📚 Case Study in Practice: Roller Coaster Physics | 案例分析实战演练：过山车物理探究
In the SQA Physics examination, case study questions ask you to apply physical principles to a realistic scenario, analyse the given data, and evaluate the result against what ideal theory predicts. This worked example follows a single roller coaster car through a complete energy analysis, covering gravitational potential energy, kinetic energy, friction and air resistance, circular motion, forces, measurement uncertainty, and exam technique.
在 SQA 物理考试中，案例分析题要求你把物理原理应用到真实情境中，分析给定数据，并把结果与理想理论的预测进行对比评估。这个示范案例将跟随一节过山车车厢完成一次完整的能量分析，涵盖重力势能、动能、摩擦与空气阻力、圆周运动、力、测量不确定度以及答题技巧。
1. Scenario and Key Data | 情景与关键数据
which is at ground level. It then rises to point C, which is 10 m above the ground. The shape of the track is smooth, but in the real system friction between the wheels and the rails, and air resistance, are not negligible. Take the gravitational field strength as g = 9.8 N/kg throughout.
一节质量为 500 kg 的过山车车厢从离地面 20 m 的 A 点由静止释放，下降到轨道最低点 B（位于地面高度），然后再上升到离地面 10 m 的 C 点。轨道形状平滑，但在真实系统中，车轮与轨道之间的摩擦以及空气阻力不可忽略。全过程中取重力场强度 g = 9.8 N/kg。
Two further pieces of information complete the case study:
本案例还补充两条关键信息：
- The curved section at B has a radius of curvature r = 12 m. | B 点处的弯曲轨道曲率半径为 r = 12 m。
- A light gate at B measures the speed of the car as it passes. | B 点处安装了一个光门，用于测量车厢通过时的速度。
The task is to compare ideal (frictionless) predictions with measured values, then explain and evaluate the differences. Write the data into a table before you start calculating, because it prevents substitution errors later.
任务是把理想（无摩擦）预测值与实际测量值进行比较，然后解释并评估二者的差异。建议动笔计算之前先把数据整理成表格，这样可以避免后续代入数值时出错。
| Quantity | 物理量 | Symbol | 符号 | Value | 数值 |
| Mass of car | 车厢质量 | m | 500 kg |
| Height of A | A 点高度 | hA | 20 m |
| Height of B | B 点高度 | hB | 0 m |
| Height of C | C 点高度 | hC | 10 m |
| Gravitational field strength | 重力场强度 | g | 9.8 N/kg |
| Radius of curvature at B | B 点曲率半径 | r | 12 m |
| Speed at B from light gate | 光门测得 B 点速度 | vB | 18.2 m/s |
2. Gravitational Potential Energy at Point A | A 点重力势能
ored at A is Eₚ = mgh, so substituting the values gives a total of 98 kJ available at the start.
重力势能是相对于某个选定的参考面来衡量的，这里最自然的选择是地面（B 点）。A 点储存的能量为 Eₚ = mgh，代入数值可得出发时可用总能量为 98 kJ。
Eₚ = mgh = 500 × 9.8 × 20 = 98 000 J = 98 kJ
Two points are worth marking in an exam answer. First, the unit N/kg is identical to m/s², so the calculation is fully consistent. Second, all of this energy is potential energy at the instant of release, because the car starts from rest and therefore has zero kinetic energy.
答题时有两个得分点值得注意。第一，单位 N/kg 与 m/s² 完全等价，因此该计算在量纲上是自洽的。第二，由于车厢由静止释放，动能为零，所以释放瞬间这 98 kJ 全部是势能。
3. Ideal Speed at Point B | 理想情况下 B 点的速度
If the track were frictionless and there were no air resistance, mechanical energy would be conserved as the car descends from A to B. All of the 98 kJ of potential energy would become kinetic energy, so we set ½mv² equal to mgh.
如果轨道无摩擦且没有空气阻力，车厢从 A 下降到 B 的过程中机械能守恒。98 kJ 的势能将全部转化为动能，因此令 ½mv² 等于 mgh。
½mv² = mgh → v = √2gh = √2 × 9.8 × 20 = √392 = 19.8 m/s
Notice that the mass cancels completely, which is a classic exam insight: the ideal speed at the bottom of a frictionless drop depends only on the vertical height fallen and the gravitational field strength, not on how heavy the car is. A 200 kg car and a 500 kg car released from the same height would arrive at B with the same speed.
注意质量被完全约掉，这是考试中的经典结论：在无摩擦的下落过程中，最低点的理想速度只取决于下落的高度和重力场强度，与车厢有多重无关。从同一高度释放的 200 kg 与 500 kg 车厢，到达 B 点时速度相同。
4. Energy Losses from A to B | A 到 B 的能量损失
In the real system, the light gate at B records 18.2 m/s, which is lower than the ideal value of 19.8 m/s. We can now calculate the actual kinetic energy and compare it with the 98 kJ available at the start.
在真实系统中，B 点光门测得的速度为 18.2 m/s，低于理想值 19.8 m/s。现在可以计算实际动能，并与出发时的 98 kJ 进行比较。
Eₖ = ½mv² = ½ × 500 × 18.2² = 82 810 J ≈ 82.8 kJ
Energy lost = 98 000 − 82 810 = 15 190 J ≈ 15.2 kJ (15.5% of the initial energy)
The missing energy has not disappeared - it has been transferred to the thermal store of the track, the wheels and the surrounding air, and a very small amount has been carried away as sound. When a question says "explain where the energy has gone", you must name these stores explicitly rather than writing only that the energy "was lost".
缺失的能量并没有消失，而是转移到了轨道、车轮和周围空气的热能储库中，另有极少量以声能形式带走。当题目问"解释能量去了哪里"时，你必须明确指出这些能量储库，而不能只写"能量损失了"。
- Friction between wheels and rails → thermal energy in the wheel bearings and track. | 车轮与轨道间的摩擦 → 轴承与轨道的热能。
- Air resistance (drag) → thermal energy in the air, plus sound. | 空气阻力 → 空气的热能，以及声能。
- Deformation and vibration of the structure → thermal energy and sound. | 结构变形与振动 → 热能与声能。
5. Climbing to Point C with Losses | 带着
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