AS Physics Insert 2 June 2022: Experimental Investigation of Free Fall | AS物理插入材料2 2022年6月：自由落体实验探究

📚 AS Physics Insert 2 June 2022: Experimental Investigation of Free Fall | AS物理插入材料2 2022年6月：自由落体实验探究

In AS level physics, the examination insert for Paper 2 in June 2022 presented a practical investigation into free fall. This article unpacks the experiment, guiding you through the theoretical principles, apparatus, data handling, graphical analysis and uncertainty evaluation. By understanding each stage, you will build the skills required to tackle similar practical-based questions in your own assessments.

在AS物理2022年6月卷二所附的材料中，给出了一项自由落体的实验探究。本文将对这一实验进行详细分析，带您梳理理论原理、实验装置、数据处理、图像分析以及不确定度评定。通过弄清每个环节，您将能够掌握应对试卷中类似实验探究题所需的关键技能。

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1. Introduction to the Insert Experiment | 插入材料实验简介

The insert described a student using a free‑fall apparatus to determine the acceleration due to gravity, g. A steel sphere was released from rest and allowed to fall between two light gates. The time taken for the sphere to travel a measured vertical distance was recorded electronically.

插入材料描述了一名学生使用自由落体装置测量重力加速度 g。一颗钢球从静止释放，垂直下落并通过两个光电门。电子系统记录球体通过一段已知竖直距离所用的时间。

The main objective was to find a reliable value for g and to analyse the associated uncertainties. The insert provided a set of raw data and asked candidates to complete the table, plot a graph and evaluate the findings.

实验的主要目标是得出可靠的 g 值并分析相关不确定度。材料提供了一组原始数据，要求考生补全表格、绘制图像并评估结果。

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2. Theoretical Background | 理论背景

For an object falling freely from rest under gravity, the vertical displacement s after time t is given by the kinematic equation s = ½gt², provided air resistance is negligible and the initial velocity u = 0. This is derived from s = ut + ½at² where a = g.

在忽略空气阻力且初速度 u = 0 的条件下，从静止开始的自由落体，其竖直位移 s 与时间 t 的关系满足运动学方程 s = ½gt²。该式由 s = ut + ½at² 且 a = g 得出。

Rearranging yields g = 2s / t². By measuring s and t for various drops, g can be calculated. If a graph of s against t² is plotted, the slope equals ½g, giving g = 2 × slope. This linear relationship forms the basis of the graphical analysis.

移项可得 g = 2s / t²。通过测量不同下落距离的 s 和 t，就能算出 g。若以 s 为纵轴、t² 为横轴作图，斜率等于 ½g，因此 g = 2 × 斜率。这种线性关系是图像分析的基础。

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3. Apparatus and Setup | 实验器材与装置

The typical free‑fall apparatus includes an electromagnet to hold and release a steel sphere, two light gates connected to a timer, a metre ruler or a set of known distances marked on a stand, and a control unit to reset the timer. The light gates are positioned so that the top gate starts the timer and the bottom gate stops it.

典型的自由落体装置包含：一个电磁铁用于固定和释放钢球、两个连接计时器的光电门、一把米尺或支架上标注好的距离刻度，以及一个用于复位计时的控制单元。光电门的位置安排为上方的门启动计时，下方的门停止计时。

The distance s between the light gates can be adjusted. The sphere must fall vertically through both gates without touching any side. A plumb line is often used to ensure alignment.

两光电门之间的距离 s 可以调整。球体必须竖直下落，穿过两个光电门且不触碰任何一侧。通常会用铅垂线来确保装置准直。

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4. Experimental Procedure | 实验步骤

First, set the upper light gate just below the electromagnet and measure the vertical separation s between the gates using a metre ruler. Take readings from eye level to avoid parallax error and record the reading with its uncertainty, typically ±1 mm or ±2 mm.

首先，将上方的光电门置于电磁铁正下方，并用米尺测量两门之间的竖直距离 s。读数时视线应与刻度齐平以避免视差，并记录读数及其不确定度，通常为 ±1 mm 或 ±2 mm。

Turn on the electromagnet, hold the sphere, then switch off to release it. The timer records the fall time t between the two gates. Repeat the measurement at least three times for each distance and calculate the mean time t̄. The insert provided a partially completed table for distances of 0.400 m, 0.600 m, 0.800 m, 1.000 m and 1.200 m.

接通电磁铁，吸住球体，再断开使其释放。计时器记录球体通过两门的时间 t。每一距离至少重复测量三次，并计算平均时间 t̄。插入材料给出了一张部分填好的表格，距离分别为 0.400 m、0.600 m、0.800 m、1.000 m 和 1.200 m。

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5. Data Collection Table | 数据收集表

The insert required students to complete the t² column and to record the uncertainty in t. A typical completed table is shown below, with realistic values for demonstration. All times are in seconds, distances in metres.

