📚 Down, up, down – energy changes | 下、上、下——能量变化
When you drop a ball and watch it bounce, you are witnessing a fascinating sequence of energy transformations: gravitational potential energy to kinetic energy, then to elastic strain energy at impact, back to kinetic energy, and finally back to gravitational potential energy as it rises. This 'down, up, down' motion illustrates one of the most fundamental concepts in A-Level Physics – the principle of conservation of energy and the reality of energy dissipation in real systems.
当你让一个球下落并观察它弹跳时，你正目睹一系列迷人的能量转化：重力势能转化为动能，然后在碰撞时转化为弹性应变能，再转化回动能，最后随着它上升又转化为重力势能。这种“下、上、下”的运动展示了A-Level物理中最基本的概念之一——能量守恒原理以及真实系统中能量耗散的实际情况。
1. Kinetic and Gravitational Potential Energy Review | 动能与重力势能回顾
Before analysing the bounce cycle, recall the two main forms of mechanical energy. Gravitational potential energy (GPE) depends on an object's mass, height, and the gravitational field strength, given by the equation:
在分析弹跳周期之前，先回顾两种主要的机械能形式。重力势能取决于物体的质量、高度和重力场强度，由以下公式给出：
GPE = mgh
where m is mass (kg), g is gravitational field strength (9.81 N kg⁻¹ on Earth), and h is height above a reference level (m). Kinetic energy (KE) is the energy of motion, expressed as:
其中 m 是质量（kg），g 是重力场强度（地球上为9.81 N kg⁻¹），h 是高于参考面的高度（m）。动能是物体运动的能量，表达式为：
KE = ½ mv²
where v is the speed of the object. Both are measured in joules (J). In an isolated system free from air resistance and other non-conservative forces, the total mechanical energy (GPE + KE) remains constant.
其中 v 是物体的速率。两者都以焦耳（J）为单位。在一个没有空气阻力和其他非保守力的孤立系统中，总机械能（GPE + KE）保持恒定。
2. Stage 1: The Drop – GPE to KE Conversion | 第一阶段：下落——重力势能转化为动能
When the ball is held at a height h₀ above the ground and released, it initially possesses maximum GPE and zero KE. As it falls, height decreases, so GPE decreases. According to the conservation of mechanical energy, this lost GPE is converted into KE, causing the ball's speed to increase. Just before impact, all the initial GPE has become KE (assuming negligible air resistance during this short fall).
当球被举至地面上方高度 h₀ 并释放时，它最初具有最大的重力势能，动能为零。下落时高度减小，因此重力势能减小。根据机械能守恒定律，这部分损失的重力势能转化为动能，使球的速度增加。在碰撞前的瞬间，所有初始的重力势能已经转化为动能（假设此次短距离下落过程中空气阻力可忽略）。
Mathematically, the speed just before striking the ground can be found by equating mgh₀ = ½ mv², giving v = √2gh₀. This relationship shows that the impact speed depends solely on the drop height, not on the mass of the ball.
数学上，撞击地面前的速率可通过使 mgh₀ = ½ mv² 相等求得，得出 v = √2gh₀。这一关系表明撞击速率只取决于下落高度，而与球的质量无关。
3. The Moment of Impact: Temporary Energy Storage | 碰撞瞬间：能量的短暂储存
As the ball hits the ground, it deforms, compressing the material. At this instant, kinetic energy is not destroyed but is temporarily stored as elastic strain energy (also called elastic potential energy) within the ball and the surface. In an ideal perfectly elastic collision, this energy would be fully recovered as KE as the ball regains its shape.
当球撞击地面时，它发生形变，压迫材料。在这一瞬间，动能并没有消失，而是暂时以弹性应变能的形式储存于球体和地面之中。在一个理想的完全弹性碰撞中，当球恢复形状时，这些能量将全部恢复为动能。
Real collisions, however, are never perfectly elastic for ordinary balls. Some of the mechanical energy is converted into internal (thermal) energy, sound, and permanent deformation. This conversion lowers the amount of KE available for the upward motion and explains why the ball does not return to its original height.
