📚 Case Study Practical Drill: Determining g by Free Fall | 案例分析实战演练：通过自由落体测定重力加速度
In A-Level Science, especially in the Edexcel Year 13 Physics specification, the ability to carry out a practical investigation and critically analyse the data is a key skill. This case study walks you through a classic experiment: determining the acceleration due to gravity, g, using a free-fall method. We will explore how to design the experiment, record and process data, plot an appropriate graph, calculate uncertainties, and evaluate the procedure. By the end, you will feel confident in tackling similar practical-based case study questions in your exams.
在A-Level科学中，特别是在Edexcel Year 13物理考试中，开展实验探究并批判性地分析数据是一项核心技能。本案例分析将带你走过一个经典实验：利用自由落体法测定重力加速度g。我们将探讨如何设计实验、记录和处理数据、绘制合适的图表、计算不确定度以及评估实验过程。读完本文后，你将更有信心在考试中应对类似的基于实验的案例分析题。
1. Understanding the Aim and Principle | 理解实验目的与原理
The aim is to determine a value for the acceleration of free fall, g. The underlying principle is that an object falling freely from rest near the Earth’s surface undergoes uniform acceleration. The distance fallen, s, is related to the time taken, t, by the equation s = ½ g t², provided air resistance is negligible. By measuring s and t for a range of heights, we can find g from the gradient of a suitable straight-line graph.
实验目的是测定自由落体加速度g的值。基本原理是，在地球表面附近，从静止开始自由下落的物体做匀加速运动。若空气阻力可忽略不计，下落距离s与所用时间t满足关系式 s = ½ g t²。通过测量不同高度的s和t，我们可以从合适直线的斜率中求出g。
2. Apparatus and Setup | 实验器材与装置
A typical setup includes a steel ball bearing, an electromagnet to release the ball, a trapdoor or impact switch at the bottom, an electronic timer (or digital stopwatch with microsecond resolution), a metre ruler, and a plumb line to ensure vertical alignment. The ball is held by the electromagnet at a measured height h above the trapdoor. When the circuit breaks, the ball falls and the timer starts; when it hits the trapdoor, the timer stops.
典型的装置包括一个钢球、一个用于释放小钢球的电磁铁、一个底部的活板门或碰撞开关、一个电子计时器（或微秒级精度的数字秒表）、一把米尺，以及用于确保垂直对齐的铅垂线。小钢球被电磁铁固定在活板门上方已测定的高度h处。电路断开时，小球下落并启动计时器；当它击中活板门时，计时器停止。
3. Variables and Data Range | 变量与数据范围
The independent variable is the drop height, s. You should record at least six different heights ranging from about 0.30 m to 1.80 m, spaced evenly. The dependent variable is the time of fall, t. Control variables include using the same ball (constant mass and shape), ensuring the electromagnet releases cleanly (no residual magnetism delay), and minimising air drafts. Repeat each height three times to obtain an average t and reduce random error.
自变量是下落高度s。你应至少记录从约0.30 m到1.80 m的六个不同高度，间隔均匀。因变量是下落时间t。控制变量包括使用同一个小球（质量和形状不变），确保电磁铁释放干脆（无剩磁延迟），并尽量减少空气流动。每个高度重复三次以获得平均时间t，从而减少随机误差。
4. Sample Data Recording Table | 示例数据记录表
Below is a typical data table a student might produce. The heights are measured with a metre ruler (±0.001 m or ±1 mm) and the times with a digital timer reading to 0.01 ms (though trigger uncertainties dominate). All values are recorded to appropriate significant figures.
下面是一张学生可能获得的典型数据表。高度用米尺测量（±0.001 m或±1 mm），时间用读数可达0.01 ms的数字计时器测量（尽管触发不确定度占主导）。所有数值均以适当的有效数字记录。
| s / m | t₁ / s | t₂ / s | t₃ / s | t (mean) / s | t² / s² |
|---|---|---|---|---|---|
| 0.400 | 0.286 | 0.285 | 0.286 | 0.2857 | 0.0816 |
| 0.600 | 0.350 | 0.349 | 0.350 | 0.3497 | 0.1223 |
| 0.800 | 0.404 | 0.404 | 0.403 | 0.4037 | 0.1630 |
| 1.000 | 0.452 | 0.451 | 0.452 | 0.4517 | 0.2040 |
| 1.200 | 0.495 | 0.494 | 0.495 | 0.4947 | 0.2447 |
| 1.400 | 0.535 | 0.534 | 0.535 | 0.5347 | 0.2859 |
5. Graph Plotting and Analysis | 图像绘制与分析
According to s = ½ g t², if we plot s on the y-axis against t² on the x-axis, the data should form a straight line passing through the origin. The gradient of this line is ½ g, so g can be calculated as 2 × gradient. Use a sharp pencil to plot points with small crosses, draw a line of best fit that balances the points, and if the line does not go through the origin, discuss systematic error.