插入材料要求学生补全 t² 列并记录 t 的不确定度。以下是一张典型的完成后的表格，数值真实合理，用于展示。所有时间单位为秒，距离单位为米。

s / m	t₁ / s	t₂ / s	t₃ / s	Mean t / s	t² / s²	Uncertainty in t / s
0.400	0.285	0.287	0.286	0.286	0.0818	0.001
0.600	0.350	0.349	0.351	0.350	0.1225	0.001
0.800	0.404	0.403	0.405	0.404	0.1632	0.001
1.000	0.452	0.451	0.453	0.452	0.2043	0.001
1.200	0.495	0.496	0.494	0.495	0.2450	0.001

The uncertainty in t was estimated from the spread of repeat readings, often given as half the range. Here the variations are small, so ±0.001 s is appropriate.

t 的不确定度由重复读数的离散程度估计，常取极差的一半。此处的读数差异很小，因此 ±0.001 s 是合理的。

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6. Data Analysis and Calculating g | 数据分析与 g 的计算

For each pair of s and t², use g = 2s / t². For instance, when s = 0.800 m and t² = 0.1632 s², g = (2 × 0.800) / 0.1632 ≈ 9.80 m s⁻². Repeat for all distances to check consistency.

对每一组 s 和 t²，用 g = 2s / t² 计算。例如，当 s = 0.800 m、t² = 0.1632 s² 时，g = (2 × 0.800) / 0.1632 ≈ 9.80 m s⁻²。对所有距离重复计算，检验数值是否一致。

Calculated g values: 0.400 m → 9.78 m s⁻², 0.600 m → 9.80 m s⁻², 0.800 m → 9.80 m s⁻², 1.000 m → 9.79 m s⁻², 1.200 m → 9.80 m s⁻². The mean g = 9.79 m s⁻². This is close to the accepted value of 9.81 m s⁻².

计算出的 g 值分别为：0.400 m → 9.78 m s⁻²、0.600 m → 9.80 m s⁻²、0.800 m → 9.80 m s⁻²、1.000 m → 9.79 m s⁻²、1.200 m → 9.80 m s⁻²。平均 g = 9.79 m s⁻²，与公认值 9.81 m s⁻² 非常接近。

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7. Graphical Analysis | 图像分析

Plot a graph of s (vertical axis) against t² (horizontal axis). According to s = ½gt², the graph should be a straight line passing through the origin. Draw a line of best fit and calculate the gradient.

绘制 s（纵轴）与 t²（横轴）的关系图。根据 s = ½gt²，图像应为过原点的直线。画出最佳拟合线并计算斜率。

Using the data above, the gradient = Δs / Δ(t²) = (1.200 – 0.400) / (0.2450 – 0.0818) = 0.800 / 0.1632 ≈ 4.90 m s⁻². Since gradient = ½g, we obtain g = 2 × 4.90 = 9.80 m s⁻².

利用上述数据，斜率 = Δs / Δ(t²) = (1.200 − 0.400) / (0.2450 − 0.0818) = 0.800 / 0.1632 ≈ 4.90 m s⁻²。由于斜率 = ½g，可得 g = 2 × 4.90 = 9.80 m s⁻²。

The graphical method is more reliable than taking a single calculation because it averages out random errors and clearly reveals any anomalous points. The intercept should be zero; a non‑zero intercept suggests a systematic error such as a misaligned starting position.

图像法比单次计算更可靠，因为它能平均掉随机误差并清晰地显示出异常数据点。截距理论上应为零；非零截距暗示存在系统误差，例如起始位置未对准。

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8. Uncertainty Analysis | 不确定度分析

The uncertainty in g can be estimated from the uncertainties in s and t. Since g = 2s / t², the fractional uncertainty in g is Δg/g = Δs/s + 2(Δt/t). For s = 1.000 m with Δs = 0.002 m and t = 0.452 s with Δt = 0.001 s, we get Δg/g = 0.002 + 2×(0.001/0.452) ≈ 0.002 + 0.0044 = 0.0064, so Δg ≈ 0.0064 × 9.79 ≈ 0.06 m s⁻².

g 的不确定度可由 s 和 t 的不确定度求得。由于 g = 2s / t²，其相对不确定度为 Δg/g = Δs/s + 2(Δt/t)。以 s = 1.000 m、Δs = 0.002 m 和 t = 0.452 s、Δt = 0.001 s 为例，Δg/g = 0.002 + 2×(0.001/0.452) ≈ 0.002 + 0.0044 = 0.0064，因此 Δg ≈ 0.0064 × 9.79 ≈ 0.06 m s⁻²。

The final result can be quoted as g = 9.79 ± 0.06 m s⁻². The accepted value 9.81 m s⁻² lies within this range, confirming the experiment’s validity.

