Work, energy, power and collisions
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| coefficient of restitution/ˌkəʊɪˈfɪʃənt ɒv rɪstɪˈtjuːʃn/ | 恢复系数 | huī fù xì shù |
What does a slower rebound tell us?
- A ball rebounds slower after hitting a wall. Momentum and kinetic energy answer different parts of the collision story.
- This lesson studies coefficient of restitution 恢复系数: The ratio of relative separation speed to relative approach speed along the line of impact.
反弹速度变慢说明了什么？
- 球撞击墙壁后反弹速度变慢。动量和动能分别解释了碰撞过程的不同方面。
- 本课研究恢复系数（coefficient of restitution）：沿撞击线方向的相对分离速度与相对接近速度之比。
Choose the mathematical structure
- Work is force times displacement in its direction. Kinetic energy is mv²/2 and gravitational potential energy is mgh. Momentum is conserved for an isolated collision; restitution gives the relative speed relation, not conservation of kinetic energy.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 功等于力乘以该力方向上的位移。动能公式为 mv²/2，重力势能公式为 mgh。孤立碰撞中动量守恒；恢复系数给出的是相对速度的关系，而非动能守恒。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines coefficient of restitution?
The ratio of relative separation speed to relative approach speed along the line of impact.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A 2 kg particle moving at 6 m/s has kinetic energy 2×6²/2=36 J. If a constant opposing force of 3 N stops it, work=3s=36, giving s=12 m. A wall bounce with approach speed 5 and e=0.6 has rebound speed 3 m/s in the opposite direction.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
一个质量为 2 kg 的粒子以 6 m/s 的速度运动，其动能为 2×6²/2=36 J。若一个大小为 3 N 的恒定阻力使其停止，则功=3s=36，解得 s=12 m。若球以 5 的速度撞墙且恢复系数 e=0.6，则反弹速度为 3 m/s，方向相反。
Work, energy, power and collisions
Work is force times displacement in its direction
Compare the model with the worked case and explain one change.
Find kinetic energy for m=2 kg,v=6 m/s.
Kinetic energy=mv²/2=2×36/2=36 J.
Test a tempting shortcut
- Conserve momentum only when the external impulse is negligible. Kinetic energy is generally lost in an inelastic collision. Power is a rate of energy transfer, not an amount of energy.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every collision conserves kinetic energy. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 仅当外力冲量可忽略时才能应用动量守恒。非弹性碰撞中通常会有动能损失。功率是能量传递的速率，而非能量的总量。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
每次碰撞都守恒动能。此说法错误。请指出它违反了哪个定义或假设。
A resisting force of 3 N removes 36 J. Find stopping distance.
The resistance removes 3 J per metre: distance=36/3=12 m.
Every collision conserves kinetic energy.
Conserve momentum only when the external impulse is negligible. Kinetic energy is generally lost in an inelastic collision. Power is a rate of energy transfer, not an amount of energy.
Interpret a new situation
- State the direction signs before writing collision equations. For a connected motion problem, choose the whole system for energy when internal tensions cancel; account for frictional work explicitly.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 在列写碰撞方程前，先规定正方向符号。对于连接体问题，选取整个系统计算能量时内部张力会相互抵消；需显式计入摩擦力所做的功。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find rebound speed from a wall when approach speed is 5 m/s and e=0.6.
For a fixed wall, rebound speed=e×approach speed=0.6×5=3 m/s.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further mathematics; official unit M2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The ratio of relative separation speed to relative approach speed along the line of impact. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- Edexcel IAL 进阶数学；官方单元 M2。其他单元的补充内容在范围审查中明确列出，并非额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
沿撞击线方向的相对分离速度与相对接近速度之比。选择正确的关系式，展示解题步骤，检查其适用假设并解释结果。