然而，对于普通球体而言，真实碰撞从来不是完全弹性的。部分机械能会转化为内能（热能）、声能和永久形变。这种转化减少了可用于向上运动的动能，也解释了为什么球无法弹回到原来的高度。
4. Stage 2: The Bounce – Reverse Transformation | 第二阶段：弹起——反向转化
Immediately after maximum compression, the stored elastic strain energy is converted back into kinetic energy, propelling the ball upward. This upwards KE then transforms into gravitational potential energy as the ball rises against gravity. At the peak of the bounce, all the remaining KE has become GPE, and the ball momentarily comes to rest before falling again.
在最大压缩之后，储存的弹性应变能立即转化回动能，推动球向上运动。随着球克服重力上升，这部分向上的动能又转化为重力势能。在弹跳的最高点，所有剩余的动能都已变成重力势能，球在再次下落前暂时静止。
If the collision were perfectly elastic, the peak bounce height would equal the original drop height h₀. In reality, the new height h₁ is always less than h₀, and the difference in height reflects the energy dissipated during the impact.
如果碰撞是完全弹性的，弹跳的最高点高度将等于初始下落高度 h₀。实际上，新的高度 h₁ 总是小于 h₀，高度差反映了碰撞过程中耗散的能量。
5. Falling Again: Evidence of Energy Loss | 再次下落：能量损失的证据
After reaching h₁, the ball experiences another 'down' phase. Its GPE at h₁ is converted to KE, it strikes the ground, and the cycle repeats. Each successive bounce reaches a lower height, demonstrating progressive energy dissipation. Plotting bounce heights against time or bounce number reveals an exponential decay-like pattern in many cases.
达到高度 h₁ 之后，球又经历一次“下”的阶段。它在 h₁ 处的重力势能被转化为动能，撞击地面，循环重复。每一次后续的弹跳都能达到更低的高度，表现出逐渐的能量耗散。在许多情况下，绘制弹跳高度随时间或弹跳次数的变化图会呈现出类似指数衰减的模式。
This repetitive 'down, up, down' sequence not only reinforces the concept of energy conservation within each fall and rise but also highlights the role of dissipative forces. The area under a force–displacement graph during each compression–expansion cycle would show the energy lost per bounce.
这种重复的“下、上、下”序列不仅强化了每次下落和上升中能量守恒的概念，还突显了耗散力的作用。在每个压缩-膨胀循环中，力-位移图下的面积可以显示每次弹跳所损失的能量。
6. Energy Conservation and Dissipation | 能量守恒与耗散
The first law of thermodynamics, applied to mechanics, states that energy cannot be created or destroyed – only converted from one form to another. In the bouncing ball, total energy is always conserved globally, but the mechanical energy (GPE + KE) is not conserved during the collision because some is transferred to the surroundings as heat and sound.
应用于力学的热力学第一定律指出，能量不能被创造或毁灭——只能从一种形式转化为另一种形式。在弹跳球中，总能量在整个过程中始终守恒，但机械能（重力势能加动能）在碰撞过程中并不守恒，因为其中一部分以热和声的形式传递到了周围环境中。
The term 'energy loss' therefore refers specifically to a decrease in useful mechanical energy. It is crucial for students to distinguish between the universal conservation of energy and the dissipation of mechanical energy within a non-ideal process. This distinction underpins the calculation of efficiency.
因此，“能量损失”一词特指有用的机械能的减少。学生必须区分普遍的能量守恒与非理想过程中机械能的耗散，这一区别是效率计算的基础。
7. Calculating Bounce Efficiency | 计算反弹效率
energy just before and just after the bounce, efficiency η = (KE after / KE before) × 100%. Since KE just before and after is proportional to the height reached, we often use:
一次弹跳的效率可以用能量或高度来定义。利用弹跳前后瞬间的动能，效率 η = (弹后动能 / 弹前动能) × 100%。由于弹跳前后的动能与所达到的高度成正比，通常使用以下公式：
η = (h₁ / h₀) × 100%
where h₀ is the release height and h₁ is the first bounce height. For a squash ball, efficiency might be around 30%–50%, while a steel ball bearing bouncing on a steel plate can exceed 90%. The table below shows example data.
其中 h₀ 为释放高度，h₁ 为第一次弹跳高度。对于壁球，效率可能在 30%–50% 左右，而一个小钢球在钢板上弹跳的效率可超过 90%。下表展示了示例数据。
| Drop height h₀ (m) | Bounce height h₁ (m) | Efficiency (%) |
| 1.00 | 0.65 | 65 |
| 1.50 | 0.98 | 65 |
| 2.00 | 1.30 | 65 |
Notice that for a given ball and surface, the efficiency often remains roughly constant over a range of drop heights, indicating a consistent percentage of energy dissipation per impact.