最终结果可表示为 g = 9.79 ± 0.06 m s⁻²。公认值 9.81 m s⁻² 落在此区间内，证实了实验的有效性。

If a graphical approach is used, the uncertainty in the gradient can be found by drawing worst‑fit lines, and then Δg = 2 × uncertainty in slope.

若使用图像法，则可通过画最差拟合线求出斜率的不确定度，进而 Δg = 2 × 斜率的不确定度。

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9. Sources of Error and Improvements | 误差来源与改进措施

Major sources of error include reaction time when releasing the sphere (if manual switch is used), parallax when measuring s, and the light gates not being exactly horizontal, causing the sphere to break the beam at an angle. Air resistance also has a small effect, especially over longer distances.

主要误差来源包括：手动释放时存在的反应时间（若使用手动开关）、测量 s 时的视差，以及光电门未能严格水平而导致球体倾斜切断光束。空气阻力也有轻微影响，尤其在较长下落距离时。

Improvements: use the electromagnet to ensure instantaneous release; align the apparatus with a plumb line; repeat measurements many times; use a digital calliper to measure the sphere diameter if the gates are triggered by the leading edge; and include a vacuum chamber to eliminate air resistance in a more advanced setup.

改进措施：利用电磁铁保证瞬时释放；用铅垂线校准装置；多次重复测量；如果光电门由前沿触发，可用数显卡尺测量球径；在更高要求的装置中可引入真空腔以消除空气阻力。

Systematic errors, such as an offset in the timer or a zero error in the ruler, must be identified. Checking the timer with a known frequency and calibrating the ruler against a standard can help.

系统误差，如计时器的零点漂移或米尺的零误差，必须加以识别。用已知频率检测计时器、用标准尺校准米尺可减少这类误差。

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10. Concluding the Investigation | 实验结论

The free‑fall experiment produced a value of g = 9.79 ± 0.06 m s⁻², which agrees with the accepted value within experimental uncertainty. The linear s–t² graph confirmed the theoretical relationship, and the analysis highlighted the importance of careful measurement and graphing skills.

本自由落体实验得出的 g 值为 9.79 ± 0.06 m s⁻²，在实验不确定度范围内与公认值一致。线性的 s–t² 图证实了理论关系，而分析过程则突显了细致测量和图像处理的重要性。

This investigation mirrors the style of the Insert 2 in the June 2022 AS Physics examination, where candidates were expected to complete a data table, plot a graph, determine g and discuss uncertainties.

这一探究与2022年6月AS物理考试插入材料2的风格完全一致，该考题要求考生补全数据表、绘制图像、求出 g 并讨论不确定度。

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11. Exam Technique and Common Pitfalls | 应试技巧与常见错误

When completing the table, always record t² to an appropriate number of significant figures. If t is given to 3 s.f., t² should be kept to 3 or 4 s.f., not rounded prematurely. Label axes with quantity and unit, and choose scales that use at least half the graph paper.

补全表格时，t² 的记录务必保持适当的有效数字位数。若 t 为三位有效数字，t² 应保留三或四位，切勿过早四舍五入。坐标轴要标注物理量和单位，并选择使图像占据至少半张图纸的比例尺。

State the equation of the line in the form y = mx, and clearly equate the gradient to ½g. Do not forget to double the gradient to obtain g. Cite your final answer with the absolute uncertainty.

应写出直线方程 y = mx 的形式，并明确将斜率等同于 ½g。千万不要忘记将斜率加倍才得到 g。最终答案要带上绝对不确定度。

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12. Practice Application | 应用练习

Try this yourself: a student drops a ball from rest and records s = 0.500 m, t = 0.319 s, with ±0.001 s uncertainty. Calculate g and its percentage uncertainty. Plot a hypothetical s–t² graph using three points and find g from the gradient. How would the results change if the top light gate was triggered by the sphere’s bottom edge?

亲自尝试：一名学生从静止释放小球，测得 s = 0.500 m，t = 0.319 s，不确定度 ±0.001 s。计算 g 及其百分不确定度。假定使用三个点绘制 s–t² 图，并从斜率求出 g。如果上方光电门是由球体底边触发，结果会如何变化？

Mastering this investigation equips you with essential data‑handling and error‑analysis skills that apply to many other AS experiments, from measuring resistivity to investigating the behaviour of springs.

掌握这一探究，你将具备处理数据和误差分析的关键技能，这些技能同样适用于从测量电阻率到探究弹簧行为的许多其他AS实验。

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