📚 A-Level Physics Experimental Investigation using Insert 5 (Jan 2022) | A-Level 物理：利用2022年1月Insert 5进行实验探究
In A-Level Physics, Insert 5 from the January 2022 examination series provides a realistic data set for students to explore an experimental scenario. Typically, this insert presents a table of measurements, such as potential difference and current for a cell, time and height for a falling object, or temperature and resistance for a thermistor. The task is not merely to plot a graph but to apply practical skills, analyse uncertainties, derive a physical quantity, and evaluate the procedure. This article guides you through a complete experimental investigation using the Insert 5 data, focusing on the determination of the acceleration of free fall, g, via a light-gate and picket fence method—an experiment closely aligned with the typical challenges found in Paper 3 Section A. We will examine data processing, graphical analysis, uncertainty treatment, and critical evaluation, all of which are essential for top marks in the practical component of your A-Level Physics exam.
在 A-Level 物理中，2022年1月考试系列的 Insert 5 为学生提供了一个真实的数据集，供其探究一个实验场景。这份插页通常提供一张测量数据表，例如电池的电压和电流、下落物体的时间和高度，或热敏电阻的温度和电阻值。任务不仅仅是绘制图表，而是要应用动手实验技能、分析不确定性、推导物理量并评价实验程序。本文将以利用 Insert 5 数据确定自由落体加速度 g 为线索——该实验通过光门和挡光栅法进行，与 A-Level 物理试卷3 A 部分的典型题型高度吻合——引导你完成一次完整的实验探究。我们将深入探讨数据处理、图形分析、不确定度处理以及批判性评估，这些都是在 A-Level 物理实验部分夺取高分的关键。
1. Understanding the Insert 5 Data Set | 理解 Insert 5 数据集
Insert 5 often contains raw experimental results that a student might collect during a practical. For a free-fall investigation, you might see two columns: Distance fallen, s (in metres), and Time taken, t (in seconds). The distance may be measured by a ruler with an uncertainty of ±1 mm, and time by electronic timers with a precision of 0.01 s. There will likely be a note saying the object was dropped from rest, so initial velocity u = 0. The data will show increasing s for increasing t, and the relationship is expected to follow s = ½gt². Before jumping into calculations, always read the candidate’s notes and any additional information printed on the insert—these give context for measurement techniques and possible systematic errors.
Insert 5 通常包含学生在实验中可能收集到的原始结果。对于自由落体实验，你可能会看到两列：下落距离 s（单位：米）和所需时间 t（单位：秒）。距离可能用一把不确定度为 ±1 mm 的直尺测量，时间则由精度为 0.01 s 的电子计时器记录。插页中很可能注明物体从静止释放，因此初速度 u = 0。数据将显示随 t 增加，s 也增加，预期关系遵循 s = ½gt²。在开始计算之前，务必阅读考生笔记及插页上给出的任何附加信息——这些提供了测量技术和可能存在的系统误差的背景。
2. Stating the Physical Theory and Equation | 阐述物理理论与方程
For an object falling freely under gravity, if air resistance is negligible, the kinematic equation s = ut + ½at² applies. With u = 0 and a = g, this simplifies to s = ½gt². This equation suggests that a graph of s against t² should produce a straight line passing through the origin, with gradient equal to ½g. Therefore, g can be determined by doubling the gradient. If we decide to plot s against t², we must calculate t² for each time measurement and also determine the absolute uncertainty in t², which will be used to draw error bars. Always explain why this graph is more appropriate than a direct s–t curve—linearising the data allows easier identification of outliers and more confident gradient determination.
对于在重力作用下自由下落的物体，若空气阻力可忽略，运动学方程 s = ut + ½at² 适用。代入 u = 0 和 a = g，简化为 s = ½gt²。该方程表明，绘制 s 对 t² 的图像应得到一条过原点的直线，斜率等于 ½g。因此，由斜率乘以 2 即可求得 g。如果我们决定绘制 s–t² 图像，则必须为每个时间测量值计算 t²，并确定 t² 的绝对不确定度，后者将用于绘制误差棒。务必解释为什么这个图像比直接的 s–t 曲线更合适——将数据线性化能更容易识别异常点，并更有把握地确定斜率。
3. Processing Time Measurements and Calculating t² | 处理时间测量值并计算 t²
Suppose Insert 5 gives t = 0.25 s, 0.37 s, 0.48 s, 0.58 s, 0.67 s for five distances. Square each value: t² = 0.0625 s², 0.1369 s², 0.2304 s², 0.3364 s², 0.4489 s². Record these to an appropriate number of significant figures (typically three or four depending on the original data). The uncertainty in t is ±0.01 s, so the fractional uncertainty in t is 0.01/t. For small fractional uncertainties, the absolute uncertainty in t² can be approximated by Δ(t²) ≈ 2t·Δt. For example, when t = 0.25 s, Δt² ≈ 2 × 0.25 × 0.01 = 0.005 s². Tabulate these processed values clearly—your exam paper will expect you to show such a table with calculated quantities and their uncertainties.