根据 s = ½ g t²，若以t²为横坐标、s为纵坐标作图，数据应形成一条通过原点的直线。这条直线的斜率等于 ½ g，因此 g = 2 × 斜率。用削尖的铅笔画小十字描点，画一条最佳拟合线使点均匀分布在两侧，如果直线不通过原点，则应讨论系统误差。
From the sample data, picking two widely separated points on the best-fit line (not necessarily data points): (0.0800, 0.390) and (0.2900, 1.410). Gradient = (1.410 – 0.390) / (0.2900 – 0.0800) = 1.020 / 0.2100 = 4.857 m s⁻². Then g = 2 × 4.857 = 9.71 m s⁻².
从示例数据中，在最佳拟合线上选取两个相隔较远的点（不一定是数据点）： (0.0800, 0.390) 和 (0.2900, 1.410)。斜率 = (1.410 – 0.390) / (0.2900 – 0.0800) = 1.020 / 0.2100 = 4.857 m s⁻²。因此 g = 2 × 4.857 = 9.71 m s⁻²。
6. Calculating Percentage Uncertainty | 计算百分不确定度
To assess the reliability of the result, you need to estimate the uncertainty in g. This comes from the uncertainty in the gradient, which in turn depends on the scatter of points. Draw error bars on the graph (e.g., using ±0.001 m for s and ±0.002 s for t, which gives larger uncertainty in t²). Then draw the steepest and shallowest acceptable lines of best fit. Suppose these give gradients m_max and m_min. Then g_max = 2 m_max, g_min = 2 m_min. The absolute uncertainty in g is Δg = (g_max – g_min)/2, and the percentage uncertainty is (Δg / g) × 100%.
为了评估结果的可靠性，你需要估计g的不确定度。这源于斜率的不确定度，而斜率的不确定度又取决于点的分散程度。在图上画误差棒（例如，s的误差为±0.001 m，t的误差为±0.002 s，这将导致t²的误差更大）。然后画出可接受的最陡和最浅的最佳拟合线。假设它们给出的斜率分别为 m_max 和 m_min，则 g_max = 2 m_max，g_min = 2 m_min。g的绝对不确定度为 Δg = (g_max – g_min)/2，百分不确定度为 (Δg / g) × 100%。
If, for example, the worst acceptable lines yield g_max = 9.97 and g_min = 9.45, then Δg = 0.26, and percentage uncertainty ≈ (0.26/9.71)×100% ≈ 2.7%. This is a typical value for this experiment. Always compare your percentage uncertainty with the accepted value (9.81 m s⁻²) to see if the result is accurate within the experimental error.
例如，如果最差可接受线得出 g_max = 9.97，g_min = 9.45，则 Δg = 0.26，百分不确定度 ≈ (0.26/9.71)×100% ≈ 2.7%。这是此实验中常见的值。始终将你的百分不确定度与公认值（9.81 m s⁻²）进行比较，以判断结果是否在实验误差范围内准确。
7. Identifying Sources of Error | 识别误差来源
The main sources of uncertainty are: (1) Reaction time or trigger uncertainty – even with electronic timing, the release and impact detection may have slight delays. (2) Parallax error in measuring the drop height using a ruler. (3) The assumption that the ball falls from rest; if the electromagnet does not release instantly, the initial velocity might not be exactly zero. (4) Air resistance, which becomes more significant at larger heights, causing t to be slightly longer and leading to an underestimation of g. (5) The trapdoor switch might not trigger precisely at the moment of impact.
不确定度的主要来源有：(1) 反应时间或触发不确定度——即使使用电子计时，释放和碰撞检测也可能存在微小延迟。(2) 用直尺测量下落高度时的视差误差。(3) 假设小球从静止下落；如果电磁铁没有瞬间释放，初速度可能不正好为零。(4) 空气阻力，在较大高度时更显著，导致时间t略长，从而使g被低估。(5) 活板门开关可能不会在撞击瞬间精确触发。
8. Evaluating the Method and Improvements | 评估方法与改进措施
One key evaluation point is whether the data truly follows a linear trend. If the intercept is not zero, it reveals a systematic error—perhaps a consistent timing offset due to the release mechanism. To improve accuracy, you could use a light gate system instead of a trapdoor; two light gates placed a measured distance apart can eliminate the need for an initial velocity of zero if you use the equation s = ut + ½ g t² and measure time between the two gates. Alternatively, use a high-speed camera with a known scale to directly capture the position at known time intervals.
