📚 Year 8 CAIE Physics: Case Study Practice – Energy & Motion | CAIE 8年级物理：案例分析实战演练 – 能量与运动
In Year 8 CAIE Physics, learning to apply concepts of energy, work and motion to real-world situations is a vital skill. This case study takes you step by step through a common scenario – a child sliding down a playground slide – and shows you how to use key formulas, energy conservation and an understanding of forces to make predictions and solve problems.
在CAIE八年级物理中，学会将能量、功和运动的概念应用到现实世界中是一项至关重要的技能。本案例研究将一步步引导你分析一个常见场景——一个孩子从游乐场滑梯上滑下——并展示如何运用关键公式、能量守恒以及对力的理解来做出预测并解决问题。
1. The Scenario: A Child on a Slide | 场景：滑梯上的孩子
Imagine a child with a mass of 30 kg who climbs to the top of a straight slide. The vertical height of the slide is 2.0 m, and the length of the sliding surface is 4.0 m. The child starts from rest and slides down. There is a constant friction force of 50 N acting along the slide surface. Once the child reaches the bottom, they continue moving horizontally across the playground surface until friction brings them to a stop.
想象一个质量为30 kg的孩子爬上一个笔直滑梯的顶端。滑梯的垂直高度为2.0 m，滑动表面的长度为4.0 m。孩子从静止开始向下滑。沿着滑梯表面有一个恒定的50 N摩擦力。当孩子到达底部后，他们继续在操场水平地面上移动，直到摩擦力让他们停下来。
We will analyse the entire motion in terms of energy transfers and forces. This is a classic case study that combines gravitational potential energy, kinetic energy, work done against friction and the law of conservation of energy.
我们将从能量转换和力的角度分析整个运动过程。这是一个结合了重力势能、动能、克服摩擦力做功以及能量守恒定律的经典案例研究。
2. Key Data and Assumptions | 关键数据与假设
Before we calculate anything, we need to list the known quantities and state our assumptions. This is an essential step in any physics problem.
在我们计算任何东西之前，需要列出已知量并说明假设。这是任何物理问题中必不可少的步骤。
Known data:
已知数据：
mass of child, m = 30 kg
孩子质量，m = 30 kg
vertical height of slide, h = 2.0 m
滑梯垂直高度，h = 2.0 m
length of slide, dslide = 4.0 m
滑梯长度，dslide = 4.0 m
friction force on the slide, Fslide = 50 N
滑梯上的摩擦力，Fslide = 50 N
gravitational field strength, g = 10 N/kg (rounded for simplicity)
重力场强度，g = 10 N/kg（为简化取近似值）
Assumptions: we ignore air resistance, treat the child as a point mass, and assume that the friction force on the horizontal ground remains constant at 60 N after leaving the slide.
假设：我们忽略空气阻力，将孩子视作质点，并假设离开滑梯后水平地面上的摩擦力恒为60 N。
3. Gravitational Potential Energy at the Top | 顶部的重力势能
At the top of the slide, the child has stored energy because of their height. This is called gravitational potential energy (GPE).
在滑梯顶端，孩子由于高度而储存了能量。这被称为重力势能（GPE）。
The GPE can be found using the formula:
GPE可以用以下公式计算：
GPE = m × g × h
Substitute the numbers: m = 30 kg, g = 10 N/kg, h = 2.0 m.
代入数值：m = 30 kg，g = 10 N/kg，h = 2.0 m。
GPE = 30 × 10 × 2 = 600 J
So the child has 600 joules of gravitational potential energy at the top of the slide. This energy is the total energy available at the start of the motion.
因此孩子在滑梯顶端具有600焦耳的重力势能。这是运动开始时所有可用的能量。
4. Ideal Case: Speed with No Friction | 理想情况：无摩擦时的速度
If there were no friction, all the GPE would convert into kinetic energy (KE) at the bottom. Kinetic energy is the energy of motion.