假设 Insert 5 给出了五个距离对应的时间 t：0.25 s、0.37 s、0.48 s、0.58 s、0.67 s。将每个值平方：t² = 0.0625 s²、0.1369 s²、0.2304 s²、0.3364 s²、0.4489 s²。记录时保留适当的有效数字（通常根据原始数据取三位或四位）。时间的不确定度为 ±0.01 s，因此 t 的相对不确定度为 0.01/t。对于较小的相对不确定度，t² 的绝对不确定度可近似为 Δ(t²) ≈ 2t·Δt。例如，当 t = 0.25 s 时，Δt² ≈ 2 × 0.25 × 0.01 = 0.005 s²。将这些处理后的数据清晰地列成表格——你的试卷会要求你展示这样一个包含计算量及其不确定度的表格。
4. Constructing a Results Table with Uncertainties | 构建带不确定度的结果表格
Your table should include columns for s / m, t / s, t² / s², and Δt² / s². Assume the distance s is measured to ±0.001 m, so an error column for s may also be included, but often the uncertainty in the independent variable (s) is plotted as horizontal error bars—in this graph, we treat s as the dependent variable on the y-axis and t² on the x-axis, so the uncertainty in s is shown vertically. Regardless, a well-organised table makes data extraction easier. Here is an example:
你的表格应包括 s / m、t / s、t² / s² 和 Δt² / s² 各列。假设距离 s 的测量精度为 ±0.001 m，因此也可以加入 s 的误差列，但通常自变量（这里我们是把 s 当作因变量放在 y 轴，t² 放在 x 轴）的不确定度以垂直误差棒显示——在这个图像中，s 的不确定度便是如此。无论如何，一个组织良好的表格能使提取数据更加容易。示例如下：
| s / m | t / s | t² / s² | Δ(t²) / s² |
|---|---|---|---|
| 0.100 | 0.25 | 0.0625 | 0.005 |
| 0.200 | 0.37 | 0.1369 | 0.007 |
| 0.300 | 0.48 | 0.2304 | 0.010 |
| 0.400 | 0.58 | 0.3364 | 0.012 |
| 0.500 | 0.67 | 0.4489 | 0.013 |
Always double-check the number of decimal places and significant figures—they should be consistent with the precision of the measuring instruments. The Δ(t²) values are given to one significant figure as this is a calculated uncertainty.
务必检查小数位数和有效数字——它们应与测量仪器的精度保持一致。Δ(t²) 值保留一位有效数字，因为这是计算得出的不确定度。
5. Plotting the Graph and Drawing the Line of Best Fit | 绘制图像并画出最佳拟合线
Using the processed data, plot a graph of s/m on the vertical axis against t²/s² on the horizontal axis. Use a sharp pencil, label axes clearly with quantities and units, and choose scales that utilise at least half the graph paper in both directions. Plot the points as small crosses or dots with circles. Then, draw a single, straight line of best fit—this line should have approximately equal numbers of points on either side and should pass through the origin if the theory predicts an intercept of zero. Because the equation is s = (½g) t², the line should ideally pass through (0,0). However, do not force it unless you have justification; if the y-intercept is small, discuss whether it arises from a systematic error like a delayed timer start.
