Down, up, down – Energy Changes | 落下、上升、再落下——能量变化

📚 Down, up, down – Energy Changes | 落下、上升、再落下——能量变化

When an object is thrown upwards, dropped from a height, or bounces after hitting the ground, its energy is constantly changing form. In CIE A Level Physics, these vertical motion problems are almost always about energy transfers between gravitational potential energy, kinetic energy, and sometimes elastic potential energy or internal energy due to drag and impact. Understanding the direction of energy flow is the key to solving both qualitative and quantitative exam questions.

当一个物体被向上抛出、从高处落下或撞击地面后弹起时，它的能量形式在不断变化。在 CIE A Level 物理中，这类竖直运动问题几乎总是涉及重力势能、动能之间的能量转移，有时还涉及弹性势能或因阻力和碰撞产生的内能。理解能量流动的方向是解答定性和定量考试题的关键。

1. The Big Picture: Energy Stores and Transfers | 能量储存与转移总览

In a closed system with no external forces such as air resistance or friction, the total mechanical energy remains constant. For an object moving vertically, mechanical energy is the sum of gravitational potential energy (GPE) and kinetic energy (KE). Energy is transferred from one store to another, but the total is conserved.

在没有空气阻力或摩擦等外力的封闭系统中，总机械能保持不变。对于竖直运动的物体，机械能是重力势能 (GPE) 和动能 (KE) 之和。能量从一个储存库转移到另一个储存库，但总量守恒。

When the object moves down, GPE decreases and KE increases. When the object moves up, KE decreases and GPE increases. If the object falls again after bouncing, the same down-transfer repeats, but each rebound usually involves less useful mechanical energy because some energy is dissipated as heat and sound.

当物体向下运动时，重力势能减少，动能增加。当物体向上运动时，动能减少，重力势能增加。如果物体在弹跳后再次下落，同样的向下能量转移会重复发生，但每次反弹通常涉及更少的有用机械能，因为部分能量以热和声的形式耗散。

2. Gravitational Potential Energy: Energy of Position | 重力势能：位置的能量

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. Near the Earth’s surface, it is calculated as:

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，它的计算公式为：

GPE = mgh

where m is mass in kg, g is gravitational field strength in N kg⁻¹, and h is vertical height above a chosen reference level in m. The choice of reference level is arbitrary, but it must be consistent throughout a calculation. In CIE exams, students should state the reference level clearly when working with GPE.

其中 m 是质量（kg），g 是重力场强度（N kg⁻¹），h 是相对于选定参考面的竖直高度（m）。参考面的选择是任意的，但在整个计算过程中必须保持一致。在 CIE 考试中，学生在使用 GPE 时应明确说明参考面。

3. Kinetic Energy: Energy of Motion | 动能：运动的能量

Kinetic energy is the energy stored in a moving object. It depends on mass and speed squared:

动能是运动物体储存的能量。它取决于质量和速度的平方：

KE = ½mv²

Because KE depends on v², doubling the speed quadruples the kinetic energy. This non-linear relationship is often tested when analysing bouncing balls or vertical projectiles. Remember that speed is a scalar quantity, so KE is always positive, even if the object is moving upwards or downwards.

由于动能取决于 v²，速度加倍会使动能变为原来的四倍。这种非线性关系在分析弹跳球或竖直抛体时经常考查。请记住，速度是标量，因此动能始终为正，无论物体是向上还是向下运动。

4. Downward Motion: Converting GPE to KE | 向下运动：重力势能转化为动能

When an object falls freely from rest, the initial GPE is converted into KE. Assuming no air resistance, the decrease in GPE equals the increase in KE:

当物体从静止自由下落时，初始的重力势能转化为动能。假设没有空气阻力，重力势能的减少量等于动能的增加量：

mgh = ½mv²

Cancelling m gives the well-known free-fall speed formula:

消去 m 后得到著名的自由落体速度公式：

v = √(2gh)

This shows that the final speed does not depend on mass. A heavy object and a light object dropped from the same height reach the same speed in the absence of air resistance. In real life, air resistance reduces the acceleration and the final speed, but the energy transfer principle still holds: GPE lost equals KE gained plus work done against drag.

这表明最终速度与质量无关。在没有空气阻力的情况下，从同一高度落下的重物和轻物达到相同的速度。在现实生活中，空气阻力会降低加速度和最终速度，但能量转移原理仍然成立：损失的重力势能等于获得的动能加上克服阻力所做的功。

5. Upward Motion: Converting KE to GPE | 向上运动：动能转化为重力势能

When an object is thrown vertically upwards, it starts with maximum kinetic energy and zero GPE if the reference level is at the launch point. As it rises, KE decreases and GPE increases. At the highest point, the object momentarily stops, so all the initial KE has been converted into GPE (assuming no air resistance).