如果没有摩擦，所有重力势能将转化为底部的动能（KE）。动能是运动的能量。
The formula for kinetic energy is:
动能公式为：
KE = ½ m v²
ervation of energy: GPE lost = KE gained, so mgh = ½ m v². Cancel mass m, and solve for v:
根据能量守恒：重力势能的减少 = 动能的增加，所以 mgh = ½ m v²。消去质量 m，解出 v：
v = √2 × g × h
Put in the numbers: v = √2 × 10 × 2 = √40 ≈ 6.32 m/s.
代入数值：v = √2 × 10 × 2 = √40 ≈ 6.32 m/s。
Without friction, the child would reach the bottom moving at about 6.32 m/s. This is the theoretical maximum speed.
如果没有摩擦，孩子到达底部时的速度约为6.32 m/s。这是理论上的最大速度。
5. Work Done Against Friction on the Slide | 滑梯上克服摩擦力所做的功
In reality, friction does negative work, removing energy from the system. The work done by friction is calculated using:
实际上，摩擦力做负功，从系统中带走能量。摩擦力所做的功用以下公式计算：
W = F × d
Here, the friction force is 50 N and the distance along the slide is 4.0 m.
这里，摩擦力为50 N，沿滑梯的距离为4.0 m。
Wfriction = 50 × 4.0 = 200 J
This means 200 J of the original 600 J is converted into thermal energy (heat) and is no longer available as kinetic energy. The energy lost due to friction is sometimes called dissipated energy.
这意味着原本600 J中的200 J转化为了热能（热），不再作为动能使用。由于摩擦而损失的能量有时称为耗散能量。
6. Actual Kinetic Energy and Speed at the Bottom | 底部实际的动能与速度
Now we can find the kinetic energy the child actually has at the bottom of the slide. We subtract the work done against friction from the original GPE:
现在我们可以求出孩子在滑梯底部的实际动能。我们从初始的重力势能中减去克服摩擦力做的功：
KEactual = GPE – Wfriction = 600 – 200 = 400 J
Use the kinetic energy formula to find the real speed v:
使用动能公式求实际速度 v：
KE = ½ m v² → v = √2 × KE / m
v = √2 × 400 / 30 = √800 / 30 = √26.67 ≈ 5.16 m/s.
v = √2 × 400 / 30 = √800 / 30 = √26.67 ≈ 5.16 m/s。
The child’s speed is reduced from the ideal 6.32 m/s to about 5.16 m/s because of the energy lost to friction.
由于摩擦造成的能量损失，孩子的速度从理想的6.32 m/s降低至约5.16 m/s。
7. Horizontal Sliding to a Stop | 水平滑行至停止
Once the child reaches the horizontal ground, they have 400 J of kinetic energy. A friction force of 60 N now acts opposite to the direction of motion. The work done by this force will bring the child to rest.
一旦孩子到达水平地面，他们具有400 J的动能。现在有一个60 N的摩擦力与运动方向相反。这个力做的功将使孩子停下来。
To find the stopping distance dground, we use the fact that all the kinetic energy is converted into work against friction:
为了求得停止距离 dground，我们利用所有动能都转化为克服摩擦做功这一事实：
KE = Fground × dground
Rearrange to dground = KE / Fground = 400 / 60 ≈ 6.67 m.
变形得 dground = KE / Fground = 400 / 60 ≈ 6.67 m。
So the child slides about 6.67 metres across the playground before coming to a complete stop. All the original gravitational potential energy has now been transferred into thermal energy in the slide, the child’s clothing and the ground.
因此孩子在完全停下来之前会在操场地面上滑行约6.67米。至此，最初所有的重力势能都已转化为滑梯、孩子衣服和地面的热能。
8. Energy Transfer Summary | 能量转移总结
It is very useful to track energy changes at each stage of the motion. The table below summarises the energy conversions:
追踪运动每个阶段的能量变化十分有用。下表总结了能量转换过程：
| Stage | 中文 | Energy involved |
| Top of slide | 滑梯顶部 | GPE = 600 J |
| During slide | 滑动过程中 | 200 J → thermal energy of slide and child |
| Bottom of slide | 滑梯底部 | KE = 400 J |
| Horizontal ground | 水平地面 | 400 J → thermal energy of ground |
| Final stop | 最终停止 | All 600 J dissipated as heat |
The total energy is conserved at every instant – it is simply transferred from one store to another or dissipated. No energy is destroyed.