利用处理后的数据，绘制 s/m（纵轴）对 t²/s²（横轴）的图像。用削尖的铅笔作图，清晰地标出轴名（含物理量及单位），并选择能令两个方向至少占据一半图纸的刻度。将数据点画成小十字或带圆圈的圆点。然后，画一条单一的最佳拟合直线——该线应使点数大致均匀分布在两侧，且如果理论预测截距为零，应通过原点。由于方程为 s = (½g) t²，理想情况下直线应过 (0,0)。但除非你有充足理由，不要强行通过原点；若 y 截距很小，可讨论其是否源于系统误差，如计时器启动延迟。
6. Calculating the Gradient and Its Uncertainty | 计算斜率及其不确定度
To find the gradient, select two points on the line of best fit that are far apart—never use data points. Read coordinates (x₁, y₁) and (x₂, y₂) from the line. Gradient m = (y₂ – y₁) / (x₂ – x₁). Then, draw the steepest and shallowest plausible lines that still fit the data points (the “worst” lines), and calculate their gradients m_max and m_min. The uncertainty in the gradient is Δm = (m_max – m_min)/2. From s = ½g t², gradient m = ½g, so g = 2m. The percentage uncertainty in g is the same as that in m. Finally, express g as g ± Δg.
要计算斜率，在最佳拟合直线上选取相距较远的两个点——切勿使用原始数据点。从直线上读取坐标 (x₁, y₁) 和 (x₂, y₂)。斜率 m = (y₂ – y₁) / (x₂ – x₁)。接着，绘制出仍能拟合数据点的最陡和最浅的合理直线（即“最差”线），并计算它们的斜率 m_max 与 m_min。斜率的不确定度为 Δm = (m_max – m_min)/2。由 s = ½g t² 知，斜率 m = ½g，因此 g = 2m。g 的百分不确定度与 m 相同。最后，将 g 表示为 g ± Δg。
7. Numerical Example of g Determination | 测定 g 的数值示例
Suppose the line of best fit gives gradient m = 4.90 m s⁻². Then g = 2 × 4.90 = 9.80 m s⁻². If the worst lines give m_max = 5.05 m s⁻² and m_min = 4.75 m s⁻², then Δm = (5.05 – 4.75)/2 = 0.15 m s⁻². Thus Δg = 2 × 0.15 = 0.30 m s⁻². The result is reported as g = 9.80 ± 0.30 m s⁻². The accepted value for g is about 9.81 m s⁻², which falls within this experimental range, indicating the measurement is accurate within experimental uncertainty. Always compare your result with the accepted value and calculate the percentage difference: |(9.80 – 9.81)| / 9.81 × 100% ≈ 0.1%.
假设最佳拟合线斜率为 m = 4.90 m s⁻²，则 g = 2 × 4.90 = 9.80 m s⁻²。若最差线的斜率分别为 m_max = 5.05 m s⁻² 和 m_min = 4.75 m s⁻²，则 Δm = (5.05 – 4.75)/2 = 0.15 m s⁻²，于是 Δg = 2 × 0.15 = 0.30 m s⁻²。结果报告为 g = 9.80 ± 0.30 m s⁻²。g 的标准值约为 9.81 m s⁻²，落在本实验的误差范围内，表明在实验不确定度内测量是准确的。务必与标准值比较，并计算百分差异：|(9.80 – 9.81)| / 9.81 × 100% ≈ 0.1%.
8. Identifying Sources of Uncertainty and Error | 识别不确定度的来源与误差
The main sources of uncertainty in this experiment typically include: (1) Reaction time or electronic timer precision—though using a light gate reduces human reaction error, the light beam interruption has a finite response time. (2) Parallax error in aligning the height scale. (3) Air resistance, which causes the actual acceleration to be slightly less than g, especially if the object is light or has a large cross-sectional area. (4) Zero error on the ruler or misplacement of the light gate. (5) The assumption that the object is released exactly from rest; any initial downward velocity would shift the intercept. Discuss how each source affects the data and whether it introduces a systematic or random error.
本实验中不确定度的主要来源通常包括：（1）反应时间或电子计时器精度——虽然使用光门减少了人体反应误差，但光束中断仍有有限的响应时间。（2）对齐高度标尺时的视差误差。（3）空气阻力，它会导致实际加速度略小于 g，尤其是当物体很轻或截面积较大时。（4）直尺的零位误差或光门位置的偏离。（5）假设物体从静止释放；任何向下的初速度都会使截距发生偏移。讨论每种来源如何影响数据，以及它引入的是系统误差还是随机误差。
9. Minimising Experimental Uncertainties | 减小实验不确定度
To improve the experiment: Use a denser, streamlined object to reduce air resistance. Ensure the light gate is positioned exactly at the start of the fall, and check its alignment. Use a higher precision timer (e.g., millisecond precision) and take multiple readings at each height to average out random fluctuations. Measure the distance s from the bottom of the object to the light gate with a vernier calliper for greater accuracy. Finally, increase the range of heights to at least five different values, and repeat the whole procedure twice to gauge repeatability. In your investigation based on Insert 5, you can suggest these improvements in the evaluation section.