当一个物体被竖直向上抛出时，如果参考面在抛出点，它开始时具有最大动能和零重力势能。随着它上升，动能减少，重力势能增加。在最高点，物体瞬间停止，因此所有初始动能都转化为重力势能（假设没有空气阻力）。

The maximum height can be found by equating initial KE to final GPE:

最大高度可以通过令初始动能等于最终重力势能来求得：

½mu² = mgh

Solving for h:

求出 h：

h = u² / 2g

Here u is the initial launch speed. If the object returns to the launch level, the same energy conversion happens in reverse: GPE at the top is converted back to KE as it falls.

这里的 u 是初始抛出速度。如果物体回到抛出高度，相同的能量转换会反向发生：最高点的重力势能在下落时又转化为动能。

6. Down Again: Bouncing and Energy Dissipation | 再次下落：弹跳与能量耗散

When a ball bounces, the sequence is down-up-down. During the first fall, GPE converts to KE. During the impact with the ground, some KE is stored temporarily as elastic potential energy in the deformed ball and then released, but a significant fraction is dissipated as internal energy (heat) and sound. The ball then rises with less KE than it had just before impact, so it reaches a lower height.

当球弹跳时，运动顺序是下落—上升—再下落。在第一次下落过程中，重力势能转化为动能。在与地面碰撞时，部分动能暂时储存为球变形产生的弹性势能，然后释放，但相当一部分以内部能量（热）和声的形式耗散。然后球以比撞击前更少的动能上升，因此达到的高度更低。

The efficiency of a bounce is described by the coefficient of restitution:

弹跳的效率用恢复系数来描述：

e = v₂ / v₁ = √(h₂ / h₁)

where v₁ is the speed just before impact, v₂ is the rebound speed, h₁ is the original drop height, and h₂ is the rebound height. For a perfectly elastic collision, e = 1; for a completely inelastic collision, e = 0. Most real balls have 0 < e < 1.

其中 v₁ 是撞击前的速度，v₂ 是反弹速度，h₁ 是原始下落高度，h₂ 是反弹高度。对于完全弹性碰撞，e = 1；对于完全非弹性碰撞，e = 0。大多数真实球的恢复系数介于 0 和 1 之间。

7. Work Done and Non-Conservative Forces | 做功与非保守力

Air resistance and contact friction are non-conservative forces. They remove mechanical energy from the system. The work-energy principle for a vertical motion can be written as:

空气阻力和接触摩擦是非保守力。它们会从系统中带走机械能。竖直运动的功-能原理可以写成：

W_{non-conservative} = ΔKE + ΔGPE

If W_{non-conservative} is negative, it means mechanical energy is lost. For example, when a ball falls through air, the work done by air resistance is negative, so the final KE is less than the initial GPE lost. This lost energy usually reappears as internal energy, raising the temperature of the object and the surrounding air.

如果 W_非保守为负，说明机械能损失了。例如，当球在空气中下落时，空气阻力做的功为负，因此最终动能小于初始重力势能的减少量。这些损失的能量通常以内能的形式重新出现，使物体和周围空气的温度升高。

In exam answers, avoid saying energy is “lost” without specifying where it goes. The correct statement is that energy is dissipated, transferred to the surroundings, or converted into internal energy.

在考试答案中，避免只说能量“丢失”而不说明去向。正确的说法是能量被耗散、转移到周围环境或转化为内能。

8. Energy–Displacement and Energy–Time Graphs | 能量-位移和能量-时间图像

CIE exams often ask students to sketch or interpret energy graphs for a bouncing ball or a vertical projectile. For an object in free fall with no air resistance, the total mechanical energy is a horizontal straight line. GPE decreases linearly with height because GPE = mgh, while KE increases by the same amount.

CIE 考试经常要求学生画出或解释弹跳球或竖直抛体的能量图像。对于无空气阻力的自由落体，总机械能是一条水平直线。重力势能随高度线性减少，因为 GPE = mgh，而动能以相同量增加。

In an energy–displacement graph for a bouncing ball, the GPE line is a straight line decreasing as height decreases, and KE is the mirror image above the GPE line, keeping total energy constant between bounces. After each bounce, the total mechanical energy line drops to a lower value because of dissipation. The GPE peak after each bounce is lower, showing the reduced rebound height.

在弹跳球的能量-位移图中，GPE 线是一条随高度降低而减少的直线，KE 线是 GPE 线上方的镜像，使两次弹跳之间的总能量保持不变。每次弹跳后，总机械能线下降到更低的值，因为存在能量耗散。每次弹跳后的 GPE 峰值更低，这反映了反弹高度的降低。

9. Vertical Projectiles: Symmetry in Energy Changes | 竖直抛体：能量变化的对称性

For an ideal vertical projectile with no air resistance, the motion is symmetric. At the same height on the way up and on the way down, the object has the same speed and therefore the same kinetic energy. The GPE is also the same at that height. This symmetry comes directly from energy conservation: the sum KE + GPE is constant.