总能量在每一时刻都是守恒的——它只是从一个储存转移到另一个储存或者被耗散。没有能量被消灭。
9. Efficiency of the Slide | 滑梯的效率
Efficiency describes how much of the input energy is converted into useful output energy. For this slide, the useful energy is the kinetic energy at the bottom that allows the child to move forward.
效率描述的是有多少输入能量转化为有用的输出能量。对这个滑梯来说，有用能量就是底部能使孩子向前运动的动能。
Efficiency = (useful energy output / total energy input) × 100%
Efficiency = (400 J / 600 J) × 100% = 66.7%
The slide is about 67% efficient in converting gravitational potential energy into kinetic energy. The remaining 33% is wasted as heat. In many real machines, some energy is always dissipated due to friction.
这个滑梯将重力势能转化为动能的效率约为67%。剩下的33%以热能形式浪费掉了。在许多真实机械中，由于摩擦总是会有一些能量被耗散。
10. Common Mistakes and Reminders | 常见错误与提醒
When solving case studies like this, take care to avoid these frequent errors:
在解决此类案例研究时，小心避免以下常见错误：
= m × g, measured in newtons. Mass is in kilograms. The GPE formula uses mass, not weight.
用重量代替质量。重量 = m × g，单位是牛顿。质量单位是千克。GPE公式使用的是质量，不是重量。
Forgetting that work done by friction depends on the distance along the surface, not the vertical height.
忘记摩擦力做功取决于沿表面的距离，而不是垂直高度。
Mixing up energy units. Energy and work are both measured in joules (J).
混淆能量单位。能量和功都用焦耳（J）来度量。
Assuming speed is the same with and without friction. Always check for energy losses.
假设有摩擦和没有摩擦时速度相同。一定要检查能量损失。
11. Real-World Application: Playground Safety | 实际应用：操场安全
Understanding these energy transfers helps designers create safer playgrounds. If the slide were steeper (greater height for the same length), the child’s speed at the bottom would be higher. If the surface were rougher (larger friction force), the speed would be lower and the stopping distance shorter.
理解这些能量转移有助于设计师打造更安全的操场。如果滑梯更陡（同样长度下高度更高），孩子到底部的速度会更大。如果表面更粗糙（更大的摩擦力），速度会更低，停车距离会更短。
Engineers also consider the material of the ground at the bottom of slides. A surface that provides more friction will stop children more quickly, reducing the risk of collisions. Energy calculations allow them to predict exactly how far a child might slide on different surfaces.
工程师还会考虑滑梯底部地面的材质。提供更大摩擦力的表面会让孩子更快停下来，降低碰撞风险。能量计算使他们能够精确预测孩子在不同表面上的滑行距离。
12. Summary and Practice Suggestions | 总结与练习建议
This case study has shown how to combine GPE, KE, work done by friction and energy conservation to analyse a complete motion sequence. Always follow a systematic approach: list the data, identify energy stores at each point, account for work done by non-conservative forces, and then apply formulas.
本案例研究展示了如何结合重力势能、动能、克服摩擦力做功以及能量守恒来分析完整的运动序列。始终遵循系统的方法：列出数据，确定各点的能量储存，考虑非保守力所做的功，然后应用公式。
You can create your own variations: change the mass, height or friction force, and recalculate the speeds and distances. Try adding a second slope or a small ramp at the bottom. Practising different scenarios will build your confidence in tackling CAIE case study questions.
你可以创建自己的变式：改变质量、高度或摩擦力，并重新计算速度和距离。尝试在底部添加第二个斜坡或一个小坡道。练习不同的场景将有助于你建立解答CAIE案例分析题的信心。
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