How does the height at which a ball is dropped affect the elasticity of its collision? | 下落高度对碰撞弹性的影响实验探究

📚 How does the height at which a ball is dropped affect the elasticity of its collision? | 下落高度对碰撞弹性的影响实验探究

When a ball bounces, the height it returns to tells us something about the energy lost during impact. The ‘elasticity’ of a collision is measured by the coefficient of restitution. This investigation explores whether changing the drop height alters that elasticity for a given ball and surface. By systematically varying the drop height and measuring the rebound, we can test if the collision becomes more inelastic at greater impact speeds, or if the material’s response remains constant.

当球弹跳时，它返回的高度能反映出撞击过程中的能量损失。碰撞的’弹性’由恢复系数来量化。本次实验探究旨在检验对于给定的球和表面，改变下落高度是否会改变这种弹性。通过系统性地改变下落高度并测量反弹高度，我们可以测试碰撞是否在更高的撞击速度下变得更非弹性，或者材料的响应是否保持恒定。

1. Understanding Elasticity in Collisions | 理解碰撞中的弹性

In an ideal perfectly elastic collision, a ball dropped from height h₁ would rebound to exactly the same height. In reality, some kinetic energy is converted to internal energy (heat), sound, and permanent deformation. The coefficient of restitution, e, quantifies the ratio of speeds before and after impact. For a ball bouncing off a stationary rigid floor, e can be conveniently expressed in terms of heights: e = √(h₂ / h₁), where h₂ is the rebound height. An e of 1 is perfectly elastic; an e of 0 means the ball does not bounce at all.

e = √(h₂ / h₁)

在理想的完全弹性碰撞中，从高度 h₁ 下落的球会反弹到完全相同的高度。实际上，部分动能会转化为内能（热）、声能和永久形变。恢复系数 e 量化了撞击前后的速度比。对于落在静止刚性地面上的球，e 可以方便地用高度表示为：e = √(h₂ / h₁)，其中 h₂ 是反弹高度。e=1 表示完全弹性；e=0 表示球完全不反弹。

2. The Role of Drop Height | 下落高度的作用

When the drop height increases, the ball’s impact velocity also increases according to v = √(2gh₁). A higher velocity means the ball undergoes more rapid and severe compression. If the ball material obeys Hooke’s law perfectly and no energy is dissipated, e should be independent of height. However, real materials exhibit hysteresis and strain-rate dependence: greater compression can lead to proportionally larger energy losses. This investigation will measure e at various drop heights to determine whether e remains constant or decreases with height.

v = √(2gh₁)

当下落高度增加时，球的撞击速度也根据 v = √(2gh₁) 而增大。更高的速度意味着球会经历更快速、更剧烈的压缩。如果球的材料完全遵循胡克定律且不耗散能量，e 应与高度无关。然而，真实材料表现出滞后和应变率依赖性：更大的压缩可能导致比例上更大的能量损失。本次探究将在不同下落高度下测量 e，以确定 e 是保持恒定还是随高度减小。

3. Experimental Variables | 实验变量

The independent variable is the drop height, h₁, measured from the bottom of the ball to the floor. The dependent variable is the rebound height, h₂, also measured to the bottom of the ball at the peak of its first bounce, from which e is calculated. Controlled variables that must be kept constant include: the type and condition of the ball (mass, size, inflation pressure if applicable), the floor surface material and its rigidity, the release method (no initial spin, no downward push), and the ambient temperature and air pressure.

自变量是下落高度 h₁，从球的底部到地面测量。因变量是反弹高度 h₂，同样是在第一次反弹最高点从球底部测量，并由此计算 e。必须保持恒定的控制变量包括：球的类型和状况（质量、大小、充气压力如适用）、地板表面材料及其刚性、释放方式（无初始旋转、无下推）以及环境温度和气压。

4. Apparatus and Setup | 器材与装置

A suitable ball (e.g., tennis ball, basketball, or solid rubber ball) is needed. A metre ruler or measuring tape is fixed vertically against a wall, with clear markings. A high-speed camera or a smartphone capable of slow-motion video (at least 120 fps) is positioned to capture the bounce without parallax. A flat, hard, and level floor serves as the impact surface. Optional: an electromagnetic release clamp to eliminate release inconsistencies, and reference markers placed on the wall at known heights.

需要一个合适的球（如网球、篮球或实心橡胶球）。一把米尺或卷尺竖直固定在墙上，并有清晰的刻度。一部高速摄像机或能够拍摄慢动作视频（至少 120 fps）的智能手机被放置好，以无视差地捕捉反弹。一个平坦坚硬的水平地面作为撞击表面。可选：一个电磁释放夹以消除释放的不一致性，以及墙上已知高度处放置的参考标记。

5. Procedure | 实验步骤

1. Secure the ruler and camera so the ball’s entire trajectory is in frame. 2. Measure and mark drop heights: e.g., 0.50 m, 1.00 m, 1.50 m, 2.00 m, 2.50 m. 3. Hold the ball with its bottom exactly at the desired h₁ and release without applying force. 4. Record the bounce; use frame-by-frame playback to identify the maximum height h₂ reached after the first impact. 5. Repeat each drop height three to five times, taking the average h₂. 6. Calculate e for each drop height using e

Published by TutorHao | IB Physics Revision Series | aleveler.com

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