📚 Slingshot Energy Conversion Case Study | 弹弓能量转换案例分析实战演练
In this case study, a student investigates how the energy stored in a stretched elastic band is converted into the kinetic energy of a small projectile. The aim is to explore the relationship between the extension of the slingshot and the horizontal range of the projectile, and to understand energy losses in a real system.
在该案例分析中，一名学生探究了橡皮筋拉伸时储存的能量如何转化为小抛射物的动能。目标是探索弹弓的拉伸长度与抛射物水平射程之间的关系，并理解真实系统中的能量损失。
1. Background and Aim | 背景与目标
A slingshot stores elastic potential energy when its rubber band is stretched. Upon release, most of this energy is transferred to the projectile as kinetic energy, causing it to fly forward. The student wants to find out how the extension length affects the horizontal distance the projectile travels, and to identify where energy is lost.
弹弓在橡皮筋被拉伸时储存弹性势能。释放时，大部分能量转化为抛射物的动能，使其向前飞出。该学生想要探究拉伸长度如何影响抛射物飞行的水平距离，并识别能量在何处损失。
2. Key Physics Concepts | 关键物理概念
Hooke's Law states that the force needed to extend a spring or elastic band is directly proportional to the extension, as long as the elastic limit is not exceeded. This can be written as:
胡克定律指出，伸长弹簧或橡皮筋所需的力与伸长量成正比，只要不超过弹性限度。可以写为：
F = k x
where F is force in newtons (N), x is extension in metres (m), and k is the spring constant (N/m).
其中 F 是力，单位为牛顿（N），x 是伸长量，单位为米（m），k 是弹簧常数（N/m）。
The elastic potential energy (Eₚ) stored in the stretched band is equal to the work done to stretch it. For a material obeying Hooke's Law, the energy is:
拉伸的橡皮筋中储存的弹性势能（Eₚ）等于拉伸时对它所做之功。对于遵守胡克定律的材料，能量为：
Eₚ = ½ F x = ½ k x²
This stored energy is then converted into kinetic energy (Eₖ) of the projectile. In an ideal world with no energy loss, all Eₚ would become Eₖ. The kinetic energy is given by:
该储存的能量随后转化为抛射物的动能（Eₖ）。在没有能量损失的理想世界中，所有 Eₚ 都将转化为 Eₖ。动能由以下公式给出：
Eₖ = ½ m v²
where m is mass in kg and v is velocity in m/s. In reality, some energy is wasted as heat, sound, and air resistance.
其中 m 为质量，单位为 kg，v 为速度，单位为 m/s。现实中，一些能量会以热、声音和空气阻力的形式损失掉。
3. Apparatus and Materials | 实验器材与材料
| Item / 器材 | Purpose / 用途 |
|---|---|
| Rubber band (identical, new) / 橡皮筋（相同的新品） | Acts as the slingshot elastic material / 作为弹弓弹性材料 |
| Small ball bearing or marble (constant mass) / 小钢
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