注意，对于给定的球和地面，效率在一定下落高度范围内往往大致保持不变，这表明每次碰撞的能量耗散百分比是稳定的。
8. Energy–Height Graphs | 能量-高度变化图
Plotting energy against height or time provides powerful visual insight. During free fall, the GPE–height graph is a straight line (GPE ∝ h), while the sum of GPE and KE forms a horizontal line if no energy is lost. In a bounce cycle including losses, the total mechanical energy line drops at each impact, creating a step-like decrease.
绘制能量随高度或时间的变化图能提供直观的洞察。在自由下落过程中，重力势能-高度图是一条直线（GPE ∝ h），而在没有能量损失的情况下，重力势能与动能之和是一条水平线。在包含能量损失的弹跳周期中，每次碰撞时总机械能线都会下跌，形成阶梯式的下降。
Another common representation is the bar chart of energy distribution at key moments: at release (all GPE), just before impact (all KE), just after impact (reduced KE), and at the bounce peak (all GPE again, but less than before). These charts reinforce the principle that energy is neither created nor destroyed, merely redistributed and partially dissipated.
另一种常见的表示方式是关键瞬间的能量分布条形图：释放时（全部为重力势能）、撞击前（全部为动能）、撞击后（减少的动能）、以及弹跳最高点时（再次全部为重力势能，但比之前少）。这些图表强化了一个原则：能量既不创生也不消失，只是重新分配并部分耗散。
9. Real-World Complexities: Air Resistance and Deformation | 真实世界的复杂性：空气阻力与形变
Air resistance acts on the ball throughout its flight, converting a small amount of mechanical energy into thermal energy of the air. For light, low-density balls such as ping-pong balls, air resistance can be significant and must be accounted for when comparing predicted and measured bounce heights. By contrast, a dense steel ball experiences negligible air drag over the same distance.
空气阻力在整个飞行过程中作用于球体，将少量机械能转化为空气的热能。对于像乒乓球这样的轻质、低密度球体，空气阻力可能相当显著，在比较预测弹跳高度与实测值时必须予以考虑。相比之下，密度大的钢球在同样距离内所受的空气阻力可忽略不计。
Furthermore, the exact nature of deformation affects the efficiency. A fully inflated basketball bounces high because the compressed gas stores energy elastically; a flat ball deforms plastically, converting more energy to heat. Understanding these material properties links energy changes to the real behaviour of objects in the CIE syllabus.
此外，形变的确切性质也会影响效率。一个充满气的篮球弹跳高，因为压缩气体能弹性储存能量；而一个瘪掉的球发生塑性形变，会转化更多的能量为热能。理解这些材料性质，能将能量变化与CIE考试大纲中物体的真实行为联系起来。
10. Experimental Investigation: Tracking Energy Changes with Data | 实验探究：用数据追踪能量变化
A typical A-Level investigation involves dropping a ball from various heights and measuring the bounce height using a metre rule, a smartphone high-speed camera, or an electronic position sensor. Students can calculate efficiency for each trial, plot h₁ against h₀, and determine the gradient, which represents the fraction of energy retained.
一个典型的A-Level实验探究包括：将球从不同高度释放，并用米尺、智能手机高速摄影或电子位置传感器测量弹跳高度。学生可以计算每次试验的效率，绘制 h₁ 与 h₀ 的关系图，并求出直线梯度，该梯度代表所保留的能量比例。
If the bounce efficiency is constant, the graph of h₁ vs h₀ will be a straight line through the origin with a gradient equal to the efficiency. Any deviation from linearity at higher speeds may indicate increased air resistance or a change in the collision dynamics. This practical work builds essential data analysis and evaluation skills assessed in Paper 3 and Paper 5 of the CIE A-Level Physics qualification.
如果反弹效率恒定，则 h₁ 与 h₀ 的关系图将是一条穿过原点的直线，其梯度等于效率。在较高速情况下若出现偏离线性的现象，则可能表明空气阻力增大或碰撞动力学发生了变化。这一实践工作培养了CIE A-Level物理考试Paper 3和Paper 5中所评估的重要数据分析和评价技能。
Published by TutorHao | Physics Revision Series | aleveler.com
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