对于没有空气阻力的理想竖直抛体，运动具有对称性。在上升和下降过程中处于同一高度时，物体的速度大小相同，因此动能相同。该高度下的重力势能也相同。这种对称性直接来自能量守恒：KE + GPE 的总和保持不变。

This means the time taken to rise to maximum height is equal to the time taken to fall back, and the launch speed equals the landing speed if landing occurs at the same level. Energy arguments are often faster than kinematic equations for comparing speeds and heights in CIE multiple-choice questions.

这意味着上升到最高点所用的时间等于落回所用的时间，如果落点与抛出点在同一水平面，抛出速度等于落地速度。在 CIE 选择题中，能量分析通常比运动学方程更快地比较速度和高度。

10. Terminal Velocity: When Downward Energy Stops Converting | 终端速度：当向下的能量转换停止时

When an object falls through air, air resistance increases with speed. Eventually the upward drag force equals the downward weight, so the resultant force becomes zero and the object stops accelerating. It continues to fall at constant terminal velocity.

当物体在空气中下落时，空气阻力随速度增加。最终向上的阻力等于向下的重力，因此合力变为零，物体停止加速。它继续以恒定的终端速度下落。

At terminal velocity, the kinetic energy is constant, but the object is still losing GPE as it falls. This lost GPE is not converted into KE; instead, it is converted into internal energy of the air and the object due to work done against drag. This is a common misunderstanding: energy conservation still holds, but mechanical energy decreases steadily while total energy remains constant.

在终端速度下，动能保持不变，但物体在下降过程中仍在损失重力势能。这些损失的重力势能没有转化为动能，而是因克服阻力做功转化为空气和物体的内能。这是一个常见的误解：能量守恒仍然成立，但机械能持续减少，而总能量保持不变。

11. Bungee and Springs: Adding Elastic Potential Energy | 蹦极与弹簧：引入弹性势能

In a vertical bungee jump or a mass on a spring, a third energy store appears: elastic potential energy (EPE). The energy changes follow the sequence down-up-down, but with more transfers. During the initial free-fall part of a bungee jump, GPE is converted to KE. Once the rope becomes taut and stretches, KE and remaining GPE are converted into EPE stored in the stretched rope.

在竖直蹦极或弹簧上的质量块运动中，出现了第三种能量储存库：弹性势能 (EPE)。能量变化遵循下落—上升—再下落的顺序，但转换过程更多。在蹦极的初始自由下落阶段，GPE 转化为 KE。一旦绳子拉紧并伸长，KE 和剩余的 GPE 被转化为储存在拉伸绳子中的 EPE。

Elastic potential energy for a spring or rope obeying Hooke’s law is given by:

对于遵守胡克定律的弹簧或绳子，弹性势能由以下公式给出：

EPE = ½kx²

where k is the spring constant and x is the extension from the natural length. At the lowest point of the jump, the kinetic energy is zero and the maximum GPE has been converted into EPE. At the highest rebound point, the EPE is zero and the system has maximum GPE, though not as high as the starting point because of energy losses in stretching and air resistance.

其中 k 是弹簧劲度系数，x 是相对于自然长度的伸长量。在跳跃的最低点，动能为零，最大重力势能已转化为弹性势能。在最高反弹点，弹性势能为零，系统具有最大重力势能，但由于伸长和空气阻力造成的能量损失，其高度不如起始点。

12. Exam Tips and Common Mistakes | 考试提示与常见错误

Always define the reference level for GPE. If it is not given in the question, state your choice clearly.

始终定义重力势能的参考面。如果题目未给出，请清楚说明你的选择。
Do not cancel mass in energy equations unless both sides of the equation contain the same mass term. In most GPE-KE conversions you can cancel m, but be careful when other energy stores are involved.

除非方程两边包含相同的质量项，否则不要在能量方程中随意约去质量。在大多数 GPE-KE 转换中可以约去 m，但当涉及其他能量储存库时要小心。
Use energy conservation only when non-conservative forces are negligible or when you account for work done against them. If drag or friction is present, write W_{non-conservative} = ΔKE + ΔGPE.

仅当非保守力可忽略时，或当你已考虑克服它们所做的功时，才使用能量守恒。如果存在阻力或摩擦，请写成 W_非保守 = ΔKE + ΔGPE。
For bouncing problems, remember that the rebound speed is less than the impact speed unless the collision is perfectly elastic. Use e = v₂/v₁ = √(h₂/h₁) to compare heights.

对于弹跳问题，请记住除非是完全弹性碰撞，否则反弹速度小于撞击速度。使用 e = v₂/v₁ = √(h₂/h₁) 来比较高度。
Be clear about energy “loss”. Energy cannot disappear; it is transferred to the surroundings as heat or sound. Use precise language in structured questions.

要清楚能量“损失”的含义。能量不会消失；它以内能或声的形式转移到周围环境。在结构化问题中使用精确的语言。

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Down, up, down – Energy Changes | 落下、上升、再落下——能量变化