-extension graph.
对于被拉伸的弹簧，力从零线性增加到 F，因此平均力为 ½ F。用平均力乘以伸长量 x 得到功 ½ F x，这与力-伸长量图像下方三角形面积一致。
In exam questions, you may be given a curved force-distance graph and asked to estimate the work done by counting squares. Each square represents a known amount of energy, and the total number of squares multiplied by the energy per square gives the area.
在考试中，你可能会遇到一条曲线力-距离图像，并被要求通过数格子的方法估算功。每个格子代表已知的能量值，总格子数乘以每格代表能量即可得到面积。
7. Conservation of Energy | 能量守恒
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. In an isolated system with no external work done, the total mechanical energy remains constant.
能量守恒定律指出，能量不能凭空产生或消失，只能从一种形式转化为另一种形式。在没有外部做功的孤立系统中，总机械能保持不变。
E_total = Eₖ + Eₚ = constant
For a pendulum, energy continuously changes between gravitational potential energy and kinetic energy. At the highest point, the pendulum has maximum potential energy and zero kinetic energy; at the lowest point, its speed is maximum and its potential energy is minimum.
对于单摆，能量在重力势能和动能之间不断转换。在最高点，摆球势能最大、动能为零；在最低点，速度最大、势能最小。
When using conservation of energy, choose two points where the energy terms can be written easily. Write the total energy at the first point, set it equal to the total energy at the second point, and solve for the unknown speed or height.
使用能量守恒时，应选择两个能量项容易写出的位置。写出第一点的总能量，令其等于第二点的总能量，然后求解未知速度或高度。
8. Energy Dissipation and Friction | 能量耗散与摩擦
Friction and air resistance are non-conservative forces. Work done against them is not stored as recoverable mechanical energy but is dissipated as internal energy, often observed as a temperature rise.
摩擦和空气阻力是非保守力。克服这些力所做的功不会以可回收的机械能形式储存，而是以内能耗散，通常表现为温度升高。
When friction is present, the total initial mechanical energy is equal to the total final mechanical energy plus the work done against friction. This can be written as: initial Eₖ + initial Eₚ = final Eₖ + final Eₚ + thermal energy.
当存在摩擦时，初始总机械能等于最终总机械能加上克服摩擦所做的功。这可以写为：初始 Eₖ + 初始 Eₚ = 末态 Eₖ + 末态 Eₚ + 热能。
For example, a block sliding down a rough slope loses some gravitational potential energy to friction. The speed at the bottom is therefore lower than the value predicted by m g h = ½ m v² for a smooth slope.
例如，物块沿粗糙斜面下滑时，一部分重力势能因摩擦而损耗。因此底部速度低于光滑斜面中 m g h = ½ m v² 预测的值。
CIE questions often ask you to calculate the work done against friction by comparing the theoretical energy transfer with the actual kinetic energy at the end of the motion.
CIE 题目经常要求通过比较理论能量转移与运动末端的实际动能来计算克服摩擦所做的功。
9. Power as the Rate of Energy Change | 功率作为能量变化率
Power is the rate at which work is done or energy is transferred. The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹.
功率是做功或能量转移的速率。功率的国际单位是瓦特 (W)，其中 1 W = 1 J s⁻¹。
P = W / t = ΔE / t
If a constant force F moves an object at constant speed v in the direction of the force, the power developed can also be written as P = F v. This is derived from P = W / t = F d / t = F v.
如果恒力 F 使物体沿力的方向以恒定速度 v 运动，则产生的功率也可写为 P = F v。这是由 P = W / t = F d / t = F v 推导得出的。
Power calculations are often linked to lifting masses, accelerating vehicles, or electrical devices. Always convert time to seconds and work or energy to joules before substituting into the power equation.
功率计算常与提升重物、车辆加速或电器相关。代入功率公式之前，必须先将时间换算为秒，将功或能量换算为焦耳。
10. Efficiency Calculations | 效率计算
Efficiency describes how much of the input energy or power is converted into useful output. In any real device, some energy is lost to friction, heat or sound, so efficiency is always less than 100%.
效率描述输入能量或功率中有多少被转化为有用输出。在任何真实设备中，部分能量因摩擦、热量或声音而损失，因此效率总是小于 100%。
η = (useful energy output / total energy input) × 100%
Efficiency can also be calculated using power: η = (useful output power / input power) × 100%. This is useful when the device operates continuously, such as an electric motor lifting a load.
效率也可用功率计算：η = (有用输出功率 / 输入功率) × 100%。当设备连续工作时，例如电动机提升负载，这种方法很有用。
In CIE calculations, make sure the useful output and total input are expressed in the same units before dividing. If energy is given in kJ and power in W, convert them to J and J s⁻¹ or use consistent time intervals.
在 CIE 计算中，相除之前必须确保有用输出和总输入使用相同单位。如果能量以 kJ 给出、功率以 W 给出，应转换为 J 和 J s⁻¹，或使用一致的时间间隔。
11. Experimental Methods for Measuring Energy Changes | 测量能量变化的实验方法
CIE practical questions often ask you to determine energy changes using simple apparatus. Kinetic energy can be measured by using light gates to record the speed of a moving card, while mass is measured with a balance.
CIE 实验题经常要求使用简单装置测量能量变化。动能可通过光门记录运动挡光片的速度来测量，质量则用天平测量。
Gravitational potential energy changes are found by measuring the vertical height moved by a known mass. The change in Eₚ is then calculated using m g h. For a falling mass, the lost potential energy can be compared with the gained kinetic energy to estimate efficiency.
重力势能变化通过测量已知质量移动的竖直高度来确定。然后用 m g h 计算 Eₚ 的变化。对于下落的质量，损失的势能可与增加的动能进行比较，以估算效率。
Power can be measured by timing how long it takes a motor or a person to lift a known weight through a measured height. The work done is m g h and the power is m g h / t.
功率可通过测量电动机或人将已知重物提升一段高度所需的时间来确定。所做的功为 m g h，功率为 m g h / t。
12. Exam Technique and Common Errors | 考试技巧与常见错误
Always convert quantities to SI units before performing energy calculations: grams to kilograms, centimetres to metres, and minutes to seconds. A common mistake is to substitute mass in grams directly into Eₖ or m g h.
进行能量计算之前，必须始终将量转换为 SI 单位：克换算为千克，厘米换算为米，分钟换算为秒。一个常见错误是把以克为单位的质量直接代入 Eₖ 或 m g h。
Do not forget to square the speed when calculating kinetic energy. Many students write Eₖ = ½ m v, which is incorrect. Also remember that energy is a scalar, so you should add energies arithmetically and not treat them as vectors.
计算动能时不要忘记将速度平方。许多学生写成 Eₖ = ½ m v，这是错误的。还要记住能量是标量，因此应按算术方式相加，不应视为矢量。
When using W = F d cos θ, identify the angle between the force and the displacement carefully. If the force is perpendicular to the displacement, no work is done even if the force is large.
使用 W = F d cos θ 时，要仔细判断力与位移之间的夹角。如果力与位移垂直，即使力很大也不做功。
Finally, define the system clearly. If you include the Earth and the object in your system, gravitational potential energy is internal. If you do work against gravity from outside the system, that work increases the system’s potential energy.
最后，要明确系统边界。如果系统包含地球和物体，重力势能是内部能量。如果从系统外部克服重力做功，则该功增加了系统的势能。
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