为改进实验：使用密度更大、流线型的物体以减小空气阻力。确保光门精确位于下落起点处，并检查其校准。使用更高精度的计时器（如毫秒级），并在每个高度进行多次读数以平均掉随机波动。用游标卡尺更精确地测量物体底部到光门的距离 s。最后，将高度范围增加到至少五个不同值，并重复整个流程两次以评估可重复性。在你基于 Insert 5 的探究中，可在评估段落提出这些改进建议。
10. Dealing with Anomalous Data Points | 处理异常数据点
If your graph shows one point lying significantly off the line of best fit, identify it as an anomaly. Do not ignore it—circle it on the graph and in your analysis, state that you excluded it from the line fitting. Explain possible reasons, such as a misread time due to a double interruption of the light beam, a wobble during release, or a recording mistake. Then, redraw the line of best fit without that point and recalculate the gradient. Compare the two gradients to assess the impact of the anomaly. This step demonstrates the critical evaluation skills examiners look for in a high-level practical investigation.
若你的图中显示某个点明显偏离最佳拟合线，应将其识别为异常点。不要忽视它——在图上用圈标出，并在分析中说明你在拟合直线时将其排除。解释可能的原因，如光线被二次中断导致的时间读数错误、释放时发生晃动或记录错误。然后，排除该点重新绘制最佳拟合线，并重新计算斜率。比较两个斜率以评估异常点的影响。这一步展现了考官在高水平实验探究中所寻求的批判性评估技能。
11. Comparing with Standard Value and Discussing Discrepancies | 与标准值比较并讨论偏差
After obtaining g = 9.80 ± 0.30 m s⁻², note that the accepted value (9.81 m s⁻²) lies within the uncertainty interval. Even if the result had been, say, 9.5 ± 0.2 m s⁻², a discrepancy would exist. In that case, discuss whether the difference is significant—check if the accepted value falls within the range g ± 2Δg (for a 95% confidence). If not, a systematic error likely dominates. Possible culprits: friction in the pulley (if used), parallax in height measurement, or a tilted release. This statistical comparison forms a key part of the evaluation for Paper 3.
在得到 g = 9.80 ± 0.30 m s⁻² 后，注意标准值（9.81 m s⁻²）落在不确定度区间内。即使结果变为，例如 g = 9.5 ± 0.2 m s⁻²，偏差便存在。此时，讨论该差异是否显著——检查标准值是否落在 g ± 2Δg 范围内（对应于 95% 置信水平）。如果不在，则可能以系统误差为主导。可能的元凶：滑轮摩擦（若使用）、高度测量中的视差、或释放时倾斜。这一统计学比较构成了试卷3评估部分的核心内容。
12. Conclusion and Exam Tips for Insert-based Questions | 结论及基于插页题型的应试技巧
Working through an Insert 5 practical investigation hones every skill required for A-Level Physics Paper 3: data tabulation, graph plotting, gradient and uncertainty determination, identification of errors, and critical evaluation. Always start by thoroughly reading the insert and noting the context, instruments, and any hints. Allocate time wisely—graph drawing and analysis often carry many marks. When calculating uncertainties, show intermediate steps clearly. Finally, link your conclusions back to the original aim, stating whether the experiment successfully determined the physical quantity and just how reliable your value is. With consistent practise on past insert papers, you will approach your exam with confidence.
完成一份基于 Insert 5 的实验探究能磨练 A-Level 物理试卷3所需的所有技能：数据制表、图像绘制、斜率与不确定度的确定、误差识别以及批判性评估。开始时一定要仔细阅读插页内容，注意实验背景、所用仪器及任何提示。合理分配时间——画图和分析通常占大量分值。在计算不确定度时，清晰地展示中间步骤。最后，将结论联系回初始目标，说明实验是否成功测定了该物理量，以及你的数值究竟有多可靠。通过反复练习历年插页题目，你将充满信心地踏入考场。
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