一个关键的评估点在于数据是否真正遵循线性趋势。如果截距不为零，则表明存在系统误差——可能是释放机构导致的恒定计时偏差。为了提高准确度，可以用光门系统代替活板门；两个相隔已知距离的光门可以消除初速度必须为零的要求，利用 s = ut + ½ g t² 并测量通过两个光门之间的时间。或者，使用带有已知标尺的高速摄像机，直接捕捉已知时间间隔下的位置。
Another improvement is to increase the range of heights and take more repeats to reduce random error. Using a heavier ball can help reduce the relative effect of air resistance. Ensuring the apparatus is perfectly vertical with a plumb line minimises cosine errors in the height measurement.
另一个改进是增大高度范围并增加重复次数以减小随机误差。使用更重的球有助于减小空气阻力的相对影响。用铅垂线确保装置完全竖直，可以最小化高度测量中的余弦误差。
9. Applying to Exam Questions | 应用于考试题型
In Edexcel Year 13 science exam papers, a case study question might present you with a data table similar to the one above, ask you to plot a graph, calculate g and its uncertainty, and then discuss the reliability and suggest improvements. You must be able to manipulate the equation, correctly identify axes, calculate gradient from a large triangle on the best-fit line, and express g to the appropriate number of significant figures. Be prepared to comment on whether the accepted value lies within the experimental uncertainty range.
在Edexcel Year 13科学试卷中，案例分析题可能会提供类似上述的数据表，要求你画图、计算g及其不确定度，然后讨论可靠性并提出改进建议。你必须能够对公式进行变换，正确确定坐标轴，从最佳拟合线上的大三角形计算斜率，并以适当的有效数字表示g。准备好评论公认值是否落在实验不确定度范围内。
A typical question might also ask: ‘Explain why the graph of s against t² is preferred over s against t.’ The answer is that s ∝ t² yields a straight line, which is easier to analyse and the gradient directly relates to g, whereas s against t would be a curve.
常见的考题还可能问：“解释为什么用s对t²作图优于s对t作图？”答案是 s ∝ t² 能得到一条直线，更容易分析，且斜率直接与g相关，而s对t作图将是一条曲线。
10. Statistical Treatment and Error Bars | 统计处理与误差棒
When you have multiple repeats at each height, you can calculate the mean t and the standard deviation (if you have many repeats) or simply use the range/2 as an estimate of the uncertainty in t. For the sample data above, the values are very consistent, so the random uncertainty in t is small. The uncertainty in t² is then calculated using the formula: Δ(t²) ≈ 2t Δt. For instance, at s = 1.400 m, t = 0.5347 s, if Δt = 0.001 s (typical instrument precision), then Δ(t²) ≈ 2 × 0.5347 × 0.001 ≈ 0.0011 s². These values can be used to draw horizontal error bars on the graph.
当每个高度有多组重复数据时，你可以计算平均时间t和标准差（如果重复次数多的话），或者简单地用（极差/2）作为t的不确定度估计值。对于上面的示例数据，各组数据非常一致，因此t的随机不确定度很小。然后使用公式 Δ(t²) ≈ 2t Δt 计算t²的不确定度。例如，在 s = 1.400 m处，t = 0.5347 s，若 Δt = 0.001 s（典型的仪器精度），则 Δ(t²) ≈ 2 × 0.5347 × 0.001 ≈ 0.0011 s²。这些值可用于在图上绘制水平误差棒。
11. Dealing with Anomalous Data | 处理异常数据
If one repeat is clearly out of line—perhaps due to a fumbled timer start—you should exclude it from the mean calculation and note it as anomalous. Always comment on the precision of the data: if repeats are very close, the experiment has high precision. A precise result may still be inaccurate if a systematic error is present. For example, a consistent 0.02 s delay would shift all points, making the line not pass through the origin and leading to an incorrect g even with precise data.
如果某次重复明显偏离其他数据——可能因为计时器启动失误——你应将其从平均值计算中剔除，并注明为异常值。始终要评论数据的精密度：如果重复值非常接近，说明实验精密度高。但如果存在系统误差，精密的结果仍可能不准确。例如，持续的0.02 s延迟会使所有点发生平移，导致直线不通过原点，即使数据很精密也会得到错误的g值。
12. Conclusion and Key Takeaways | 结论与要点总结
This case study illustrates a full cycle of practical data analysis: from raw measurements to a final value with uncertainty. Remember that g derived from a free-fall experiment on Earth should be close to 9.81 m s⁻². If your result deviates by more than the combined experimental uncertainty, you must identify a systematic cause. Mastering this process prepares you not only for exam questions but also for the practical endorsement requirements of A-Level science.
本案例分析展示了实验数据分析的完整循环：从原始测量到带有不确定度的最终结果。记住，在地球上通过自由落体实验得出的g应当接近9.81 m s⁻²。如果你的结果偏离超过综合实验不确定度，就必须找出系统原因。掌握这一过程不仅为考试题目做好准备，也为A-Level科学的实践认证要求打下基础。
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